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case hβ.refine'_1
n : β
n_big : 2 < n
no_prime : Β¬β p, Nat.Prime p β§ n < p β§ p β€ 2 * n
n_pos : 0 < n
n2_pos : 1 β€ 2 * n
S : Finset β := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : β β β := fun x => x ^ (Nat.factorization (centralBinom n)) x
this : β x in S, f x = β x in Finset.range (2 * n / 3 + 1), f x
p : β
xβ : p β Finset.filter (fun x => x β€ sqrt (2 * n)) S
β’ f p β€ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· | exact pow_factorization_choose_le (mul_pos two_pos n_pos) | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· | Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) | Mathlib_NumberTheory_Bertrand |
case hβ.refine'_2
n : β
n_big : 2 < n
no_prime : Β¬β p, Nat.Prime p β§ n < p β§ p β€ 2 * n
n_pos : 0 < n
n2_pos : 1 β€ 2 * n
S : Finset β := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : β β β := fun x => x ^ (Nat.factorization (centralBinom n)) x
this : β x in S, f x = β x in Finset.range (2 * n / 3 + 1), f x
β’ β i in Finset.filter (fun x => x β€ sqrt (2 * n)) S, 2 * n β€ (2 * n) ^ sqrt (2 * n) | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
| have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel] | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
| Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) | Mathlib_NumberTheory_Bertrand |
n : β
n_big : 2 < n
no_prime : Β¬β p, Nat.Prime p β§ n < p β§ p β€ 2 * n
n_pos : 0 < n
n2_pos : 1 β€ 2 * n
S : Finset β := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : β β β := fun x => x ^ (Nat.factorization (centralBinom n)) x
this : β x in S, f x = β x in Finset.range (2 * n / 3 + 1), f x
β’ Finset.card (Finset.Icc 1 (sqrt (2 * n))) = sqrt (2 * n) | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by | rw [card_Icc, Nat.add_sub_cancel] | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by | Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) | Mathlib_NumberTheory_Bertrand |
case hβ.refine'_2
n : β
n_big : 2 < n
no_prime : Β¬β p, Nat.Prime p β§ n < p β§ p β€ 2 * n
n_pos : 0 < n
n2_pos : 1 β€ 2 * n
S : Finset β := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : β β β := fun x => x ^ (Nat.factorization (centralBinom n)) x
thisβ : β x in S, f x = β x in Finset.range (2 * n / 3 + 1), f x
this : Finset.card (Finset.Icc 1 (sqrt (2 * n))) = sqrt (2 * n)
β’ β i in Finset.filter (fun x => x β€ sqrt (2 * n)) S, 2 * n β€ (2 * n) ^ sqrt (2 * n) | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
| rw [Finset.prod_const] | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
| Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) | Mathlib_NumberTheory_Bertrand |
case hβ.refine'_2
n : β
n_big : 2 < n
no_prime : Β¬β p, Nat.Prime p β§ n < p β§ p β€ 2 * n
n_pos : 0 < n
n2_pos : 1 β€ 2 * n
S : Finset β := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : β β β := fun x => x ^ (Nat.factorization (centralBinom n)) x
thisβ : β x in S, f x = β x in Finset.range (2 * n / 3 + 1), f x
this : Finset.card (Finset.Icc 1 (sqrt (2 * n))) = sqrt (2 * n)
β’ (2 * n) ^ Finset.card (Finset.filter (fun x => x β€ sqrt (2 * n)) S) β€ (2 * n) ^ sqrt (2 * n) | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
| refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le) | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
| Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) | Mathlib_NumberTheory_Bertrand |
case hβ.refine'_2
n : β
n_big : 2 < n
no_prime : Β¬β p, Nat.Prime p β§ n < p β§ p β€ 2 * n
n_pos : 0 < n
n2_pos : 1 β€ 2 * n
S : Finset β := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : β β β := fun x => x ^ (Nat.factorization (centralBinom n)) x
thisβ : β x in S, f x = β x in Finset.range (2 * n / 3 + 1), f x
this : Finset.card (Finset.Icc 1 (sqrt (2 * n))) = sqrt (2 * n)
x : β
hx : x β Finset.filter (fun x => x β€ sqrt (2 * n)) S
β’ x β Finset.Icc 1 (sqrt (2 * n)) | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
| obtain β¨h1, h2β© := Finset.mem_filter.1 hx | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
| Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) | Mathlib_NumberTheory_Bertrand |
case hβ.refine'_2.intro
n : β
n_big : 2 < n
no_prime : Β¬β p, Nat.Prime p β§ n < p β§ p β€ 2 * n
n_pos : 0 < n
n2_pos : 1 β€ 2 * n
S : Finset β := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : β β β := fun x => x ^ (Nat.factorization (centralBinom n)) x
thisβ : β x in S, f x = β x in Finset.range (2 * n / 3 + 1), f x
this : Finset.card (Finset.Icc 1 (sqrt (2 * n))) = sqrt (2 * n)
x : β
hx : x β Finset.filter (fun x => x β€ sqrt (2 * n)) S
h1 : x β S
h2 : x β€ sqrt (2 * n)
β’ x β Finset.Icc 1 (sqrt (2 * n)) | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
| exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β© | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
| Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) | Mathlib_NumberTheory_Bertrand |
case hβ
n : β
n_big : 2 < n
no_prime : Β¬β p, Nat.Prime p β§ n < p β§ p β€ 2 * n
n_pos : 0 < n
n2_pos : 1 β€ 2 * n
S : Finset β := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : β β β := fun x => x ^ (Nat.factorization (centralBinom n)) x
this : β x in S, f x = β x in Finset.range (2 * n / 3 + 1), f x
β’ β x in Finset.filter (fun x => Β¬x β€ sqrt (2 * n)) S, f x β€ 4 ^ (2 * n / 3) | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· | refine' le_trans _ (primorial_le_4_pow (2 * n / 3)) | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· | Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) | Mathlib_NumberTheory_Bertrand |
case hβ
n : β
n_big : 2 < n
no_prime : Β¬β p, Nat.Prime p β§ n < p β§ p β€ 2 * n
n_pos : 0 < n
n2_pos : 1 β€ 2 * n
S : Finset β := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : β β β := fun x => x ^ (Nat.factorization (centralBinom n)) x
this : β x in S, f x = β x in Finset.range (2 * n / 3 + 1), f x
β’ β x in Finset.filter (fun x => Β¬x β€ sqrt (2 * n)) S, f x β€ primorial (2 * n / 3) | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
| refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _ | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
| Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) | Mathlib_NumberTheory_Bertrand |
case hβ.refine'_1
n : β
n_big : 2 < n
no_prime : Β¬β p, Nat.Prime p β§ n < p β§ p β€ 2 * n
n_pos : 0 < n
n2_pos : 1 β€ 2 * n
S : Finset β := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : β β β := fun x => x ^ (Nat.factorization (centralBinom n)) x
this : β x in S, f x = β x in Finset.range (2 * n / 3 + 1), f x
p : β
hp : p β Finset.filter (fun x => Β¬x β€ sqrt (2 * n)) S
β’ f p β€ p | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· | obtain β¨h1, h2β© := Finset.mem_filter.1 hp | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· | Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) | Mathlib_NumberTheory_Bertrand |
case hβ.refine'_1.intro
n : β
n_big : 2 < n
no_prime : Β¬β p, Nat.Prime p β§ n < p β§ p β€ 2 * n
n_pos : 0 < n
n2_pos : 1 β€ 2 * n
S : Finset β := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : β β β := fun x => x ^ (Nat.factorization (centralBinom n)) x
this : β x in S, f x = β x in Finset.range (2 * n / 3 + 1), f x
p : β
hp : p β Finset.filter (fun x => Β¬x β€ sqrt (2 * n)) S
h1 : p β S
h2 : Β¬p β€ sqrt (2 * n)
β’ f p β€ p | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
| refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
| Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) | Mathlib_NumberTheory_Bertrand |
case hβ.refine'_1.intro
n : β
n_big : 2 < n
no_prime : Β¬β p, Nat.Prime p β§ n < p β§ p β€ 2 * n
n_pos : 0 < n
n2_pos : 1 β€ 2 * n
S : Finset β := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : β β β := fun x => x ^ (Nat.factorization (centralBinom n)) x
this : β x in S, f x = β x in Finset.range (2 * n / 3 + 1), f x
p : β
hp : p β Finset.filter (fun x => Β¬x β€ sqrt (2 * n)) S
h1 : p β S
h2 : Β¬p β€ sqrt (2 * n)
β’ (Nat.factorization (centralBinom n)) p β€ 1 | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
| exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2) | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
| Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) | Mathlib_NumberTheory_Bertrand |
case hβ.refine'_2
n : β
n_big : 2 < n
no_prime : Β¬β p, Nat.Prime p β§ n < p β§ p β€ 2 * n
n_pos : 0 < n
n2_pos : 1 β€ 2 * n
S : Finset β := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : β β β := fun x => x ^ (Nat.factorization (centralBinom n)) x
this : β x in S, f x = β x in Finset.range (2 * n / 3 + 1), f x
β’ β i in Finset.filter (fun x => Β¬x β€ sqrt (2 * n)) S, i β€ primorial (2 * n / 3) | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
| refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _ | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
| Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) | Mathlib_NumberTheory_Bertrand |
case hβ.refine'_2
n : β
n_big : 2 < n
no_prime : Β¬β p, Nat.Prime p β§ n < p β§ p β€ 2 * n
n_pos : 0 < n
n2_pos : 1 β€ 2 * n
S : Finset β := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))
f : β β β := fun x => x ^ (Nat.factorization (centralBinom n)) x
this : β x in S, f x = β x in Finset.range (2 * n / 3 + 1), f x
β’ β i β Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1)), i β Finset.filter (fun x => Β¬x β€ sqrt (2 * n)) S β 1 β€ i | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _
| exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _
| Mathlib.NumberTheory.Bertrand.155_0.gJXoOT9Ce2wC0xc | /-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) | Mathlib_NumberTheory_Bertrand |
n : β
n_big : 512 β€ n
β’ β p, Prime p β§ n < p β§ p β€ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _
exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le
#align central_binom_le_of_no_bertrand_prime centralBinom_le_of_no_bertrand_prime
namespace Nat
/-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
