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case h.e'_3
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m n : ℕ
⊢ T R ((m + 2) * n) + T R (m * n) = T R (2 * n + m * n) + T R (m * n) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
/-!
# Chebyshev polynomials
The Chebyshev polynomials are two families of polynomials indexed by `ℕ`,
with integral coefficients.
## Main definitions
* `Polynomial.Chebyshev.T`: the Chebyshev polynomials of the first kind.
* `Polynomial.Chebyshev.U`: the Chebyshev polynomials of the second kind.
## Main statements
* The formal derivative of the Chebyshev polynomials of the first kind is a scalar multiple of the
Chebyshev polynomials of the second kind.
* `Polynomial.Chebyshev.mul_T`, the product of the `m`-th and `(m + k)`-th Chebyshev polynomials of
the first kind is the sum of the `(2 * m + k)`-th and `k`-th Chebyshev polynomials of the first
kind.
* `Polynomial.Chebyshev.T_mul`, the `(m * n)`-th Chebyshev polynomial of the first kind is the
composition of the `m`-th and `n`-th Chebyshev polynomials of the first kind.
## Implementation details
Since Chebyshev polynomials have interesting behaviour over the complex numbers and modulo `p`,
we define them to have coefficients in an arbitrary commutative ring, even though
technically `ℤ` would suffice.
The benefit of allowing arbitrary coefficient rings, is that the statements afterwards are clean,
and do not have `map (Int.castRingHom R)` interfering all the time.
## References
[Lionel Ponton, _Roots of the Chebyshev polynomials: A purely algebraic approach_]
[ponton2020chebyshev]
## TODO
* Redefine and/or relate the definition of Chebyshev polynomials to `LinearRecurrence`.
* Add explicit formula involving square roots for Chebyshev polynomials
* Compute zeroes and extrema of Chebyshev polynomials.
* Prove that the roots of the Chebyshev polynomials (except 0) are irrational.
* Prove minimax properties of Chebyshev polynomials.
-/
noncomputable section
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
open Polynomial
variable (R S : Type*) [CommRing R] [CommRing S]
/-- `T n` is the `n`-th Chebyshev polynomial of the first kind -/
noncomputable def T : ℕ → R[X]
| 0 => 1
| 1 => X
| n + 2 => 2 * X * T (n + 1) - T n
#align polynomial.chebyshev.T Polynomial.Chebyshev.T
@[simp]
theorem T_zero : T R 0 = 1 := rfl
#align polynomial.chebyshev.T_zero Polynomial.Chebyshev.T_zero
@[simp]
theorem T_one : T R 1 = X := rfl
#align polynomial.chebyshev.T_one Polynomial.Chebyshev.T_one
@[simp]
theorem T_add_two (n : ℕ) : T R (n + 2) = 2 * X * T R (n + 1) - T R n := by rw [T]
#align polynomial.chebyshev.T_add_two Polynomial.Chebyshev.T_add_two
theorem T_two : T R 2 = 2 * X ^ 2 - 1 := by simp only [T, sub_left_inj, sq, mul_assoc]
#align polynomial.chebyshev.T_two Polynomial.Chebyshev.T_two
theorem T_of_two_le (n : ℕ) (h : 2 ≤ n) : T R n = 2 * X * T R (n - 1) - T R (n - 2) := by
obtain ⟨n, rfl⟩ := Nat.exists_eq_add_of_le h
rw [add_comm]
exact T_add_two R n
#align polynomial.chebyshev.T_of_two_le Polynomial.Chebyshev.T_of_two_le
/-- `U n` is the `n`-th Chebyshev polynomial of the second kind -/
noncomputable def U : ℕ → R[X]
| 0 => 1
| 1 => 2 * X
| n + 2 => 2 * X * U (n + 1) - U n
#align polynomial.chebyshev.U Polynomial.Chebyshev.U
@[simp]
theorem U_zero : U R 0 = 1 := rfl
#align polynomial.chebyshev.U_zero Polynomial.Chebyshev.U_zero
@[simp]
theorem U_one : U R 1 = 2 * X := rfl
#align polynomial.chebyshev.U_one Polynomial.Chebyshev.U_one
@[simp]
theorem U_add_two (n : ℕ) : U R (n + 2) = 2 * X * U R (n + 1) - U R n := by rw [U]
#align polynomial.chebyshev.U_add_two Polynomial.Chebyshev.U_add_two
theorem U_two : U R 2 = 4 * X ^ 2 - 1 := by
simp only [U]
ring
#align polynomial.chebyshev.U_two Polynomial.Chebyshev.U_two
theorem U_of_two_le (n : ℕ) (h : 2 ≤ n) : U R n = 2 * X * U R (n - 1) - U R (n - 2) := by
obtain ⟨n, rfl⟩ := Nat.exists_eq_add_of_le h
rw [add_comm]
exact U_add_two R n
#align polynomial.chebyshev.U_of_two_le Polynomial.Chebyshev.U_of_two_le
theorem U_eq_X_mul_U_add_T : ∀ n : ℕ, U R (n + 1) = X * U R n + T R (n + 1)
| 0 => by simp only [T, U, two_mul, mul_one]
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
U R (n + 2 + 1) = 2 * X * (X * U R (n + 1) + T R (n + 2)) - (X * U R n + T R (n + 1)) := by
rw [U_add_two, U_eq_X_mul_U_add_T n, U_eq_X_mul_U_add_T (n + 1), U_eq_X_mul_U_add_T n]
_ = X * (2 * X * U R (n + 1) - U R n) + (2 * X * T R (n + 2) - T R (n + 1)) := by ring
_ = X * U R (n + 2) + T R (n + 2 + 1) := by simp only [U_add_two, T_add_two]
#align polynomial.chebyshev.U_eq_X_mul_U_add_T Polynomial.Chebyshev.U_eq_X_mul_U_add_T
theorem T_eq_U_sub_X_mul_U (n : ℕ) : T R (n + 1) = U R (n + 1) - X * U R n := by
rw [U_eq_X_mul_U_add_T, add_comm (X * U R n), add_sub_cancel]
#align polynomial.chebyshev.T_eq_U_sub_X_mul_U Polynomial.Chebyshev.T_eq_U_sub_X_mul_U
theorem T_eq_X_mul_T_sub_pol_U : ∀ n : ℕ, T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n
| 0 => by simp only [T, U]; ring
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
T R (n + 2 + 2) = 2 * X * T R (n + 2 + 1) - T R (n + 2) := T_add_two _ _
_ = 2 * X * (X * T R (n + 2) - (1 - X ^ 2) * U R (n + 1)) -
(X * T R (n + 1) - (1 - X ^ 2) * U R n) :=
by simp only [T_eq_X_mul_T_sub_pol_U]
_ = X * (2 * X * T R (n + 2) - T R (n + 1)) - (1 - X ^ 2) * (2 * X * U R (n + 1) - U R n) :=
by ring
_ = X * T R (n + 2 + 1) - (1 - X ^ 2) * U R (n + 2) := by rw [T_add_two _ (n + 1), U_add_two]
#align polynomial.chebyshev.T_eq_X_mul_T_sub_pol_U Polynomial.Chebyshev.T_eq_X_mul_T_sub_pol_U
theorem one_sub_X_sq_mul_U_eq_pol_in_T (n : ℕ) :
(1 - X ^ 2) * U R n = X * T R (n + 1) - T R (n + 2) := by
rw [T_eq_X_mul_T_sub_pol_U, ← sub_add, sub_self, zero_add]
#align polynomial.chebyshev.one_sub_X_sq_mul_U_eq_pol_in_T Polynomial.Chebyshev.one_sub_X_sq_mul_U_eq_pol_in_T
variable {R S}
@[simp]
theorem map_T (f : R →+* S) : ∀ n : ℕ, map f (T R n) = T S n
| 0 => by simp only [T_zero, Polynomial.map_one]
| 1 => by simp only [T_one, map_X]
| n + 2 => by
simp only [T_add_two, Polynomial.map_mul, Polynomial.map_sub, map_X, Polynomial.map_add,
Polynomial.map_one, Polynomial.map_ofNat, map_T f (n + 1), map_T f n]
#align polynomial.chebyshev.map_T Polynomial.Chebyshev.map_T
@[simp]
theorem map_U (f : R →+* S) : ∀ n : ℕ, map f (U R n) = U S n
| 0 => by simp only [U_zero, Polynomial.map_one]
| 1 => by
simp [U_one, map_X, Polynomial.map_mul, Polynomial.map_add, Polynomial.map_one]
| n + 2 => by
simp only [U_add_two, Polynomial.map_mul, Polynomial.map_sub, map_X, Polynomial.map_add,
Polynomial.map_one, map_U f (n + 1), map_U f n]
norm_num
#align polynomial.chebyshev.map_U Polynomial.Chebyshev.map_U
theorem T_derivative_eq_U : ∀ n : ℕ, derivative (T R (n + 1)) = (n + 1) * U R n
| 0 => by simp only [T_one, U_zero, derivative_X, Nat.cast_zero, zero_add, mul_one]
| 1 => by
simp [T_two, U_one, derivative_sub, derivative_one, derivative_mul, derivative_X_pow, add_mul]
| n + 2 =>
calc
derivative (T R (n + 2 + 1)) =
2 * T R (n + 2) + 2 * X * derivative (T R (n + 1 + 1)) - derivative (T R (n + 1)) := by
rw [T_add_two _ (n + 1), derivative_sub, derivative_mul, derivative_mul, derivative_X,
derivative_ofNat]
ring_nf
_ = 2 * (U R (n + 1 + 1) - X * U R (n + 1)) + 2 * X * (((n + 1 + 1) : R[X]) * U R (n + 1))
- ((n + 1) : R[X]) * U R n := by
rw_mod_cast [T_derivative_eq_U (n + 1), T_derivative_eq_U n, T_eq_U_sub_X_mul_U _ (n + 1)]
_ = (n + 1 : R[X]) * (2 * X * U R (n + 1) - U R n) + 2 * U R (n + 2) := by ring
_ = (n + 1) * U R (n + 2) + 2 * U R (n + 2) := by rw [U_add_two]
_ = (n + 2 + 1) * U R (n + 2) := by ring
_ = (↑(n + 2) + 1) * U R (n + 2) := by norm_cast
#align polynomial.chebyshev.T_derivative_eq_U Polynomial.Chebyshev.T_derivative_eq_U
theorem one_sub_X_sq_mul_derivative_T_eq_poly_in_T (n : ℕ) :
(1 - X ^ 2) * derivative (T R (n + 1)) = (n + 1 : R[X]) * (T R n - X * T R (n + 1)) :=
calc
(1 - X ^ 2) * derivative (T R (n + 1)) = (1 - X ^ 2) * ((n + 1 : R[X]) * U R n) := by
rw [T_derivative_eq_U]
_ = (n + 1 : R[X]) * ((1 - X ^ 2) * U R n) := by ring
_ = (n + 1 : R[X]) * (X * T R (n + 1) - (2 * X * T R (n + 1) - T R n)) := by
rw [one_sub_X_sq_mul_U_eq_pol_in_T, T_add_two]
_ = (n + 1 : R[X]) * (T R n - X * T R (n + 1)) := by ring
#align polynomial.chebyshev.one_sub_X_sq_mul_derivative_T_eq_poly_in_T Polynomial.Chebyshev.one_sub_X_sq_mul_derivative_T_eq_poly_in_T
theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) := by
have h : derivative (T R (n + 2)) = U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) +
2 * X * U R n - (1 - X ^ 2) * derivative (U R n) := by
conv_lhs => rw [T_eq_X_mul_T_sub_pol_U]
simp only [derivative_sub, derivative_mul, derivative_X, derivative_one, derivative_X_pow,
one_mul, T_derivative_eq_U]
rw [T_eq_U_sub_X_mul_U, C_eq_nat_cast]
ring
calc
((n : R[X]) + 1) * T R (n + 1) =
((n : R[X]) + 1 + 1) * (X * U R n + T R (n + 1)) - X * ((n + 1 : R[X]) * U R n) -
(X * U R n + T R (n + 1)) :=
by ring
_ = derivative (T R (n + 2)) - X * derivative (T R (n + 1)) - U R (n + 1) := by
rw [← U_eq_X_mul_U_add_T, ← T_derivative_eq_U, ← Nat.cast_one, ← Nat.cast_add, Nat.cast_one, ←
T_derivative_eq_U (n + 1)]
_ = U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) + 2 * X * U R n -
(1 - X ^ 2) * derivative (U R n) -
X * derivative (T R (n + 1)) -
U R (n + 1) :=
by rw [h]
_ = X * U R n - (1 - X ^ 2) * derivative (U R n) := by ring
#align polynomial.chebyshev.add_one_mul_T_eq_poly_in_U Polynomial.Chebyshev.add_one_mul_T_eq_poly_in_U
variable (R)
/-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k by
have h_nat₁ : 2 * (m + 2) + k = 2 * m + k + 4 := by ring
have h_nat₂ : m + 2 + k = m + k + 2 := by ring
simpa [h_nat₁, h_nat₂] using this
-- clean up the `T` nat indices in the inductive hypothesis applied to `m + 1` and `k + 1`
have H₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1) := by
have h_nat₁ : m + 1 + (k + 1) = m + k + 2 := by ring
have h_nat₂ : 2 * (m + 1) + (k + 1) = 2 * m + k + 3 := by ring
simpa [h_nat₁, h_nat₂] using mul_T (m + 1) (k + 1)
-- clean up the `T` nat indices in the inductive hypothesis applied to `m` and `k + 2`
have H₂ : 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2) := by
have h_nat₁ : 2 * m + (k + 2) = 2 * m + k + 2 := by simp [add_assoc]
have h_nat₂ : m + (k + 2) = m + k + 2 := by simp [add_assoc]
simpa [h_nat₁, h_nat₂] using mul_T m (k + 2)
-- state the `T` recurrence relation for a few useful indices
have h₁ := T_add_two R m
have h₂ : T R (2 * m + k + 4) = 2 * X * T R (2 * m + k + 3) - T R (2 * m + k + 2) :=
T_add_two R (2 * m + k + 2)
have h₃ := T_add_two R k
-- the desired identity is an appropriate linear combination of H₁, H₂, h₁, h₂, h₃
linear_combination 2 * T R (m + k + 2) * h₁ + 2 * (X : R[X]) * H₁ - H₂ - h₂ - h₃
#align polynomial.chebyshev.mul_T Polynomial.Chebyshev.mul_T
/-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/
theorem T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n)
| 0 => by simp
| 1 => by simp
| m + 2 => by
intro n
have : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) := by
convert mul_T R n (m * n) using 1 <;> | ring_nf | /-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/
theorem T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n)
| 0 => by simp
| 1 => by simp
| m + 2 => by
intro n
have : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) := by
convert mul_T R n (m * n) using 1 <;> | Mathlib.RingTheory.Polynomial.Chebyshev.268_0.SRy1jgYRAFbFJky | /-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/
theorem T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n)
| 0 => by simp
| 1 => by simp
| m + 2 => by
intro n
have : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) | Mathlib_RingTheory_Polynomial_Chebyshev |
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
m n : ℕ
this : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n)
⊢ T R ((m + 2) * n) = comp (T R (m + 2)) (T R n) | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
/-!