| by_contra no_prime | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
| Mathlib.NumberTheory.Bertrand.195_0.gJXoOT9Ce2wC0xc | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n | Mathlib_NumberTheory_Bertrand |
n : β
n_big : 512 β€ n
no_prime : Β¬β p, Prime p β§ n < p β§ p β€ 2 * n
β’ False | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _
exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le
#align central_binom_le_of_no_bertrand_prime centralBinom_le_of_no_bertrand_prime
namespace Nat
/-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
| have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
| Mathlib.NumberTheory.Bertrand.195_0.gJXoOT9Ce2wC0xc | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n | Mathlib_NumberTheory_Bertrand |
n : β
n_big : 512 β€ n
no_prime : Β¬β p, Prime p β§ n < p β§ p β€ 2 * n
H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n
β’ False | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _
exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le
#align central_binom_le_of_no_bertrand_prime centralBinom_le_of_no_bertrand_prime
namespace Nat
/-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
| have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big) | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
| Mathlib.NumberTheory.Bertrand.195_0.gJXoOT9Ce2wC0xc | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n | Mathlib_NumberTheory_Bertrand |
n : β
n_big : 512 β€ n
no_prime : Β¬β p, Prime p β§ n < p β§ p β€ 2 * n
H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n
β’ 4 β€ 512 | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _
exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le
#align central_binom_le_of_no_bertrand_prime centralBinom_le_of_no_bertrand_prime
namespace Nat
/-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by | norm_num1 | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by | Mathlib.NumberTheory.Bertrand.195_0.gJXoOT9Ce2wC0xc | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n | Mathlib_NumberTheory_Bertrand |
n : β
n_big : 512 β€ n
no_prime : Β¬β p, Prime p β§ n < p β§ p β€ 2 * n
H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n
H2 : 4 ^ n < n * centralBinom n
β’ False | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _
exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le
#align central_binom_le_of_no_bertrand_prime centralBinom_le_of_no_bertrand_prime
namespace Nat
/-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big)
| have H3 : n.centralBinom β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) :=
centralBinom_le_of_no_bertrand_prime n (lt_of_lt_of_le (by norm_num1) n_big) no_prime | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big)
| Mathlib.NumberTheory.Bertrand.195_0.gJXoOT9Ce2wC0xc | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n | Mathlib_NumberTheory_Bertrand |
n : β
n_big : 512 β€ n
no_prime : Β¬β p, Prime p β§ n < p β§ p β€ 2 * n
H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n
H2 : 4 ^ n < n * centralBinom n
β’ 2 < 512 | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _
exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le
#align central_binom_le_of_no_bertrand_prime centralBinom_le_of_no_bertrand_prime
namespace Nat
/-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big)
have H3 : n.centralBinom β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) :=
centralBinom_le_of_no_bertrand_prime n (lt_of_lt_of_le (by | norm_num1 | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big)
have H3 : n.centralBinom β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) :=
centralBinom_le_of_no_bertrand_prime n (lt_of_lt_of_le (by | Mathlib.NumberTheory.Bertrand.195_0.gJXoOT9Ce2wC0xc | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n | Mathlib_NumberTheory_Bertrand |
n : β
n_big : 512 β€ n
no_prime : Β¬β p, Prime p β§ n < p β§ p β€ 2 * n
H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n
H2 : 4 ^ n < n * centralBinom n
H3 : centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)
β’ False | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _
exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le
#align central_binom_le_of_no_bertrand_prime centralBinom_le_of_no_bertrand_prime
namespace Nat
/-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big)
have H3 : n.centralBinom β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) :=
centralBinom_le_of_no_bertrand_prime n (lt_of_lt_of_le (by norm_num1) n_big) no_prime
| rw [mul_assoc] at H1 | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big)
have H3 : n.centralBinom β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) :=
centralBinom_le_of_no_bertrand_prime n (lt_of_lt_of_le (by norm_num1) n_big) no_prime
| Mathlib.NumberTheory.Bertrand.195_0.gJXoOT9Ce2wC0xc | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n | Mathlib_NumberTheory_Bertrand |
n : β
n_big : 512 β€ n
no_prime : Β¬β p, Prime p β§ n < p β§ p β€ 2 * n
H1 : n * ((2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)) β€ 4 ^ n
H2 : 4 ^ n < n * centralBinom n
H3 : centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)
β’ False | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _
exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le
#align central_binom_le_of_no_bertrand_prime centralBinom_le_of_no_bertrand_prime
namespace Nat
/-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big)
have H3 : n.centralBinom β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) :=
centralBinom_le_of_no_bertrand_prime n (lt_of_lt_of_le (by norm_num1) n_big) no_prime
rw [mul_assoc] at H1; | exact not_le.2 H2 ((mul_le_mul_left' H3 n).trans H1) | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big)
have H3 : n.centralBinom β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) :=
centralBinom_le_of_no_bertrand_prime n (lt_of_lt_of_le (by norm_num1) n_big) no_prime
rw [mul_assoc] at H1; | Mathlib.NumberTheory.Bertrand.195_0.gJXoOT9Ce2wC0xc | /-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n | Mathlib_NumberTheory_Bertrand |
n q p : β
prime_p : Prime p
covering : p β€ 2 * q
H : n < q β β p, Prime p β§ n < p β§ p β€ 2 * n
hn : n < p
β’ β p, Prime p β§ n < p β§ p β€ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _
exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le
#align central_binom_le_of_no_bertrand_prime centralBinom_le_of_no_bertrand_prime
namespace Nat
/-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big)
have H3 : n.centralBinom β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) :=
centralBinom_le_of_no_bertrand_prime n (lt_of_lt_of_le (by norm_num1) n_big) no_prime
rw [mul_assoc] at H1; exact not_le.2 H2 ((mul_le_mul_left' H3 n).trans H1)
#align nat.exists_prime_lt_and_le_two_mul_eventually Nat.exists_prime_lt_and_le_two_mul_eventually
/-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number β€ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : β} (prime_p : Nat.Prime p)
(covering : p β€ 2 * q) (H : n < q β β p : β, p.Prime β§ n < p β§ p β€ 2 * n) (hn : n < p) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
| by_cases h : p β€ 2 * n | /-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number β€ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : β} (prime_p : Nat.Prime p)
(covering : p β€ 2 * q) (H : n < q β β p : β, p.Prime β§ n < p β§ p β€ 2 * n) (hn : n < p) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
| Mathlib.NumberTheory.Bertrand.212_0.gJXoOT9Ce2wC0xc | /-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number β€ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : β} (prime_p : Nat.Prime p)
(covering : p β€ 2 * q) (H : n < q β β p : β, p.Prime β§ n < p β§ p β€ 2 * n) (hn : n < p) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n | Mathlib_NumberTheory_Bertrand |
case pos
n q p : β
prime_p : Prime p
covering : p β€ 2 * q
H : n < q β β p, Prime p β§ n < p β§ p β€ 2 * n
hn : n < p
h : p β€ 2 * n
β’ β p, Prime p β§ n < p β§ p β€ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _
exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le
#align central_binom_le_of_no_bertrand_prime centralBinom_le_of_no_bertrand_prime
namespace Nat
/-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big)
have H3 : n.centralBinom β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) :=
centralBinom_le_of_no_bertrand_prime n (lt_of_lt_of_le (by norm_num1) n_big) no_prime
rw [mul_assoc] at H1; exact not_le.2 H2 ((mul_le_mul_left' H3 n).trans H1)
#align nat.exists_prime_lt_and_le_two_mul_eventually Nat.exists_prime_lt_and_le_two_mul_eventually
/-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number β€ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : β} (prime_p : Nat.Prime p)
(covering : p β€ 2 * q) (H : n < q β β p : β, p.Prime β§ n < p β§ p β€ 2 * n) (hn : n < p) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
by_cases h : p β€ 2 * n; Β· | exact β¨p, prime_p, hn, hβ© | /-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number β€ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : β} (prime_p : Nat.Prime p)
(covering : p β€ 2 * q) (H : n < q β β p : β, p.Prime β§ n < p β§ p β€ 2 * n) (hn : n < p) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
by_cases h : p β€ 2 * n; Β· | Mathlib.NumberTheory.Bertrand.212_0.gJXoOT9Ce2wC0xc | /-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number β€ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : β} (prime_p : Nat.Prime p)
(covering : p β€ 2 * q) (H : n < q β β p : β, p.Prime β§ n < p β§ p β€ 2 * n) (hn : n < p) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n | Mathlib_NumberTheory_Bertrand |
case neg
n q p : β
prime_p : Prime p
covering : p β€ 2 * q
H : n < q β β p, Prime p β§ n < p β§ p β€ 2 * n
hn : n < p
h : Β¬p β€ 2 * n
β’ β p, Prime p β§ n < p β§ p β€ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _
exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le
#align central_binom_le_of_no_bertrand_prime centralBinom_le_of_no_bertrand_prime
namespace Nat