# Chebyshev polynomials
The Chebyshev polynomials are two families of polynomials indexed by `ℕ`,
with integral coefficients.
## Main definitions
* `Polynomial.Chebyshev.T`: the Chebyshev polynomials of the first kind.
* `Polynomial.Chebyshev.U`: the Chebyshev polynomials of the second kind.
## Main statements
* The formal derivative of the Chebyshev polynomials of the first kind is a scalar multiple of the
Chebyshev polynomials of the second kind.
* `Polynomial.Chebyshev.mul_T`, the product of the `m`-th and `(m + k)`-th Chebyshev polynomials of
the first kind is the sum of the `(2 * m + k)`-th and `k`-th Chebyshev polynomials of the first
kind.
* `Polynomial.Chebyshev.T_mul`, the `(m * n)`-th Chebyshev polynomial of the first kind is the
composition of the `m`-th and `n`-th Chebyshev polynomials of the first kind.
## Implementation details
Since Chebyshev polynomials have interesting behaviour over the complex numbers and modulo `p`,
we define them to have coefficients in an arbitrary commutative ring, even though
technically `ℤ` would suffice.
The benefit of allowing arbitrary coefficient rings, is that the statements afterwards are clean,
and do not have `map (Int.castRingHom R)` interfering all the time.
## References
[Lionel Ponton, _Roots of the Chebyshev polynomials: A purely algebraic approach_]
[ponton2020chebyshev]
## TODO
* Redefine and/or relate the definition of Chebyshev polynomials to `LinearRecurrence`.
* Add explicit formula involving square roots for Chebyshev polynomials
* Compute zeroes and extrema of Chebyshev polynomials.
* Prove that the roots of the Chebyshev polynomials (except 0) are irrational.
* Prove minimax properties of Chebyshev polynomials.
-/
noncomputable section
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
open Polynomial
variable (R S : Type*) [CommRing R] [CommRing S]
/-- `T n` is the `n`-th Chebyshev polynomial of the first kind -/
noncomputable def T : ℕ → R[X]
| 0 => 1
| 1 => X
| n + 2 => 2 * X * T (n + 1) - T n
#align polynomial.chebyshev.T Polynomial.Chebyshev.T
@[simp]
theorem T_zero : T R 0 = 1 := rfl
#align polynomial.chebyshev.T_zero Polynomial.Chebyshev.T_zero
@[simp]
theorem T_one : T R 1 = X := rfl
#align polynomial.chebyshev.T_one Polynomial.Chebyshev.T_one
@[simp]
theorem T_add_two (n : ℕ) : T R (n + 2) = 2 * X * T R (n + 1) - T R n := by rw [T]
#align polynomial.chebyshev.T_add_two Polynomial.Chebyshev.T_add_two
theorem T_two : T R 2 = 2 * X ^ 2 - 1 := by simp only [T, sub_left_inj, sq, mul_assoc]
#align polynomial.chebyshev.T_two Polynomial.Chebyshev.T_two
theorem T_of_two_le (n : ℕ) (h : 2 ≤ n) : T R n = 2 * X * T R (n - 1) - T R (n - 2) := by
obtain ⟨n, rfl⟩ := Nat.exists_eq_add_of_le h
rw [add_comm]
exact T_add_two R n
#align polynomial.chebyshev.T_of_two_le Polynomial.Chebyshev.T_of_two_le
/-- `U n` is the `n`-th Chebyshev polynomial of the second kind -/
noncomputable def U : ℕ → R[X]
| 0 => 1
| 1 => 2 * X
| n + 2 => 2 * X * U (n + 1) - U n
#align polynomial.chebyshev.U Polynomial.Chebyshev.U
@[simp]
theorem U_zero : U R 0 = 1 := rfl
#align polynomial.chebyshev.U_zero Polynomial.Chebyshev.U_zero
@[simp]
theorem U_one : U R 1 = 2 * X := rfl
#align polynomial.chebyshev.U_one Polynomial.Chebyshev.U_one
@[simp]
theorem U_add_two (n : ℕ) : U R (n + 2) = 2 * X * U R (n + 1) - U R n := by rw [U]
#align polynomial.chebyshev.U_add_two Polynomial.Chebyshev.U_add_two
theorem U_two : U R 2 = 4 * X ^ 2 - 1 := by
simp only [U]
ring
#align polynomial.chebyshev.U_two Polynomial.Chebyshev.U_two
theorem U_of_two_le (n : ℕ) (h : 2 ≤ n) : U R n = 2 * X * U R (n - 1) - U R (n - 2) := by
obtain ⟨n, rfl⟩ := Nat.exists_eq_add_of_le h
rw [add_comm]
exact U_add_two R n
#align polynomial.chebyshev.U_of_two_le Polynomial.Chebyshev.U_of_two_le
theorem U_eq_X_mul_U_add_T : ∀ n : ℕ, U R (n + 1) = X * U R n + T R (n + 1)
| 0 => by simp only [T, U, two_mul, mul_one]
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
U R (n + 2 + 1) = 2 * X * (X * U R (n + 1) + T R (n + 2)) - (X * U R n + T R (n + 1)) := by
rw [U_add_two, U_eq_X_mul_U_add_T n, U_eq_X_mul_U_add_T (n + 1), U_eq_X_mul_U_add_T n]
_ = X * (2 * X * U R (n + 1) - U R n) + (2 * X * T R (n + 2) - T R (n + 1)) := by ring
_ = X * U R (n + 2) + T R (n + 2 + 1) := by simp only [U_add_two, T_add_two]
#align polynomial.chebyshev.U_eq_X_mul_U_add_T Polynomial.Chebyshev.U_eq_X_mul_U_add_T
theorem T_eq_U_sub_X_mul_U (n : ℕ) : T R (n + 1) = U R (n + 1) - X * U R n := by
rw [U_eq_X_mul_U_add_T, add_comm (X * U R n), add_sub_cancel]
#align polynomial.chebyshev.T_eq_U_sub_X_mul_U Polynomial.Chebyshev.T_eq_U_sub_X_mul_U
theorem T_eq_X_mul_T_sub_pol_U : ∀ n : ℕ, T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n
| 0 => by simp only [T, U]; ring
| 1 => by simp only [T, U]; ring
| n + 2 =>
calc
T R (n + 2 + 2) = 2 * X * T R (n + 2 + 1) - T R (n + 2) := T_add_two _ _
_ = 2 * X * (X * T R (n + 2) - (1 - X ^ 2) * U R (n + 1)) -
(X * T R (n + 1) - (1 - X ^ 2) * U R n) :=
by simp only [T_eq_X_mul_T_sub_pol_U]
_ = X * (2 * X * T R (n + 2) - T R (n + 1)) - (1 - X ^ 2) * (2 * X * U R (n + 1) - U R n) :=
by ring
_ = X * T R (n + 2 + 1) - (1 - X ^ 2) * U R (n + 2) := by rw [T_add_two _ (n + 1), U_add_two]
#align polynomial.chebyshev.T_eq_X_mul_T_sub_pol_U Polynomial.Chebyshev.T_eq_X_mul_T_sub_pol_U
theorem one_sub_X_sq_mul_U_eq_pol_in_T (n : ℕ) :
(1 - X ^ 2) * U R n = X * T R (n + 1) - T R (n + 2) := by
rw [T_eq_X_mul_T_sub_pol_U, ← sub_add, sub_self, zero_add]
#align polynomial.chebyshev.one_sub_X_sq_mul_U_eq_pol_in_T Polynomial.Chebyshev.one_sub_X_sq_mul_U_eq_pol_in_T
variable {R S}
@[simp]
theorem map_T (f : R →+* S) : ∀ n : ℕ, map f (T R n) = T S n
| 0 => by simp only [T_zero, Polynomial.map_one]
| 1 => by simp only [T_one, map_X]
| n + 2 => by
simp only [T_add_two, Polynomial.map_mul, Polynomial.map_sub, map_X, Polynomial.map_add,
Polynomial.map_one, Polynomial.map_ofNat, map_T f (n + 1), map_T f n]
#align polynomial.chebyshev.map_T Polynomial.Chebyshev.map_T
@[simp]
theorem map_U (f : R →+* S) : ∀ n : ℕ, map f (U R n) = U S n
| 0 => by simp only [U_zero, Polynomial.map_one]
| 1 => by
simp [U_one, map_X, Polynomial.map_mul, Polynomial.map_add, Polynomial.map_one]
| n + 2 => by
simp only [U_add_two, Polynomial.map_mul, Polynomial.map_sub, map_X, Polynomial.map_add,
Polynomial.map_one, map_U f (n + 1), map_U f n]
norm_num
#align polynomial.chebyshev.map_U Polynomial.Chebyshev.map_U
theorem T_derivative_eq_U : ∀ n : ℕ, derivative (T R (n + 1)) = (n + 1) * U R n
| 0 => by simp only [T_one, U_zero, derivative_X, Nat.cast_zero, zero_add, mul_one]
| 1 => by
simp [T_two, U_one, derivative_sub, derivative_one, derivative_mul, derivative_X_pow, add_mul]
| n + 2 =>
calc
derivative (T R (n + 2 + 1)) =
2 * T R (n + 2) + 2 * X * derivative (T R (n + 1 + 1)) - derivative (T R (n + 1)) := by
rw [T_add_two _ (n + 1), derivative_sub, derivative_mul, derivative_mul, derivative_X,
derivative_ofNat]
ring_nf
_ = 2 * (U R (n + 1 + 1) - X * U R (n + 1)) + 2 * X * (((n + 1 + 1) : R[X]) * U R (n + 1))
- ((n + 1) : R[X]) * U R n := by
rw_mod_cast [T_derivative_eq_U (n + 1), T_derivative_eq_U n, T_eq_U_sub_X_mul_U _ (n + 1)]
_ = (n + 1 : R[X]) * (2 * X * U R (n + 1) - U R n) + 2 * U R (n + 2) := by ring
_ = (n + 1) * U R (n + 2) + 2 * U R (n + 2) := by rw [U_add_two]
_ = (n + 2 + 1) * U R (n + 2) := by ring
_ = (↑(n + 2) + 1) * U R (n + 2) := by norm_cast
#align polynomial.chebyshev.T_derivative_eq_U Polynomial.Chebyshev.T_derivative_eq_U
theorem one_sub_X_sq_mul_derivative_T_eq_poly_in_T (n : ℕ) :
(1 - X ^ 2) * derivative (T R (n + 1)) = (n + 1 : R[X]) * (T R n - X * T R (n + 1)) :=
calc
(1 - X ^ 2) * derivative (T R (n + 1)) = (1 - X ^ 2) * ((n + 1 : R[X]) * U R n) := by
rw [T_derivative_eq_U]
_ = (n + 1 : R[X]) * ((1 - X ^ 2) * U R n) := by ring
_ = (n + 1 : R[X]) * (X * T R (n + 1) - (2 * X * T R (n + 1) - T R n)) := by
rw [one_sub_X_sq_mul_U_eq_pol_in_T, T_add_two]
_ = (n + 1 : R[X]) * (T R n - X * T R (n + 1)) := by ring
#align polynomial.chebyshev.one_sub_X_sq_mul_derivative_T_eq_poly_in_T Polynomial.Chebyshev.one_sub_X_sq_mul_derivative_T_eq_poly_in_T
theorem add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) := by
have h : derivative (T R (n + 2)) = U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) +
2 * X * U R n - (1 - X ^ 2) * derivative (U R n) := by
conv_lhs => rw [T_eq_X_mul_T_sub_pol_U]
simp only [derivative_sub, derivative_mul, derivative_X, derivative_one, derivative_X_pow,
one_mul, T_derivative_eq_U]
rw [T_eq_U_sub_X_mul_U, C_eq_nat_cast]
ring
calc
((n : R[X]) + 1) * T R (n + 1) =
((n : R[X]) + 1 + 1) * (X * U R n + T R (n + 1)) - X * ((n + 1 : R[X]) * U R n) -
(X * U R n + T R (n + 1)) :=
by ring
_ = derivative (T R (n + 2)) - X * derivative (T R (n + 1)) - U R (n + 1) := by
rw [← U_eq_X_mul_U_add_T, ← T_derivative_eq_U, ← Nat.cast_one, ← Nat.cast_add, Nat.cast_one, ←
T_derivative_eq_U (n + 1)]
_ = U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) + 2 * X * U R n -
(1 - X ^ 2) * derivative (U R n) -
X * derivative (T R (n + 1)) -
U R (n + 1) :=
by rw [h]
_ = X * U R n - (1 - X ^ 2) * derivative (U R n) := by ring
#align polynomial.chebyshev.add_one_mul_T_eq_poly_in_U Polynomial.Chebyshev.add_one_mul_T_eq_poly_in_U
variable (R)
/-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/
theorem mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k
| 0 => by simp [two_mul, add_mul]
| 1 => by simp [add_comm]
| m + 2 => by
intro k
-- clean up the `T` nat indices in the goal
suffices 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k by
have h_nat₁ : 2 * (m + 2) + k = 2 * m + k + 4 := by ring
have h_nat₂ : m + 2 + k = m + k + 2 := by ring
simpa [h_nat₁, h_nat₂] using this
-- clean up the `T` nat indices in the inductive hypothesis applied to `m + 1` and `k + 1`
have H₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1) := by
have h_nat₁ : m + 1 + (k + 1) = m + k + 2 := by ring
have h_nat₂ : 2 * (m + 1) + (k + 1) = 2 * m + k + 3 := by ring
simpa [h_nat₁, h_nat₂] using mul_T (m + 1) (k + 1)
-- clean up the `T` nat indices in the inductive hypothesis applied to `m` and `k + 2`
have H₂ : 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2) := by
have h_nat₁ : 2 * m + (k + 2) = 2 * m + k + 2 := by simp [add_assoc]
have h_nat₂ : m + (k + 2) = m + k + 2 := by simp [add_assoc]
simpa [h_nat₁, h_nat₂] using mul_T m (k + 2)
-- state the `T` recurrence relation for a few useful indices
have h₁ := T_add_two R m
have h₂ : T R (2 * m + k + 4) = 2 * X * T R (2 * m + k + 3) - T R (2 * m + k + 2) :=
T_add_two R (2 * m + k + 2)
have h₃ := T_add_two R k
-- the desired identity is an appropriate linear combination of H₁, H₂, h₁, h₂, h₃
linear_combination 2 * T R (m + k + 2) * h₁ + 2 * (X : R[X]) * H₁ - H₂ - h₂ - h₃
#align polynomial.chebyshev.mul_T Polynomial.Chebyshev.mul_T
/-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/
theorem T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n)
| 0 => by simp
| 1 => by simp
| m + 2 => by
intro n
have : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) := by
convert mul_T R n (m * n) using 1 <;> ring_nf
| simp [this, T_mul m, ← T_mul (m + 1)] | /-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/
theorem T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n)
| 0 => by simp
| 1 => by simp
| m + 2 => by
intro n
have : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) := by
convert mul_T R n (m * n) using 1 <;> ring_nf
| Mathlib.RingTheory.Polynomial.Chebyshev.268_0.SRy1jgYRAFbFJky | /-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/
theorem T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n)
| 0 => by simp
| 1 => by simp
| m + 2 => by
intro n
have : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) | Mathlib_RingTheory_Polynomial_Chebyshev |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
⊢ Linear k (FdRep k G) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by | infer_instance | instance : Linear k (FdRep k G) := by | Mathlib.RepresentationTheory.FdRep.62_0.ADbOgJGW1JDvdmK | instance : Linear k (FdRep k G) | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
V : FdRep k G
⊢ AddCommGroup (CoeSort.coe V) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
| change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj | instance (V : FdRep k G) : AddCommGroup V := by
| Mathlib.RepresentationTheory.FdRep.67_0.ADbOgJGW1JDvdmK | instance (V : FdRep k G) : AddCommGroup V | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
V : FdRep k G
⊢ AddCommGroup ↑((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; | infer_instance | instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; | Mathlib.RepresentationTheory.FdRep.67_0.ADbOgJGW1JDvdmK | instance (V : FdRep k G) : AddCommGroup V | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
V : FdRep k G
⊢ Module k (CoeSort.coe V) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : Module k V := by
| change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj | instance (V : FdRep k G) : Module k V := by
| Mathlib.RepresentationTheory.FdRep.70_0.ADbOgJGW1JDvdmK | instance (V : FdRep k G) : Module k V | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