/-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big)
have H3 : n.centralBinom β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) :=
centralBinom_le_of_no_bertrand_prime n (lt_of_lt_of_le (by norm_num1) n_big) no_prime
rw [mul_assoc] at H1; exact not_le.2 H2 ((mul_le_mul_left' H3 n).trans H1)
#align nat.exists_prime_lt_and_le_two_mul_eventually Nat.exists_prime_lt_and_le_two_mul_eventually
/-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number β€ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : β} (prime_p : Nat.Prime p)
(covering : p β€ 2 * q) (H : n < q β β p : β, p.Prime β§ n < p β§ p β€ 2 * n) (hn : n < p) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
by_cases h : p β€ 2 * n; Β· exact β¨p, prime_p, hn, hβ©
| exact H (lt_of_mul_lt_mul_left' (lt_of_lt_of_le (not_le.1 h) covering)) | /-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number β€ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : β} (prime_p : Nat.Prime p)
(covering : p β€ 2 * q) (H : n < q β β p : β, p.Prime β§ n < p β§ p β€ 2 * n) (hn : n < p) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
by_cases h : p β€ 2 * n; Β· exact β¨p, prime_p, hn, hβ©
| Mathlib.NumberTheory.Bertrand.212_0.gJXoOT9Ce2wC0xc | /-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number β€ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : β} (prime_p : Nat.Prime p)
(covering : p β€ 2 * q) (H : n < q β β p : β, p.Prime β§ n < p β§ p β€ 2 * n) (hn : n < p) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n | Mathlib_NumberTheory_Bertrand |
n : β
hn0 : n β 0
β’ β p, Prime p β§ n < p β§ p β€ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _
exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le
#align central_binom_le_of_no_bertrand_prime centralBinom_le_of_no_bertrand_prime
namespace Nat
/-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big)
have H3 : n.centralBinom β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) :=
centralBinom_le_of_no_bertrand_prime n (lt_of_lt_of_le (by norm_num1) n_big) no_prime
rw [mul_assoc] at H1; exact not_le.2 H2 ((mul_le_mul_left' H3 n).trans H1)
#align nat.exists_prime_lt_and_le_two_mul_eventually Nat.exists_prime_lt_and_le_two_mul_eventually
/-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number β€ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : β} (prime_p : Nat.Prime p)
(covering : p β€ 2 * q) (H : n < q β β p : β, p.Prime β§ n < p β§ p β€ 2 * n) (hn : n < p) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
by_cases h : p β€ 2 * n; Β· exact β¨p, prime_p, hn, hβ©
exact H (lt_of_mul_lt_mul_left' (lt_of_lt_of_le (not_le.1 h) covering))
#align nat.exists_prime_lt_and_le_two_mul_succ Nat.exists_prime_lt_and_le_two_mul_succ
/--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n := by
-- Split into cases whether `n` is large or small
| cases' lt_or_le 511 n with h h | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n := by
-- Split into cases whether `n` is large or small
| Mathlib.NumberTheory.Bertrand.223_0.gJXoOT9Ce2wC0xc | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n | Mathlib_NumberTheory_Bertrand |
case inl
n : β
hn0 : n β 0
h : 511 < n
β’ β p, Prime p β§ n < p β§ p β€ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _
exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le
#align central_binom_le_of_no_bertrand_prime centralBinom_le_of_no_bertrand_prime
namespace Nat
/-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big)
have H3 : n.centralBinom β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) :=
centralBinom_le_of_no_bertrand_prime n (lt_of_lt_of_le (by norm_num1) n_big) no_prime
rw [mul_assoc] at H1; exact not_le.2 H2 ((mul_le_mul_left' H3 n).trans H1)
#align nat.exists_prime_lt_and_le_two_mul_eventually Nat.exists_prime_lt_and_le_two_mul_eventually
/-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number β€ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : β} (prime_p : Nat.Prime p)
(covering : p β€ 2 * q) (H : n < q β β p : β, p.Prime β§ n < p β§ p β€ 2 * n) (hn : n < p) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
by_cases h : p β€ 2 * n; Β· exact β¨p, prime_p, hn, hβ©
exact H (lt_of_mul_lt_mul_left' (lt_of_lt_of_le (not_le.1 h) covering))
#align nat.exists_prime_lt_and_le_two_mul_succ Nat.exists_prime_lt_and_le_two_mul_succ
/--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_le 511 n with h h
-- If `n` is large, apply the lemma derived from the inequalities on the central binomial
-- coefficient.
Β· | exact exists_prime_lt_and_le_two_mul_eventually n h | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_le 511 n with h h
-- If `n` is large, apply the lemma derived from the inequalities on the central binomial
-- coefficient.
Β· | Mathlib.NumberTheory.Bertrand.223_0.gJXoOT9Ce2wC0xc | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n | Mathlib_NumberTheory_Bertrand |
case inr
n : β
hn0 : n β 0
h : n β€ 511
β’ β p, Prime p β§ n < p β§ p β€ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _
exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le
#align central_binom_le_of_no_bertrand_prime centralBinom_le_of_no_bertrand_prime
namespace Nat
/-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big)
have H3 : n.centralBinom β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) :=
centralBinom_le_of_no_bertrand_prime n (lt_of_lt_of_le (by norm_num1) n_big) no_prime
rw [mul_assoc] at H1; exact not_le.2 H2 ((mul_le_mul_left' H3 n).trans H1)
#align nat.exists_prime_lt_and_le_two_mul_eventually Nat.exists_prime_lt_and_le_two_mul_eventually
/-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number β€ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : β} (prime_p : Nat.Prime p)
(covering : p β€ 2 * q) (H : n < q β β p : β, p.Prime β§ n < p β§ p β€ 2 * n) (hn : n < p) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
by_cases h : p β€ 2 * n; Β· exact β¨p, prime_p, hn, hβ©
exact H (lt_of_mul_lt_mul_left' (lt_of_lt_of_le (not_le.1 h) covering))
#align nat.exists_prime_lt_and_le_two_mul_succ Nat.exists_prime_lt_and_le_two_mul_succ
/--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_le 511 n with h h
-- If `n` is large, apply the lemma derived from the inequalities on the central binomial
-- coefficient.
Β· exact exists_prime_lt_and_le_two_mul_eventually n h
| replace h : n < 521 := h.trans_lt (by norm_num1) | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_le 511 n with h h
-- If `n` is large, apply the lemma derived from the inequalities on the central binomial
-- coefficient.
Β· exact exists_prime_lt_and_le_two_mul_eventually n h
| Mathlib.NumberTheory.Bertrand.223_0.gJXoOT9Ce2wC0xc | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n | Mathlib_NumberTheory_Bertrand |
n : β
hn0 : n β 0
h : n β€ 511
β’ 511 < 521 | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _
exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le
#align central_binom_le_of_no_bertrand_prime centralBinom_le_of_no_bertrand_prime
namespace Nat
/-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big)
have H3 : n.centralBinom β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) :=
centralBinom_le_of_no_bertrand_prime n (lt_of_lt_of_le (by norm_num1) n_big) no_prime
rw [mul_assoc] at H1; exact not_le.2 H2 ((mul_le_mul_left' H3 n).trans H1)
#align nat.exists_prime_lt_and_le_two_mul_eventually Nat.exists_prime_lt_and_le_two_mul_eventually
/-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number β€ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : β} (prime_p : Nat.Prime p)
(covering : p β€ 2 * q) (H : n < q β β p : β, p.Prime β§ n < p β§ p β€ 2 * n) (hn : n < p) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
by_cases h : p β€ 2 * n; Β· exact β¨p, prime_p, hn, hβ©
exact H (lt_of_mul_lt_mul_left' (lt_of_lt_of_le (not_le.1 h) covering))
#align nat.exists_prime_lt_and_le_two_mul_succ Nat.exists_prime_lt_and_le_two_mul_succ
/--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_le 511 n with h h
-- If `n` is large, apply the lemma derived from the inequalities on the central binomial
-- coefficient.
Β· exact exists_prime_lt_and_le_two_mul_eventually n h
replace h : n < 521 := h.trans_lt (by | norm_num1 | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_le 511 n with h h
-- If `n` is large, apply the lemma derived from the inequalities on the central binomial
-- coefficient.
Β· exact exists_prime_lt_and_le_two_mul_eventually n h
replace h : n < 521 := h.trans_lt (by | Mathlib.NumberTheory.Bertrand.223_0.gJXoOT9Ce2wC0xc | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n | Mathlib_NumberTheory_Bertrand |
case inr
n : β
hn0 : n β 0
h : n < 521
β’ β p, Prime p β§ n < p β§ p β€ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _
exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le
#align central_binom_le_of_no_bertrand_prime centralBinom_le_of_no_bertrand_prime
namespace Nat
/-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big)
have H3 : n.centralBinom β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) :=
centralBinom_le_of_no_bertrand_prime n (lt_of_lt_of_le (by norm_num1) n_big) no_prime
rw [mul_assoc] at H1; exact not_le.2 H2 ((mul_le_mul_left' H3 n).trans H1)
#align nat.exists_prime_lt_and_le_two_mul_eventually Nat.exists_prime_lt_and_le_two_mul_eventually
/-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number β€ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : β} (prime_p : Nat.Prime p)
(covering : p β€ 2 * q) (H : n < q β β p : β, p.Prime β§ n < p β§ p β€ 2 * n) (hn : n < p) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
by_cases h : p β€ 2 * n; Β· exact β¨p, prime_p, hn, hβ©
exact H (lt_of_mul_lt_mul_left' (lt_of_lt_of_le (not_le.1 h) covering))
#align nat.exists_prime_lt_and_le_two_mul_succ Nat.exists_prime_lt_and_le_two_mul_succ
/--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_le 511 n with h h
-- If `n` is large, apply the lemma derived from the inequalities on the central binomial
-- coefficient.