V : FdRep k G
⊢ Module k ↑((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : Module k V := by
change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; | infer_instance | instance (V : FdRep k G) : Module k V := by
change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; | Mathlib.RepresentationTheory.FdRep.70_0.ADbOgJGW1JDvdmK | instance (V : FdRep k G) : Module k V | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
V : FdRep k G
⊢ FiniteDimensional k (CoeSort.coe V) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : Module k V := by
change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : FiniteDimensional k V := by
| change FiniteDimensional k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V) | instance (V : FdRep k G) : FiniteDimensional k V := by
| Mathlib.RepresentationTheory.FdRep.73_0.ADbOgJGW1JDvdmK | instance (V : FdRep k G) : FiniteDimensional k V | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
V : FdRep k G
⊢ FiniteDimensional k ↑((forget₂ (FdRep k G) (FGModuleCat k)).obj V) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : Module k V := by
change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : FiniteDimensional k V := by
change FiniteDimensional k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V); | infer_instance | instance (V : FdRep k G) : FiniteDimensional k V := by
change FiniteDimensional k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V); | Mathlib.RepresentationTheory.FdRep.73_0.ADbOgJGW1JDvdmK | instance (V : FdRep k G) : FiniteDimensional k V | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
V W : FdRep k G
i : V ≅ W
g : G
⊢ (ρ W) g = (LinearEquiv.conj (isoToLinearEquiv i)) ((ρ V) g) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : Module k V := by
change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : FiniteDimensional k V := by
change FiniteDimensional k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V); infer_instance
/-- All hom spaces are finite dimensional. -/
instance (V W : FdRep k G) : FiniteDimensional k (V ⟶ W) :=
FiniteDimensional.of_injective ((forget₂ (FdRep k G) (FGModuleCat k)).mapLinearMap k)
(Functor.map_injective (forget₂ (FdRep k G) (FGModuleCat k)))
/-- The monoid homomorphism corresponding to the action of `G` onto `V : FdRep k G`. -/
def ρ (V : FdRep k G) : G →* V →ₗ[k] V :=
Action.ρ V
#align fdRep.ρ FdRep.ρ
/-- The underlying `LinearEquiv` of an isomorphism of representations. -/
def isoToLinearEquiv {V W : FdRep k G} (i : V ≅ W) : V ≃ₗ[k] W :=
FGModuleCat.isoToLinearEquiv ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)
#align fdRep.iso_to_linear_equiv FdRep.isoToLinearEquiv
theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
| erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply] | theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
| Mathlib.RepresentationTheory.FdRep.91_0.ADbOgJGW1JDvdmK | theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
V W : FdRep k G
i : V ≅ W
g : G
⊢ (ρ W) g =
((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).inv ≫
(ρ V) g ≫ ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).hom | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : Module k V := by
change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : FiniteDimensional k V := by
change FiniteDimensional k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V); infer_instance
/-- All hom spaces are finite dimensional. -/
instance (V W : FdRep k G) : FiniteDimensional k (V ⟶ W) :=
FiniteDimensional.of_injective ((forget₂ (FdRep k G) (FGModuleCat k)).mapLinearMap k)
(Functor.map_injective (forget₂ (FdRep k G) (FGModuleCat k)))
/-- The monoid homomorphism corresponding to the action of `G` onto `V : FdRep k G`. -/
def ρ (V : FdRep k G) : G →* V →ₗ[k] V :=
Action.ρ V
#align fdRep.ρ FdRep.ρ
/-- The underlying `LinearEquiv` of an isomorphism of representations. -/
def isoToLinearEquiv {V W : FdRep k G} (i : V ≅ W) : V ≃ₗ[k] W :=
FGModuleCat.isoToLinearEquiv ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)
#align fdRep.iso_to_linear_equiv FdRep.isoToLinearEquiv
theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
| rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)] | theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
| Mathlib.RepresentationTheory.FdRep.91_0.ADbOgJGW1JDvdmK | theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
V W : FdRep k G
i : V ≅ W
g : G
⊢ ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).hom ≫ (ρ W) g =
(ρ V) g ≫ ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).hom | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : Module k V := by
change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : FiniteDimensional k V := by
change FiniteDimensional k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V); infer_instance
/-- All hom spaces are finite dimensional. -/
instance (V W : FdRep k G) : FiniteDimensional k (V ⟶ W) :=
FiniteDimensional.of_injective ((forget₂ (FdRep k G) (FGModuleCat k)).mapLinearMap k)
(Functor.map_injective (forget₂ (FdRep k G) (FGModuleCat k)))
/-- The monoid homomorphism corresponding to the action of `G` onto `V : FdRep k G`. -/
def ρ (V : FdRep k G) : G →* V →ₗ[k] V :=
Action.ρ V
#align fdRep.ρ FdRep.ρ
/-- The underlying `LinearEquiv` of an isomorphism of representations. -/
def isoToLinearEquiv {V W : FdRep k G} (i : V ≅ W) : V ≃ₗ[k] W :=
FGModuleCat.isoToLinearEquiv ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)
#align fdRep.iso_to_linear_equiv FdRep.isoToLinearEquiv
theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]
| exact (i.hom.comm g).symm | theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]
| Mathlib.RepresentationTheory.FdRep.91_0.ADbOgJGW1JDvdmK | theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
V : FdRep k G
⊢ Rep.ρ ((forget₂ (FdRep k G) (Rep k G)).obj V) = ρ V | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : Module k V := by
change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : FiniteDimensional k V := by
change FiniteDimensional k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V); infer_instance
/-- All hom spaces are finite dimensional. -/
instance (V W : FdRep k G) : FiniteDimensional k (V ⟶ W) :=
FiniteDimensional.of_injective ((forget₂ (FdRep k G) (FGModuleCat k)).mapLinearMap k)
(Functor.map_injective (forget₂ (FdRep k G) (FGModuleCat k)))
/-- The monoid homomorphism corresponding to the action of `G` onto `V : FdRep k G`. -/
def ρ (V : FdRep k G) : G →* V →ₗ[k] V :=
Action.ρ V
#align fdRep.ρ FdRep.ρ
/-- The underlying `LinearEquiv` of an isomorphism of representations. -/
def isoToLinearEquiv {V W : FdRep k G} (i : V ≅ W) : V ≃ₗ[k] W :=
FGModuleCat.isoToLinearEquiv ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)
#align fdRep.iso_to_linear_equiv FdRep.isoToLinearEquiv
theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]
exact (i.hom.comm g).symm
#align fdRep.iso.conj_ρ FdRep.Iso.conj_ρ
/-- Lift an unbundled representation to `FdRep`. -/
@[simps ρ]
def of {V : Type u} [AddCommGroup V] [Module k V] [FiniteDimensional k V]
(ρ : Representation k G V) : FdRep k G :=
⟨FGModuleCat.of k V, ρ⟩
#align fdRep.of FdRep.of
instance : HasForget₂ (FdRep k G) (Rep k G) where
forget₂ := (forget₂ (FGModuleCat k) (ModuleCat k)).mapAction (MonCat.of G)
theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ := by
| ext g v | theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ := by
| Mathlib.RepresentationTheory.FdRep.109_0.ADbOgJGW1JDvdmK | theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ | Mathlib_RepresentationTheory_FdRep |
case h.h
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
V : FdRep k G
g : G
v : CoeSort.coe ((forget₂ (FdRep k G) (Rep k G)).obj V)
⊢ ((Rep.ρ ((forget₂ (FdRep k G) (Rep k G)).obj V)) g) v = ((ρ V) g) v | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : Module k V := by
change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : FiniteDimensional k V := by
change FiniteDimensional k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V); infer_instance
/-- All hom spaces are finite dimensional. -/
instance (V W : FdRep k G) : FiniteDimensional k (V ⟶ W) :=
FiniteDimensional.of_injective ((forget₂ (FdRep k G) (FGModuleCat k)).mapLinearMap k)
(Functor.map_injective (forget₂ (FdRep k G) (FGModuleCat k)))
/-- The monoid homomorphism corresponding to the action of `G` onto `V : FdRep k G`. -/
def ρ (V : FdRep k G) : G →* V →ₗ[k] V :=
Action.ρ V
#align fdRep.ρ FdRep.ρ
/-- The underlying `LinearEquiv` of an isomorphism of representations. -/
def isoToLinearEquiv {V W : FdRep k G} (i : V ≅ W) : V ≃ₗ[k] W :=
FGModuleCat.isoToLinearEquiv ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)
#align fdRep.iso_to_linear_equiv FdRep.isoToLinearEquiv
theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]
exact (i.hom.comm g).symm
#align fdRep.iso.conj_ρ FdRep.Iso.conj_ρ
/-- Lift an unbundled representation to `FdRep`. -/
@[simps ρ]
def of {V : Type u} [AddCommGroup V] [Module k V] [FiniteDimensional k V]
(ρ : Representation k G V) : FdRep k G :=
⟨FGModuleCat.of k V, ρ⟩
#align fdRep.of FdRep.of
instance : HasForget₂ (FdRep k G) (Rep k G) where
forget₂ := (forget₂ (FGModuleCat k) (ModuleCat k)).mapAction (MonCat.of G)
theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ := by
ext g v; | rfl | theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ := by
ext g v; | Mathlib.RepresentationTheory.FdRep.109_0.ADbOgJGW1JDvdmK | theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
⊢ MonoidalCategory (FdRep k G) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : Module k V := by
change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : FiniteDimensional k V := by
change FiniteDimensional k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V); infer_instance
/-- All hom spaces are finite dimensional. -/
instance (V W : FdRep k G) : FiniteDimensional k (V ⟶ W) :=
FiniteDimensional.of_injective ((forget₂ (FdRep k G) (FGModuleCat k)).mapLinearMap k)
(Functor.map_injective (forget₂ (FdRep k G) (FGModuleCat k)))
/-- The monoid homomorphism corresponding to the action of `G` onto `V : FdRep k G`. -/
def ρ (V : FdRep k G) : G →* V →ₗ[k] V :=
Action.ρ V
#align fdRep.ρ FdRep.ρ
/-- The underlying `LinearEquiv` of an isomorphism of representations. -/
def isoToLinearEquiv {V W : FdRep k G} (i : V ≅ W) : V ≃ₗ[k] W :=
FGModuleCat.isoToLinearEquiv ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)
#align fdRep.iso_to_linear_equiv FdRep.isoToLinearEquiv
theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]
exact (i.hom.comm g).symm
#align fdRep.iso.conj_ρ FdRep.Iso.conj_ρ
/-- Lift an unbundled representation to `FdRep`. -/
@[simps ρ]
def of {V : Type u} [AddCommGroup V] [Module k V] [FiniteDimensional k V]
(ρ : Representation k G V) : FdRep k G :=
⟨FGModuleCat.of k V, ρ⟩
#align fdRep.of FdRep.of
instance : HasForget₂ (FdRep k G) (Rep k G) where
forget₂ := (forget₂ (FGModuleCat k) (ModuleCat k)).mapAction (MonCat.of G)
theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ := by
ext g v; rfl
#align fdRep.forget₂_ρ FdRep.forget₂_ρ
-- Verify that the monoidal structure is available.
example : MonoidalCategory (FdRep k G) := by | infer_instance | example : MonoidalCategory (FdRep k G) := by | Mathlib.RepresentationTheory.FdRep.114_0.ADbOgJGW1JDvdmK | example : MonoidalCategory (FdRep k G) | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
⊢ MonoidalPreadditive (FdRep k G) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : Module k V := by
change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : FiniteDimensional k V := by
change FiniteDimensional k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V); infer_instance
/-- All hom spaces are finite dimensional. -/
instance (V W : FdRep k G) : FiniteDimensional k (V ⟶ W) :=
FiniteDimensional.of_injective ((forget₂ (FdRep k G) (FGModuleCat k)).mapLinearMap k)
(Functor.map_injective (forget₂ (FdRep k G) (FGModuleCat k)))
/-- The monoid homomorphism corresponding to the action of `G` onto `V : FdRep k G`. -/
def ρ (V : FdRep k G) : G →* V →ₗ[k] V :=
Action.ρ V
#align fdRep.ρ FdRep.ρ
/-- The underlying `LinearEquiv` of an isomorphism of representations. -/
def isoToLinearEquiv {V W : FdRep k G} (i : V ≅ W) : V ≃ₗ[k] W :=
FGModuleCat.isoToLinearEquiv ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)
#align fdRep.iso_to_linear_equiv FdRep.isoToLinearEquiv
theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]
exact (i.hom.comm g).symm
#align fdRep.iso.conj_ρ FdRep.Iso.conj_ρ
/-- Lift an unbundled representation to `FdRep`. -/
@[simps ρ]
def of {V : Type u} [AddCommGroup V] [Module k V] [FiniteDimensional k V]
(ρ : Representation k G V) : FdRep k G :=
⟨FGModuleCat.of k V, ρ⟩
#align fdRep.of FdRep.of
instance : HasForget₂ (FdRep k G) (Rep k G) where
forget₂ := (forget₂ (FGModuleCat k) (ModuleCat k)).mapAction (MonCat.of G)
theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ := by
ext g v; rfl
#align fdRep.forget₂_ρ FdRep.forget₂_ρ
-- Verify that the monoidal structure is available.