Β· exact exists_prime_lt_and_le_two_mul_eventually n h
replace h : n < 521 := h.trans_lt (by norm_num1)
| revert h | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_le 511 n with h h
-- If `n` is large, apply the lemma derived from the inequalities on the central binomial
-- coefficient.
Β· exact exists_prime_lt_and_le_two_mul_eventually n h
replace h : n < 521 := h.trans_lt (by norm_num1)
| Mathlib.NumberTheory.Bertrand.223_0.gJXoOT9Ce2wC0xc | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n | Mathlib_NumberTheory_Bertrand |
case inr
n : β
hn0 : n β 0
β’ n < 521 β β p, Prime p β§ n < p β§ p β€ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _
exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le
#align central_binom_le_of_no_bertrand_prime centralBinom_le_of_no_bertrand_prime
namespace Nat
/-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big)
have H3 : n.centralBinom β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) :=
centralBinom_le_of_no_bertrand_prime n (lt_of_lt_of_le (by norm_num1) n_big) no_prime
rw [mul_assoc] at H1; exact not_le.2 H2 ((mul_le_mul_left' H3 n).trans H1)
#align nat.exists_prime_lt_and_le_two_mul_eventually Nat.exists_prime_lt_and_le_two_mul_eventually
/-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number β€ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : β} (prime_p : Nat.Prime p)
(covering : p β€ 2 * q) (H : n < q β β p : β, p.Prime β§ n < p β§ p β€ 2 * n) (hn : n < p) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
by_cases h : p β€ 2 * n; Β· exact β¨p, prime_p, hn, hβ©
exact H (lt_of_mul_lt_mul_left' (lt_of_lt_of_le (not_le.1 h) covering))
#align nat.exists_prime_lt_and_le_two_mul_succ Nat.exists_prime_lt_and_le_two_mul_succ
/--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_le 511 n with h h
-- If `n` is large, apply the lemma derived from the inequalities on the central binomial
-- coefficient.
Β· exact exists_prime_lt_and_le_two_mul_eventually n h
replace h : n < 521 := h.trans_lt (by norm_num1)
revert h
-- For small `n`, supply a list of primes to cover the initial cases.
| open Lean Elab Tactic in
run_tac do
for i in [317, 163, 83, 43, 23, 13, 7, 5, 3, 2] do
let i : Term := quote i
evalTactic <| β
`(tactic| refine' exists_prime_lt_and_le_two_mul_succ $i (by norm_num1) (by norm_num1) _)
exact fun h2 => β¨2, prime_two, h2, Nat.mul_le_mul_left 2 (Nat.pos_of_ne_zero hn0)β© | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_le 511 n with h h
-- If `n` is large, apply the lemma derived from the inequalities on the central binomial
-- coefficient.
Β· exact exists_prime_lt_and_le_two_mul_eventually n h
replace h : n < 521 := h.trans_lt (by norm_num1)
revert h
-- For small `n`, supply a list of primes to cover the initial cases.
| Mathlib.NumberTheory.Bertrand.223_0.gJXoOT9Ce2wC0xc | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n | Mathlib_NumberTheory_Bertrand |
case inr
n : β
hn0 : n β 0
β’ n < 521 β β p, Prime p β§ n < p β§ p β€ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _
exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le
#align central_binom_le_of_no_bertrand_prime centralBinom_le_of_no_bertrand_prime
namespace Nat
/-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big)
have H3 : n.centralBinom β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) :=
centralBinom_le_of_no_bertrand_prime n (lt_of_lt_of_le (by norm_num1) n_big) no_prime
rw [mul_assoc] at H1; exact not_le.2 H2 ((mul_le_mul_left' H3 n).trans H1)
#align nat.exists_prime_lt_and_le_two_mul_eventually Nat.exists_prime_lt_and_le_two_mul_eventually
/-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number β€ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : β} (prime_p : Nat.Prime p)
(covering : p β€ 2 * q) (H : n < q β β p : β, p.Prime β§ n < p β§ p β€ 2 * n) (hn : n < p) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
by_cases h : p β€ 2 * n; Β· exact β¨p, prime_p, hn, hβ©
exact H (lt_of_mul_lt_mul_left' (lt_of_lt_of_le (not_le.1 h) covering))
#align nat.exists_prime_lt_and_le_two_mul_succ Nat.exists_prime_lt_and_le_two_mul_succ
/--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_le 511 n with h h
-- If `n` is large, apply the lemma derived from the inequalities on the central binomial
-- coefficient.
Β· exact exists_prime_lt_and_le_two_mul_eventually n h
replace h : n < 521 := h.trans_lt (by norm_num1)
revert h
-- For small `n`, supply a list of primes to cover the initial cases.
open Lean Elab Tactic in
| run_tac do
for i in [317, 163, 83, 43, 23, 13, 7, 5, 3, 2] do
let i : Term := quote i
evalTactic <| β
`(tactic| refine' exists_prime_lt_and_le_two_mul_succ $i (by norm_num1) (by norm_num1) _) | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_le 511 n with h h
-- If `n` is large, apply the lemma derived from the inequalities on the central binomial
-- coefficient.
Β· exact exists_prime_lt_and_le_two_mul_eventually n h
replace h : n < 521 := h.trans_lt (by norm_num1)
revert h
-- For small `n`, supply a list of primes to cover the initial cases.
open Lean Elab Tactic in
| Mathlib.NumberTheory.Bertrand.223_0.gJXoOT9Ce2wC0xc | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n | Mathlib_NumberTheory_Bertrand |
case inr
n : β
hn0 : n β 0
β’ n < 521 β β p, Prime p β§ n < p β§ p β€ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _
exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le
#align central_binom_le_of_no_bertrand_prime centralBinom_le_of_no_bertrand_prime
namespace Nat
/-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big)
have H3 : n.centralBinom β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) :=
centralBinom_le_of_no_bertrand_prime n (lt_of_lt_of_le (by norm_num1) n_big) no_prime
rw [mul_assoc] at H1; exact not_le.2 H2 ((mul_le_mul_left' H3 n).trans H1)
#align nat.exists_prime_lt_and_le_two_mul_eventually Nat.exists_prime_lt_and_le_two_mul_eventually
/-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number β€ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : β} (prime_p : Nat.Prime p)
(covering : p β€ 2 * q) (H : n < q β β p : β, p.Prime β§ n < p β§ p β€ 2 * n) (hn : n < p) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
by_cases h : p β€ 2 * n; Β· exact β¨p, prime_p, hn, hβ©
exact H (lt_of_mul_lt_mul_left' (lt_of_lt_of_le (not_le.1 h) covering))
#align nat.exists_prime_lt_and_le_two_mul_succ Nat.exists_prime_lt_and_le_two_mul_succ
/--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_le 511 n with h h
-- If `n` is large, apply the lemma derived from the inequalities on the central binomial
-- coefficient.
Β· exact exists_prime_lt_and_le_two_mul_eventually n h
replace h : n < 521 := h.trans_lt (by norm_num1)
revert h
-- For small `n`, supply a list of primes to cover the initial cases.
open Lean Elab Tactic in
| run_tac do
for i in [317, 163, 83, 43, 23, 13, 7, 5, 3, 2] do
let i : Term := quote i
evalTactic <| β
`(tactic| refine' exists_prime_lt_and_le_two_mul_succ $i (by norm_num1) (by norm_num1) _) | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_le 511 n with h h
-- If `n` is large, apply the lemma derived from the inequalities on the central binomial
-- coefficient.
Β· exact exists_prime_lt_and_le_two_mul_eventually n h
replace h : n < 521 := h.trans_lt (by norm_num1)
revert h
-- For small `n`, supply a list of primes to cover the initial cases.
open Lean Elab Tactic in
| Mathlib.NumberTheory.Bertrand.223_0.gJXoOT9Ce2wC0xc | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n | Mathlib_NumberTheory_Bertrand |
case inr
n : β
hn0 : n β 0
β’ n < 2 β β p, Prime p β§ n < p β§ p β€ 2 * n | /-
Copyright (c) 2020 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Bolton Bailey
-/
import Mathlib.Data.Nat.Choose.Factorization
import Mathlib.Data.Nat.PrimeNormNum
import Mathlib.NumberTheory.Primorial
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
#align_import number_theory.bertrand from "leanprover-community/mathlib"@"a16665637b378379689c566204817ae792ac8b39"
/-!
# Bertrand's Postulate
This file contains a proof of Bertrand's postulate: That between any positive number and its
double there is a prime.
The proof follows the outline of the ErdΕs proof presented in "Proofs from THE BOOK": One considers
the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various
groups, then upper bounds the contribution of each group. This upper bounds the central binomial
coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower
bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n`
an explicit list of primes is provided which covers the remaining cases.
As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from
[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central
binomial coefficient given in `Nat.four_pow_lt_mul_centralBinom`.
## References
* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs]
* [S. Tochiori, _Considering the Proof of βThere is a Prime between n and 2nβ_][tochiori_bertrand]
* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic]
## Tags
Bertrand, prime, binomial coefficients
-/
open scoped BigOperators
section Real
open Real
namespace Bertrand
/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 β€ x.
-/
theorem real_main_inequality {x : β} (n_large : (512 : β) β€ x) :
x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) β€ 4 ^ x := by
let f : β β β := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
have hf' : β x, 0 < x β 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
have hf : β x, 0 < x β f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
intro x h5
have h6 := mul_pos (zero_lt_two' β) h5
have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
log_rpow h6, log_rpow zero_lt_four, β mul_div_right_comm, β mul_div, mul_comm x]
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
rw [β div_le_one (rpow_pos_of_pos four_pos x), β div_div_eq_mul_div, β rpow_sub four_pos, β
mul_div 2 x, mul_div_left_comm, β mul_one_sub, (by norm_num1 : (1 : β) - 2 / 3 = 1 / 3),
mul_one_div, β log_nonpos_iff (hf' x h5), β hf x h5]
-- porting note: the proof was rewritten, because it was too slow
have h : ConcaveOn β (Set.Ioi 0.5) f := by
apply ConcaveOn.sub
apply ConcaveOn.add
exact strictConcaveOn_log_Ioi.concaveOn.subset
(Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
((2 : β) β’ LinearMap.id))) using 1
Β· ext x
simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
LinearMap.id_coe, id_eq, smul_eq_mul]
rw [β mul_lt_mul_left (two_pos)]
norm_num1
rfl
apply ConvexOn.smul
refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
exact convexOn_id (convex_Ioi (0.5 : β))
suffices β x1 x2, 0.5 < x1 β§ x1 < x2 β§ x2 β€ x β§ 0 β€ f x1 β§ f x2 β€ 0 by
obtain β¨x1, x2, h1, h2, h0, h3, h4β© := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
refine' β¨18, 512, by norm_num1, by norm_num1, n_large, _, _β©
Β· have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonneg_iff, this]
rw [one_le_div] <;> norm_num1
apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
apply Real.rpow_le_rpow <;> norm_num1
apply rpow_nonneg_of_nonneg; norm_num1
apply rpow_pos_of_pos; norm_num1
apply hf' 18; norm_num1
norm_num1
Β· have : sqrt (2 * 512) = 32 :=
(sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one] <;> norm_num1
Β· conv in 512 => equals 2 ^ 9 => norm_num1
conv in 1024 => equals 2 ^ 10 => norm_num1
conv in 32 => rw [β Nat.cast_ofNat]
rw [rpow_nat_cast, β pow_mul, β pow_add]
conv in 4 => equals 2 ^ (2 : β) => rw [rpow_two]; norm_num1
rw [β rpow_mul, β rpow_nat_cast]
apply rpow_le_rpow_of_exponent_le
all_goals norm_num1
Β· apply rpow_pos_of_pos four_pos
#align bertrand.real_main_inequality Bertrand.real_main_inequality
end Bertrand
end Real
section Nat
open Nat
/-- The inequality which contradicts Bertrand's postulate, for large enough `n`.
-/
theorem bertrand_main_inequality {n : β} (n_large : 512 β€ n) :
n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := by
rw [β @cast_le β]
simp only [cast_add, cast_one, cast_mul, cast_pow, β Real.rpow_nat_cast]
refine' _root_.trans ?_ (Bertrand.real_main_inequality (by exact_mod_cast n_large))
gcongr
Β· have n2_pos : 0 < 2 * n := by positivity
exact mod_cast n2_pos
Β· exact_mod_cast Real.nat_sqrt_le_real_sqrt
Β· norm_num1
Β· exact cast_div_le.trans (by norm_cast)
#align bertrand_main_inequality bertrand_main_inequality
/-- A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime
factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`.
-/
theorem centralBinom_factorization_small (n : β) (n_large : 2 < n)
(no_prime : Β¬β p : β, p.Prime β§ n < p β§ p β€ 2 * n) :
centralBinom n = β p in Finset.range (2 * n / 3 + 1), p ^ (centralBinom n).factorization p := by
refine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm
apply Finset.prod_subset
Β· exact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)
intro x hx h2x
rw [Finset.mem_range, lt_succ_iff] at hx h2x
rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x
replace no_prime := not_exists.mp no_prime x
rw [β and_assoc, not_and', not_and_or, not_lt] at no_prime
cases' no_prime hx with h h
Β· rw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]
Β· rw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]
#align central_binom_factorization_small centralBinom_factorization_small
/-- An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate.
The bound splits the prime factors of `centralBinom n` into those
1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime.
2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total.
3. Between `2 * n / 3` and `n`, which do not exist.
4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false.
5. Above `2 * n`, which do not exist.
-/
theorem centralBinom_le_of_no_bertrand_prime (n : β) (n_big : 2 < n)
(no_prime : Β¬β p : β, Nat.Prime p β§ n < p β§ p β€ 2 * n) :
centralBinom n β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := by
have n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big
have n2_pos : 1 β€ 2 * n := mul_pos (zero_lt_two' β) n_pos
let S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime
let f x := x ^ n.centralBinom.factorization x
have : β x : β in S, f x = β x : β in Finset.range (2 * n / 3 + 1), f x := by
refine' Finset.prod_filter_of_ne fun p _ h => _
contrapose! h; dsimp only
rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]
rw [centralBinom_factorization_small n n_big no_prime, β this, β
Finset.prod_filter_mul_prod_filter_not S (Β· β€ sqrt (2 * n))]
apply mul_le_mul'
Β· refine' (Finset.prod_le_prod' fun p _ => (_ : f p β€ 2 * n)).trans _
Β· exact pow_factorization_choose_le (mul_pos two_pos n_pos)
have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
rw [Finset.prod_const]
refine' pow_le_pow_right n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)
obtain β¨h1, h2β© := Finset.mem_filter.1 hx
exact Finset.mem_Icc.mpr β¨(Finset.mem_filter.1 h1).2.one_lt.le, h2β©
Β· refine' le_trans _ (primorial_le_4_pow (2 * n / 3))
refine' (Finset.prod_le_prod' fun p hp => (_ : f p β€ p)).trans _
Β· obtain β¨h1, h2β© := Finset.mem_filter.1 hp
refine' (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le
exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
refine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _
exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le
#align central_binom_le_of_no_bertrand_prime centralBinom_le_of_no_bertrand_prime
namespace Nat
/-- Proves that **Bertrand's postulate** holds for all sufficiently large `n`.
-/
theorem exists_prime_lt_and_le_two_mul_eventually (n : β) (n_big : 512 β€ n) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
-- Assume there is no prime in the range.
by_contra no_prime
-- Then we have the above sub-exponential bound on the size of this central binomial coefficient.
-- We now couple this bound with an exponential lower bound on the central binomial coefficient,
-- yielding an inequality which we have seen is false for large enough n.
have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) β€ 4 ^ n := bertrand_main_inequality n_big
have H2 : 4 ^ n < n * n.centralBinom :=
Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big)
have H3 : n.centralBinom β€ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) :=
centralBinom_le_of_no_bertrand_prime n (lt_of_lt_of_le (by norm_num1) n_big) no_prime
rw [mul_assoc] at H1; exact not_le.2 H2 ((mul_le_mul_left' H3 n).trans H1)
#align nat.exists_prime_lt_and_le_two_mul_eventually Nat.exists_prime_lt_and_le_two_mul_eventually
/-- Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a
descending list of primes, each no more than twice the next, such that the list contains a witness
for each number β€ n.
-/
theorem exists_prime_lt_and_le_two_mul_succ {n} (q) {p : β} (prime_p : Nat.Prime p)
(covering : p β€ 2 * q) (H : n < q β β p : β, p.Prime β§ n < p β§ p β€ 2 * n) (hn : n < p) :
β p : β, p.Prime β§ n < p β§ p β€ 2 * n := by
by_cases h : p β€ 2 * n; Β· exact β¨p, prime_p, hn, hβ©
exact H (lt_of_mul_lt_mul_left' (lt_of_lt_of_le (not_le.1 h) covering))
#align nat.exists_prime_lt_and_le_two_mul_succ Nat.exists_prime_lt_and_le_two_mul_succ
/--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_le 511 n with h h
-- If `n` is large, apply the lemma derived from the inequalities on the central binomial
-- coefficient.
Β· exact exists_prime_lt_and_le_two_mul_eventually n h
replace h : n < 521 := h.trans_lt (by norm_num1)
revert h
-- For small `n`, supply a list of primes to cover the initial cases.
open Lean Elab Tactic in
run_tac do
for i in [317, 163, 83, 43, 23, 13, 7, 5, 3, 2] do
let i : Term := quote i
evalTactic <| β
`(tactic| refine' exists_prime_lt_and_le_two_mul_succ $i (by norm_num1) (by norm_num1) _)
| exact fun h2 => β¨2, prime_two, h2, Nat.mul_le_mul_left 2 (Nat.pos_of_ne_zero hn0)β© | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n := by
-- Split into cases whether `n` is large or small
cases' lt_or_le 511 n with h h
-- If `n` is large, apply the lemma derived from the inequalities on the central binomial
-- coefficient.