example : MonoidalCategory (FdRep k G) := by infer_instance
example : MonoidalPreadditive (FdRep k G) := by | infer_instance | example : MonoidalPreadditive (FdRep k G) := by | Mathlib.RepresentationTheory.FdRep.116_0.ADbOgJGW1JDvdmK | example : MonoidalPreadditive (FdRep k G) | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
⊢ MonoidalLinear k (FdRep k G) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : Module k V := by
change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : FiniteDimensional k V := by
change FiniteDimensional k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V); infer_instance
/-- All hom spaces are finite dimensional. -/
instance (V W : FdRep k G) : FiniteDimensional k (V ⟶ W) :=
FiniteDimensional.of_injective ((forget₂ (FdRep k G) (FGModuleCat k)).mapLinearMap k)
(Functor.map_injective (forget₂ (FdRep k G) (FGModuleCat k)))
/-- The monoid homomorphism corresponding to the action of `G` onto `V : FdRep k G`. -/
def ρ (V : FdRep k G) : G →* V →ₗ[k] V :=
Action.ρ V
#align fdRep.ρ FdRep.ρ
/-- The underlying `LinearEquiv` of an isomorphism of representations. -/
def isoToLinearEquiv {V W : FdRep k G} (i : V ≅ W) : V ≃ₗ[k] W :=
FGModuleCat.isoToLinearEquiv ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)
#align fdRep.iso_to_linear_equiv FdRep.isoToLinearEquiv
theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]
exact (i.hom.comm g).symm
#align fdRep.iso.conj_ρ FdRep.Iso.conj_ρ
/-- Lift an unbundled representation to `FdRep`. -/
@[simps ρ]
def of {V : Type u} [AddCommGroup V] [Module k V] [FiniteDimensional k V]
(ρ : Representation k G V) : FdRep k G :=
⟨FGModuleCat.of k V, ρ⟩
#align fdRep.of FdRep.of
instance : HasForget₂ (FdRep k G) (Rep k G) where
forget₂ := (forget₂ (FGModuleCat k) (ModuleCat k)).mapAction (MonCat.of G)
theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ := by
ext g v; rfl
#align fdRep.forget₂_ρ FdRep.forget₂_ρ
-- Verify that the monoidal structure is available.
example : MonoidalCategory (FdRep k G) := by infer_instance
example : MonoidalPreadditive (FdRep k G) := by infer_instance
example : MonoidalLinear k (FdRep k G) := by | infer_instance | example : MonoidalLinear k (FdRep k G) := by | Mathlib.RepresentationTheory.FdRep.118_0.ADbOgJGW1JDvdmK | example : MonoidalLinear k (FdRep k G) | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
⊢ HasKernels (FdRep k G) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : Module k V := by
change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : FiniteDimensional k V := by
change FiniteDimensional k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V); infer_instance
/-- All hom spaces are finite dimensional. -/
instance (V W : FdRep k G) : FiniteDimensional k (V ⟶ W) :=
FiniteDimensional.of_injective ((forget₂ (FdRep k G) (FGModuleCat k)).mapLinearMap k)
(Functor.map_injective (forget₂ (FdRep k G) (FGModuleCat k)))
/-- The monoid homomorphism corresponding to the action of `G` onto `V : FdRep k G`. -/
def ρ (V : FdRep k G) : G →* V →ₗ[k] V :=
Action.ρ V
#align fdRep.ρ FdRep.ρ
/-- The underlying `LinearEquiv` of an isomorphism of representations. -/
def isoToLinearEquiv {V W : FdRep k G} (i : V ≅ W) : V ≃ₗ[k] W :=
FGModuleCat.isoToLinearEquiv ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)
#align fdRep.iso_to_linear_equiv FdRep.isoToLinearEquiv
theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]
exact (i.hom.comm g).symm
#align fdRep.iso.conj_ρ FdRep.Iso.conj_ρ
/-- Lift an unbundled representation to `FdRep`. -/
@[simps ρ]
def of {V : Type u} [AddCommGroup V] [Module k V] [FiniteDimensional k V]
(ρ : Representation k G V) : FdRep k G :=
⟨FGModuleCat.of k V, ρ⟩
#align fdRep.of FdRep.of
instance : HasForget₂ (FdRep k G) (Rep k G) where
forget₂ := (forget₂ (FGModuleCat k) (ModuleCat k)).mapAction (MonCat.of G)
theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ := by
ext g v; rfl
#align fdRep.forget₂_ρ FdRep.forget₂_ρ
-- Verify that the monoidal structure is available.
example : MonoidalCategory (FdRep k G) := by infer_instance
example : MonoidalPreadditive (FdRep k G) := by infer_instance
example : MonoidalLinear k (FdRep k G) := by infer_instance
open FiniteDimensional
open scoped Classical
-- We need to provide this instance explicitely as otherwise `finrank_hom_simple_simple` gives a
-- deterministic timeout.
instance : HasKernels (FdRep k G) := by | infer_instance | instance : HasKernels (FdRep k G) := by | Mathlib.RepresentationTheory.FdRep.126_0.ADbOgJGW1JDvdmK | instance : HasKernels (FdRep k G) | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
X Y : FdRep k G
x✝ : (forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y
⊢ (fun f => Action.Hom.mk ((forget₂ (FGModuleCat k) (ModuleCat k)).map f.hom))
(AddHom.toFun
{
toAddHom :=
{ toFun := fun f => Action.Hom.mk f.hom,
map_add' :=
(_ :
∀ (x x_1 : (forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y),
(fun f => Action.Hom.mk f.hom) (x + x_1) = (fun f => Action.Hom.mk f.hom) (x + x_1)) },
map_smul' :=
(_ :
∀ (x : k) (x_1 : (forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y),
AddHom.toFun
{ toFun := fun f => Action.Hom.mk f.hom,
map_add' :=
(_ :
∀ (x x_2 : (forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y),
(fun f => Action.Hom.mk f.hom) (x + x_2) = (fun f => Action.Hom.mk f.hom) (x + x_2)) }
(x • x_1) =
AddHom.toFun
{ toFun := fun f => Action.Hom.mk f.hom,
map_add' :=
(_ :
∀ (x x_2 : (forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y),
(fun f => Action.Hom.mk f.hom) (x + x_2) = (fun f => Action.Hom.mk f.hom) (x + x_2)) }
(x • x_1)) }.toAddHom
x✝) =
x✝ | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : Module k V := by
change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : FiniteDimensional k V := by
change FiniteDimensional k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V); infer_instance
/-- All hom spaces are finite dimensional. -/
instance (V W : FdRep k G) : FiniteDimensional k (V ⟶ W) :=
FiniteDimensional.of_injective ((forget₂ (FdRep k G) (FGModuleCat k)).mapLinearMap k)
(Functor.map_injective (forget₂ (FdRep k G) (FGModuleCat k)))
/-- The monoid homomorphism corresponding to the action of `G` onto `V : FdRep k G`. -/
def ρ (V : FdRep k G) : G →* V →ₗ[k] V :=
Action.ρ V
#align fdRep.ρ FdRep.ρ
/-- The underlying `LinearEquiv` of an isomorphism of representations. -/
def isoToLinearEquiv {V W : FdRep k G} (i : V ≅ W) : V ≃ₗ[k] W :=
FGModuleCat.isoToLinearEquiv ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)
#align fdRep.iso_to_linear_equiv FdRep.isoToLinearEquiv
theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]
exact (i.hom.comm g).symm
#align fdRep.iso.conj_ρ FdRep.Iso.conj_ρ
/-- Lift an unbundled representation to `FdRep`. -/
@[simps ρ]
def of {V : Type u} [AddCommGroup V] [Module k V] [FiniteDimensional k V]
(ρ : Representation k G V) : FdRep k G :=
⟨FGModuleCat.of k V, ρ⟩
#align fdRep.of FdRep.of
instance : HasForget₂ (FdRep k G) (Rep k G) where
forget₂ := (forget₂ (FGModuleCat k) (ModuleCat k)).mapAction (MonCat.of G)
theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ := by
ext g v; rfl
#align fdRep.forget₂_ρ FdRep.forget₂_ρ
-- Verify that the monoidal structure is available.