Β· exact exists_prime_lt_and_le_two_mul_eventually n h
replace h : n < 521 := h.trans_lt (by norm_num1)
revert h
-- For small `n`, supply a list of primes to cover the initial cases.
open Lean Elab Tactic in
run_tac do
for i in [317, 163, 83, 43, 23, 13, 7, 5, 3, 2] do
let i : Term := quote i
evalTactic <| β
`(tactic| refine' exists_prime_lt_and_le_two_mul_succ $i (by norm_num1) (by norm_num1) _)
| Mathlib.NumberTheory.Bertrand.223_0.gJXoOT9Ce2wC0xc | /--
**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than
it, but no more than twice as large.
-/
theorem exists_prime_lt_and_le_two_mul (n : β) (hn0 : n β 0) :
β p, Nat.Prime p β§ n < p β§ p β€ 2 * n | Mathlib_NumberTheory_Bertrand |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
i j : β
β’ I ^ i β’ N F j β€ N F (i + j) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
| induction' i with _ ih | theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
| Mathlib.RingTheory.Filtration.64_0.wQ6WBws0g3n9213 | theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) | Mathlib_RingTheory_Filtration |
case zero
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
j : β
β’ I ^ Nat.zero β’ N F j β€ N F (Nat.zero + j) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· | simp | theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· | Mathlib.RingTheory.Filtration.64_0.wQ6WBws0g3n9213 | theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) | Mathlib_RingTheory_Filtration |
case succ
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
j nβ : β
ih : I ^ nβ β’ N F j β€ N F (nβ + j)
β’ I ^ Nat.succ nβ β’ N F j β€ N F (Nat.succ nβ + j) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· | rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc] | theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· | Mathlib.RingTheory.Filtration.64_0.wQ6WBws0g3n9213 | theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) | Mathlib_RingTheory_Filtration |
case succ
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
j nβ : β
ih : I ^ nβ β’ N F j β€ N F (nβ + j)
β’ I β’ I ^ nβ β’ N F j β€ N F (nβ + j + 1) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
| exact (Submodule.smul_mono_right ih).trans (F.smul_le _) | theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
| Mathlib.RingTheory.Filtration.64_0.wQ6WBws0g3n9213 | theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
i j k : β
β’ I ^ (i + k) β’ N F j β€ I ^ k β’ N F (i + j) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
| rw [add_comm, pow_add, mul_smul] | theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
| Mathlib.RingTheory.Filtration.71_0.wQ6WBws0g3n9213 | theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
i j k : β
β’ I ^ k β’ I ^ i β’ N F j β€ I ^ k β’ N F (i + j) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
| exact Submodule.smul_mono_right (F.pow_smul_le i j) | theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
| Mathlib.RingTheory.Filtration.71_0.wQ6WBws0g3n9213 | theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
S : Set (Filtration I M)
i : β
β’ sSup (N '' S) (i + 1) β€ sSup (N '' S) i | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
| apply sSup_le_sSup_of_forall_exists_le _ | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
| Mathlib.RingTheory.Filtration.94_0.wQ6WBws0g3n9213 | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
S : Set (Filtration I M)
i : β
β’ β x β Set.range fun f => βf (i + 1), β y β Set.range fun f => βf i, x β€ y | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
| rintro _ β¨β¨_, F, hF, rflβ©, rflβ© | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
| Mathlib.RingTheory.Filtration.94_0.wQ6WBws0g3n9213 | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) | Mathlib_RingTheory_Filtration |
case intro.mk.intro.intro
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
Fβ F' : Filtration I M
S : Set (Filtration I M)
i : β
F : Filtration I M
hF : F β S
β’ β y β Set.range fun f => βf i, (fun f => βf (i + 1)) { val := F.N, property := (_ : β a β S, a.N = F.N) } β€ y | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
| exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ© | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
| Mathlib.RingTheory.Filtration.94_0.wQ6WBws0g3n9213 | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
S : Set (Filtration I M)
i : β
β’ I β’ sSup (N '' S) i β€ sSup (N '' S) (i + 1) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
| rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
| Mathlib.RingTheory.Filtration.94_0.wQ6WBws0g3n9213 | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
S : Set (Filtration I M)
i : β
β’ β¨ i_1, I β’ βi_1 i β€ β¨ i_1, βi_1 (i + 1) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
| apply iSup_mono _ | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
| Mathlib.RingTheory.Filtration.94_0.wQ6WBws0g3n9213 | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
S : Set (Filtration I M)
i : β
β’ β (i_1 : β(N '' S)), I β’ βi_1 i β€ βi_1 (i + 1) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
| rintro β¨_, F, hF, rflβ© | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
| Mathlib.RingTheory.Filtration.94_0.wQ6WBws0g3n9213 | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) | Mathlib_RingTheory_Filtration |
case mk.intro.intro
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
Fβ F' : Filtration I M
S : Set (Filtration I M)
i : β
F : Filtration I M
hF : F β S
β’ I β’ β{ val := F.N, property := (_ : β a β S, a.N = F.N) } i β€
β{ val := F.N, property := (_ : β a β S, a.N = F.N) } (i + 1) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
| exact F.smul_le i | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
| Mathlib.RingTheory.Filtration.94_0.wQ6WBws0g3n9213 | /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
S : Set (Filtration I M)
i : β
β’ sInf (N '' S) (i + 1) β€ sInf (N '' S) i | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
| apply sInf_le_sInf_of_forall_exists_le _ | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
| Mathlib.RingTheory.Filtration.114_0.wQ6WBws0g3n9213 | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
S : Set (Filtration I M)
i : β
β’ β x β Set.range fun f => βf i, β y β Set.range fun f => βf (i + 1), y β€ x | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
| rintro _ β¨β¨_, F, hF, rflβ©, rflβ© | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
| Mathlib.RingTheory.Filtration.114_0.wQ6WBws0g3n9213 | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) | Mathlib_RingTheory_Filtration |
case intro.mk.intro.intro
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
Fβ F' : Filtration I M
S : Set (Filtration I M)
i : β
F : Filtration I M
hF : F β S
β’ β y β Set.range fun f => βf (i + 1), y β€ (fun f => βf i) { val := F.N, property := (_ : β a β S, a.N = F.N) } | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
| exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ© | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
| Mathlib.RingTheory.Filtration.114_0.wQ6WBws0g3n9213 | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
S : Set (Filtration I M)
i : β
β’ I β’ sInf (N '' S) i β€ sInf (N '' S) (i + 1) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
| rw [sInf_eq_iInf', iInf_apply, iInf_apply] | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
| Mathlib.RingTheory.Filtration.114_0.wQ6WBws0g3n9213 | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
S : Set (Filtration I M)
i : β
β’ I β’ β¨
i_1, βi_1 i β€ β¨
i_1, βi_1 (i + 1) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
| refine' Submodule.smul_iInf_le.trans _ | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
| Mathlib.RingTheory.Filtration.114_0.wQ6WBws0g3n9213 | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
S : Set (Filtration I M)
i : β
β’ β¨
i_1, I β’ βi_1 i β€ β¨
i_1, βi_1 (i + 1) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
| apply iInf_mono _ | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
| Mathlib.RingTheory.Filtration.114_0.wQ6WBws0g3n9213 | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
S : Set (Filtration I M)
i : β
β’ β (i_1 : β(N '' S)), I β’ βi_1 i β€ βi_1 (i + 1) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
| rintro β¨_, F, hF, rflβ© | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
| Mathlib.RingTheory.Filtration.114_0.wQ6WBws0g3n9213 | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) | Mathlib_RingTheory_Filtration |
case mk.intro.intro
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
Fβ F' : Filtration I M
S : Set (Filtration I M)
i : β
F : Filtration I M
hF : F β S
β’ I β’ β{ val := F.N, property := (_ : β a β S, a.N = F.N) } i β€
β{ val := F.N, property := (_ : β a β S, a.N = F.N) } (i + 1) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
| exact F.smul_le i | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
| Mathlib.RingTheory.Filtration.114_0.wQ6WBws0g3n9213 | /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
Iβ : Ideal R
F F' : Filtration Iβ M
I : Ideal R
N : Submodule R M
i : β
β’ (fun i => I ^ i β’ N) (i + 1) β€ (fun i => I ^ i β’ N) i | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by | dsimp only | /-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by | Mathlib.RingTheory.Filtration.195_0.wQ6WBws0g3n9213 | /-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
Iβ : Ideal R
F F' : Filtration Iβ M
I : Ideal R
N : Submodule R M
i : β
β’ I ^ (i + 1) β’ N β€ I ^ i β’ N | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; | rw [add_comm, pow_add, mul_smul] | /-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; | Mathlib.RingTheory.Filtration.195_0.wQ6WBws0g3n9213 | /-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
Iβ : Ideal R
F F' : Filtration Iβ M
I : Ideal R
N : Submodule R M
i : β
β’ I ^ 1 β’ I ^ i β’ N β€ I ^ i β’ N | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; | exact Submodule.smul_le_right | /-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; | Mathlib.RingTheory.Filtration.195_0.wQ6WBws0g3n9213 | /-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
Iβ : Ideal R
F F' : Filtration Iβ M
I : Ideal R
N : Submodule R M
i : β
β’ I β’ (fun i => I ^ i β’ N) i β€ (fun i => I ^ i β’ N) (i + 1) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by | dsimp only | /-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by | Mathlib.RingTheory.Filtration.195_0.wQ6WBws0g3n9213 | /-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
Iβ : Ideal R
F F' : Filtration Iβ M
I : Ideal R
N : Submodule R M
i : β
β’ I β’ I ^ i β’ N β€ I ^ (i + 1) β’ N | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; | rw [add_comm, pow_add, mul_smul, pow_one] | /-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; | Mathlib.RingTheory.Filtration.195_0.wQ6WBws0g3n9213 | /-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
Iβ : Ideal R
F F' : Filtration Iβ M
I : Ideal R
N : Submodule R M
β’ Stable (stableFiltration I N) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
| use 0 | theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
| Mathlib.RingTheory.Filtration.203_0.wQ6WBws0g3n9213 | theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable | Mathlib_RingTheory_Filtration |
case h
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
Iβ : Ideal R
F F' : Filtration Iβ M
I : Ideal R
N : Submodule R M
β’ β n β₯ 0, I β’ Ideal.Filtration.N (stableFiltration I N) n = Ideal.Filtration.N (stableFiltration I N) (n + 1) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
| intro n _ | theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
| Mathlib.RingTheory.Filtration.203_0.wQ6WBws0g3n9213 | theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable | Mathlib_RingTheory_Filtration |
case h
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
Iβ : Ideal R
F F' : Filtration Iβ M
I : Ideal R
N : Submodule R M
n : β
aβ : n β₯ 0
β’ I β’ Ideal.Filtration.N (stableFiltration I N) n = Ideal.Filtration.N (stableFiltration I N) (n + 1) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
| dsimp | theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
| Mathlib.RingTheory.Filtration.203_0.wQ6WBws0g3n9213 | theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable | Mathlib_RingTheory_Filtration |
case h
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
Iβ : Ideal R
F F' : Filtration Iβ M
I : Ideal R
N : Submodule R M
n : β
aβ : n β₯ 0
β’ I β’ I ^ n β’ N = I ^ (n + 1) β’ N | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
| rw [add_comm, pow_add, mul_smul, pow_one] | theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
| Mathlib.RingTheory.Filtration.203_0.wQ6WBws0g3n9213 | theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
β’ β nβ, β (k : β), N F (nβ + k) = I ^ k β’ N F nβ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
| obtain β¨nβ, hnβ© := h | theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
| Mathlib.RingTheory.Filtration.213_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ | Mathlib_RingTheory_Filtration |
case intro
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
nβ : β
hn : β n β₯ nβ, I β’ N F n = N F (n + 1)
β’ β nβ, β (k : β), N F (nβ + k) = I ^ k β’ N F nβ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
| use nβ | theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
| Mathlib.RingTheory.Filtration.213_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ | Mathlib_RingTheory_Filtration |
case h
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
nβ : β
hn : β n β₯ nβ, I β’ N F n = N F (n + 1)
β’ β (k : β), N F (nβ + k) = I ^ k β’ N F nβ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
| intro k | theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
| Mathlib.RingTheory.Filtration.213_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ | Mathlib_RingTheory_Filtration |
case h
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
nβ : β
hn : β n β₯ nβ, I β’ N F n = N F (n + 1)
k : β
β’ N F (nβ + k) = I ^ k β’ N F nβ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
| induction' k with _ ih | theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
| Mathlib.RingTheory.Filtration.213_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ | Mathlib_RingTheory_Filtration |
case h.zero
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
nβ : β
hn : β n β₯ nβ, I β’ N F n = N F (n + 1)
β’ N F (nβ + Nat.zero) = I ^ Nat.zero β’ N F nβ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· | simp | theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· | Mathlib.RingTheory.Filtration.213_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ | Mathlib_RingTheory_Filtration |
case h.succ
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
nβ : β
hn : β n β₯ nβ, I β’ N F n = N F (n + 1)
nβ : β
ih : N F (nβ + nβ) = I ^ nβ β’ N F nβ
β’ N F (nβ + Nat.succ nβ) = I ^ Nat.succ nβ β’ N F nβ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· | rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one] | theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· | Mathlib.RingTheory.Filtration.213_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ | Mathlib_RingTheory_Filtration |
case h.succ.a
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
nβ : β
hn : β n β₯ nβ, I β’ N F n = N F (n + 1)
nβ : β
ih : N F (nβ + nβ) = I ^ nβ β’ N F nβ
β’ nβ + nβ β₯ nβ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
| linarith | theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
| Mathlib.RingTheory.Filtration.213_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
β’ β nβ, β n β₯ nβ, N F n = I ^ (n - nβ) β’ N F nβ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
| obtain β¨nβ, hnββ© := h.exists_pow_smul_eq | theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
| Mathlib.RingTheory.Filtration.223_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ | Mathlib_RingTheory_Filtration |
case intro
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
nβ : β
hnβ : β (k : β), N F (nβ + k) = I ^ k β’ N F nβ
β’ β nβ, β n β₯ nβ, N F n = I ^ (n - nβ) β’ N F nβ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
| use nβ | theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
| Mathlib.RingTheory.Filtration.223_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ | Mathlib_RingTheory_Filtration |
case h
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
nβ : β
hnβ : β (k : β), N F (nβ + k) = I ^ k β’ N F nβ
β’ β n β₯ nβ, N F n = I ^ (n - nβ) β’ N F nβ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
| intro n hn | theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
| Mathlib.RingTheory.Filtration.223_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ | Mathlib_RingTheory_Filtration |
case h
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
nβ : β
hnβ : β (k : β), N F (nβ + k) = I ^ k β’ N F nβ
n : β
hn : n β₯ nβ
β’ N F n = I ^ (n - nβ) β’ N F nβ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
| convert hnβ (n - nβ) | theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
| Mathlib.RingTheory.Filtration.223_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ | Mathlib_RingTheory_Filtration |
case h.e'_2.h.e'_8
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
nβ : β
hnβ : β (k : β), N F (nβ + k) = I ^ k β’ N F nβ
n : β
hn : n β₯ nβ
β’ n = nβ + (n - nβ) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
| rw [add_comm, tsub_add_cancel_of_le hn] | theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
| Mathlib.RingTheory.Filtration.223_0.wQ6WBws0g3n9213 | theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
β’ Stable F β β nβ, β n β₯ nβ, N F n = I ^ (n - nβ) β’ N F nβ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
| refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β© | theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
| Mathlib.RingTheory.Filtration.231_0.wQ6WBws0g3n9213 | theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
hβ : Stable F
h : β nβ, β n β₯ nβ, N F n = I ^ (n - nβ) β’ N F nβ
n : β
hn : n β₯ Exists.choose h
β’ I β’ N F n = N F (n + 1) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
| rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn] | theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
| Mathlib.RingTheory.Filtration.231_0.wQ6WBws0g3n9213 | theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
hβ : Stable F
h : β nβ, β n β₯ nβ, N F n = I ^ (n - nβ) β’ N F nβ
n : β
hn : n β₯ Exists.choose h
β’ n + 1 β₯ Exists.choose h | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by | linarith | theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by | Mathlib.RingTheory.Filtration.231_0.wQ6WBws0g3n9213 | theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
hβ h : Stable F
e : N F 0 β€ N F' 0
β’ β nβ, β (n : β), N F (n + nβ) β€ N F' n | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
| obtain β¨nβ, hFβ© := h | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
| Mathlib.RingTheory.Filtration.238_0.wQ6WBws0g3n9213 | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n | Mathlib_RingTheory_Filtration |
case intro
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
e : N F 0 β€ N F' 0
nβ : β
hF : β n β₯ nβ, I β’ N F n = N F (n + 1)
β’ β nβ, β (n : β), N F (n + nβ) β€ N F' n | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
| use nβ | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
| Mathlib.RingTheory.Filtration.238_0.wQ6WBws0g3n9213 | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n | Mathlib_RingTheory_Filtration |
case h
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
e : N F 0 β€ N F' 0
nβ : β
hF : β n β₯ nβ, I β’ N F n = N F (n + 1)
β’ β (n : β), N F (n + nβ) β€ N F' n | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
| intro n | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
| Mathlib.RingTheory.Filtration.238_0.wQ6WBws0g3n9213 | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n | Mathlib_RingTheory_Filtration |
case h
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
e : N F 0 β€ N F' 0
nβ : β
hF : β n β₯ nβ, I β’ N F n = N F (n + 1)
n : β
β’ N F (n + nβ) β€ N F' n | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
| induction' n with n hn | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
| Mathlib.RingTheory.Filtration.238_0.wQ6WBws0g3n9213 | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n | Mathlib_RingTheory_Filtration |
case h.zero
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
e : N F 0 β€ N F' 0
nβ : β
hF : β n β₯ nβ, I β’ N F n = N F (n + 1)
β’ N F (Nat.zero + nβ) β€ N F' Nat.zero | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· | refine' (F.antitone _).trans e | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· | Mathlib.RingTheory.Filtration.238_0.wQ6WBws0g3n9213 | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n | Mathlib_RingTheory_Filtration |
case h.zero
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
e : N F 0 β€ N F' 0
nβ : β
hF : β n β₯ nβ, I β’ N F n = N F (n + 1)
β’ 0 β€ Nat.zero + nβ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; | simp | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; | Mathlib.RingTheory.Filtration.238_0.wQ6WBws0g3n9213 | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n | Mathlib_RingTheory_Filtration |
case h.succ
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
e : N F 0 β€ N F' 0
nβ : β
hF : β n β₯ nβ, I β’ N F n = N F (n + 1)
n : β
hn : N F (n + nβ) β€ N F' n
β’ N F (Nat.succ n + nβ) β€ N F' (Nat.succ n) | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· | rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF] | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· | Mathlib.RingTheory.Filtration.238_0.wQ6WBws0g3n9213 | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n | Mathlib_RingTheory_Filtration |
case h.succ
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
e : N F 0 β€ N F' 0
nβ : β
hF : β n β₯ nβ, I β’ N F n = N F (n + 1)
n : β
hn : N F (n + nβ) β€ N F' n
β’ I β’ N F (n + nβ) β€ N F' (n + 1)
case h.succ.a
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
e : N F 0 β€ N F' 0
nβ : β
hF : β n β₯ nβ, I β’ N F n = N F (n + 1)
n : β
hn : N F (n + nβ) β€ N F' n
β’ n + nβ β₯ nβ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
| exact (Submodule.smul_mono_right hn).trans (F'.smul_le _) | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
| Mathlib.RingTheory.Filtration.238_0.wQ6WBws0g3n9213 | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n | Mathlib_RingTheory_Filtration |
case h.succ.a
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
e : N F 0 β€ N F' 0
nβ : β
hF : β n β₯ nβ, I β’ N F n = N F (n + 1)
n : β
hn : N F (n + nβ) β€ N F' n
β’ n + nβ β₯ nβ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
| simp | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
| Mathlib.RingTheory.Filtration.238_0.wQ6WBws0g3n9213 | theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
hβ h : Stable F
h' : Stable F'
e : N F 0 = N F' 0
β’ β nβ, β (n : β), N F (n + nβ) β€ N F' n β§ N F' (n + nβ) β€ N F n | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
| obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e) | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
| Mathlib.RingTheory.Filtration.250_0.wQ6WBws0g3n9213 | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n | Mathlib_RingTheory_Filtration |
case intro
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
hβ h : Stable F
h' : Stable F'
e : N F 0 = N F' 0
nβ : β
hβ : β (n : β), N F (n + nβ) β€ N F' n
β’ β nβ, β (n : β), N F (n + nβ) β€ N F' n β§ N F' (n + nβ) β€ N F n | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
| obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm) | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
| Mathlib.RingTheory.Filtration.250_0.wQ6WBws0g3n9213 | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n | Mathlib_RingTheory_Filtration |
case intro.intro
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
hβ h : Stable F
h' : Stable F'
e : N F 0 = N F' 0
nβ : β
hβ : β (n : β), N F (n + nβ) β€ N F' n
nβ : β
hβ : β (n : β), N F' (n + nβ) β€ N F n
β’ β nβ, β (n : β), N F (n + nβ) β€ N F' n β§ N F' (n + nβ) β€ N F n | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
| use max nβ nβ | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
| Mathlib.RingTheory.Filtration.250_0.wQ6WBws0g3n9213 | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n | Mathlib_RingTheory_Filtration |
case h
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
hβ h : Stable F
h' : Stable F'
e : N F 0 = N F' 0
nβ : β
hβ : β (n : β), N F (n + nβ) β€ N F' n
nβ : β
hβ : β (n : β), N F' (n + nβ) β€ N F n
β’ β (n : β), N F (n + max nβ nβ) β€ N F' n β§ N F' (n + max nβ nβ) β€ N F n | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
| intro n | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
| Mathlib.RingTheory.Filtration.250_0.wQ6WBws0g3n9213 | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n | Mathlib_RingTheory_Filtration |
case h
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
hβ h : Stable F
h' : Stable F'
e : N F 0 = N F' 0
nβ : β
hβ : β (n : β), N F (n + nβ) β€ N F' n
nβ : β
hβ : β (n : β), N F' (n + nβ) β€ N F n
n : β
β’ N F (n + max nβ nβ) β€ N F' n β§ N F' (n + max nβ nβ) β€ N F n | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
| refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
| Mathlib.RingTheory.Filtration.250_0.wQ6WBws0g3n9213 | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n | Mathlib_RingTheory_Filtration |
case h.refine'_1
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
hβ h : Stable F
h' : Stable F'
e : N F 0 = N F' 0
nβ : β
hβ : β (n : β), N F (n + nβ) β€ N F' n
nβ : β
hβ : β (n : β), N F' (n + nβ) β€ N F n
n : β
β’ n + nβ β€ n + max nβ nβ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> | simp | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> | Mathlib.RingTheory.Filtration.250_0.wQ6WBws0g3n9213 | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n | Mathlib_RingTheory_Filtration |
case h.refine'_2
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
hβ h : Stable F
h' : Stable F'
e : N F 0 = N F' 0
nβ : β
hβ : β (n : β), N F (n + nβ) β€ N F' n
nβ : β
hβ : β (n : β), N F' (n + nβ) β€ N F n
n : β
β’ n + nβ β€ n + max nβ nβ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> | simp | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> | Mathlib.RingTheory.Filtration.250_0.wQ6WBws0g3n9213 | theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
r : β₯(reesAlgebra I)
f : PolynomialModule R M
hf :
f β
{
toAddSubsemigroup :=
{ carrier := {f | β (i : β), f i β N F i},
add_mem' :=
(_ :
β {a b : PolynomialModule R M},
a β {f | β (i : β), f i β N F i} β
b β {f | β (i : β), f i β N F i} β β (i : β), a i + b i β N F i) },
zero_mem' := (_ : β (i : β), 0 β N F i) }.toAddSubsemigroup.carrier
i : β
β’ (r β’ f) i β N F i | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
| rw [Subalgebra.smul_def, PolynomialModule.smul_apply] | /-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
| Mathlib.RingTheory.Filtration.263_0.wQ6WBws0g3n9213 | /-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier | Mathlib_RingTheory_Filtration |
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
r : β₯(reesAlgebra I)
f : PolynomialModule R M
hf :
f β
{
toAddSubsemigroup :=
{ carrier := {f | β (i : β), f i β N F i},
add_mem' :=
(_ :
β {a b : PolynomialModule R M},
a β {f | β (i : β), f i β N F i} β
b β {f | β (i : β), f i β N F i} β β (i : β), a i + b i β N F i) },
zero_mem' := (_ : β (i : β), 0 β N F i) }.toAddSubsemigroup.carrier
i : β
β’ β x in Finset.antidiagonal i, coeff (βr) x.1 β’ f x.2 β N F i | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
| apply Submodule.sum_mem | /-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
| Mathlib.RingTheory.Filtration.263_0.wQ6WBws0g3n9213 | /-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier | Mathlib_RingTheory_Filtration |
case a
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
r : β₯(reesAlgebra I)
f : PolynomialModule R M
hf :
f β
{
toAddSubsemigroup :=
{ carrier := {f | β (i : β), f i β N F i},
add_mem' :=
(_ :
β {a b : PolynomialModule R M},
a β {f | β (i : β), f i β N F i} β
b β {f | β (i : β), f i β N F i} β β (i : β), a i + b i β N F i) },
zero_mem' := (_ : β (i : β), 0 β N F i) }.toAddSubsemigroup.carrier
i : β
β’ β c β Finset.antidiagonal i, coeff (βr) c.1 β’ f c.2 β N F i | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
| rintro β¨j, kβ© e | /-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
| Mathlib.RingTheory.Filtration.263_0.wQ6WBws0g3n9213 | /-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier | Mathlib_RingTheory_Filtration |
case a.mk
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
r : β₯(reesAlgebra I)
f : PolynomialModule R M
hf :
f β
{
toAddSubsemigroup :=
{ carrier := {f | β (i : β), f i β N F i},
add_mem' :=
(_ :
β {a b : PolynomialModule R M},
a β {f | β (i : β), f i β N F i} β
b β {f | β (i : β), f i β N F i} β β (i : β), a i + b i β N F i) },
zero_mem' := (_ : β (i : β), 0 β N F i) }.toAddSubsemigroup.carrier
i j k : β
e : (j, k) β Finset.antidiagonal i
β’ coeff βr (j, k).1 β’ f (j, k).2 β N F i | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
| rw [Finset.mem_antidiagonal] at e | /-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
| Mathlib.RingTheory.Filtration.263_0.wQ6WBws0g3n9213 | /-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier | Mathlib_RingTheory_Filtration |
case a.mk
R M : Type u
instβΒ² : CommRing R
instβΒΉ : AddCommGroup M
instβ : Module R M
I : Ideal R
F F' : Filtration I M
h : Stable F
r : β₯(reesAlgebra I)
f : PolynomialModule R M
hf :
f β
{
toAddSubsemigroup :=
{ carrier := {f | β (i : β), f i β N F i},
add_mem' :=
(_ :
β {a b : PolynomialModule R M},
a β {f | β (i : β), f i β N F i} β
b β {f | β (i : β), f i β N F i} β β (i : β), a i + b i β N F i) },
zero_mem' := (_ : β (i : β), 0 β N F i) }.toAddSubsemigroup.carrier
i j k : β
e : (j, k).1 + (j, k).2 = i
β’ coeff βr (j, k).1 β’ f (j, k).2 β N F i | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
| subst e | /-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
| Mathlib.RingTheory.Filtration.263_0.wQ6WBws0g3n9213 | /-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier | Mathlib_RingTheory_Filtration |
Subsets and Splits