example : MonoidalCategory (FdRep k G) := by infer_instance
example : MonoidalPreadditive (FdRep k G) := by infer_instance
example : MonoidalLinear k (FdRep k G) := by infer_instance
open FiniteDimensional
open scoped Classical
-- We need to provide this instance explicitely as otherwise `finrank_hom_simple_simple` gives a
-- deterministic timeout.
instance : HasKernels (FdRep k G) := by infer_instance
-- Verify that Schur's lemma applies out of the box.
theorem finrank_hom_simple_simple [IsAlgClosed k] (V W : FdRep k G) [Simple V] [Simple W] :
finrank k (V ⟶ W) = if Nonempty (V ≅ W) then 1 else 0 :=
CategoryTheory.finrank_hom_simple_simple k V W
#align fdRep.finrank_hom_simple_simple FdRep.finrank_hom_simple_simple
/-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f := ⟨f.hom, f.comm⟩
map_add' _ _ := rfl
map_smul' _ _ := rfl
invFun f := ⟨(forget₂ (FGModuleCat k) (ModuleCat k)).map f.hom, f.comm⟩
left_inv _ := by | ext | /-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f := ⟨f.hom, f.comm⟩
map_add' _ _ := rfl
map_smul' _ _ := rfl
invFun f := ⟨(forget₂ (FGModuleCat k) (ModuleCat k)).map f.hom, f.comm⟩
left_inv _ := by | Mathlib.RepresentationTheory.FdRep.134_0.ADbOgJGW1JDvdmK | /-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f | Mathlib_RepresentationTheory_FdRep |
case h.h
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
X Y : FdRep k G
x✝¹ : (forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y
x✝ : ↑((forget₂ (FdRep k G) (Rep k G)).obj X).V
⊢ ((fun f => Action.Hom.mk ((forget₂ (FGModuleCat k) (ModuleCat k)).map f.hom))
(AddHom.toFun
{
toAddHom :=
{ toFun := fun f => Action.Hom.mk f.hom,
map_add' :=
(_ :
∀ (x x_1 : (forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y),
(fun f => Action.Hom.mk f.hom) (x + x_1) = (fun f => Action.Hom.mk f.hom) (x + x_1)) },
map_smul' :=
(_ :
∀ (x : k) (x_1 : (forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y),
AddHom.toFun
{ toFun := fun f => Action.Hom.mk f.hom,
map_add' :=
(_ :
∀
(x x_2 :
(forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y),
(fun f => Action.Hom.mk f.hom) (x + x_2) = (fun f => Action.Hom.mk f.hom) (x + x_2)) }
(x • x_1) =
AddHom.toFun
{ toFun := fun f => Action.Hom.mk f.hom,
map_add' :=
(_ :
∀
(x x_2 :
(forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y),
(fun f => Action.Hom.mk f.hom) (x + x_2) = (fun f => Action.Hom.mk f.hom) (x + x_2)) }
(x • x_1)) }.toAddHom
x✝¹)).hom
x✝ =
x✝¹.hom x✝ | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : Module k V := by
change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : FiniteDimensional k V := by
change FiniteDimensional k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V); infer_instance
/-- All hom spaces are finite dimensional. -/
instance (V W : FdRep k G) : FiniteDimensional k (V ⟶ W) :=
FiniteDimensional.of_injective ((forget₂ (FdRep k G) (FGModuleCat k)).mapLinearMap k)
(Functor.map_injective (forget₂ (FdRep k G) (FGModuleCat k)))
/-- The monoid homomorphism corresponding to the action of `G` onto `V : FdRep k G`. -/
def ρ (V : FdRep k G) : G →* V →ₗ[k] V :=
Action.ρ V
#align fdRep.ρ FdRep.ρ
/-- The underlying `LinearEquiv` of an isomorphism of representations. -/
def isoToLinearEquiv {V W : FdRep k G} (i : V ≅ W) : V ≃ₗ[k] W :=
FGModuleCat.isoToLinearEquiv ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)
#align fdRep.iso_to_linear_equiv FdRep.isoToLinearEquiv
theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]
exact (i.hom.comm g).symm
#align fdRep.iso.conj_ρ FdRep.Iso.conj_ρ
/-- Lift an unbundled representation to `FdRep`. -/
@[simps ρ]
def of {V : Type u} [AddCommGroup V] [Module k V] [FiniteDimensional k V]
(ρ : Representation k G V) : FdRep k G :=
⟨FGModuleCat.of k V, ρ⟩
#align fdRep.of FdRep.of
instance : HasForget₂ (FdRep k G) (Rep k G) where
forget₂ := (forget₂ (FGModuleCat k) (ModuleCat k)).mapAction (MonCat.of G)
theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ := by
ext g v; rfl
#align fdRep.forget₂_ρ FdRep.forget₂_ρ
-- Verify that the monoidal structure is available.
example : MonoidalCategory (FdRep k G) := by infer_instance
example : MonoidalPreadditive (FdRep k G) := by infer_instance
example : MonoidalLinear k (FdRep k G) := by infer_instance
open FiniteDimensional
open scoped Classical
-- We need to provide this instance explicitely as otherwise `finrank_hom_simple_simple` gives a
-- deterministic timeout.
instance : HasKernels (FdRep k G) := by infer_instance
-- Verify that Schur's lemma applies out of the box.
theorem finrank_hom_simple_simple [IsAlgClosed k] (V W : FdRep k G) [Simple V] [Simple W] :
finrank k (V ⟶ W) = if Nonempty (V ≅ W) then 1 else 0 :=
CategoryTheory.finrank_hom_simple_simple k V W
#align fdRep.finrank_hom_simple_simple FdRep.finrank_hom_simple_simple
/-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f := ⟨f.hom, f.comm⟩
map_add' _ _ := rfl
map_smul' _ _ := rfl
invFun f := ⟨(forget₂ (FGModuleCat k) (ModuleCat k)).map f.hom, f.comm⟩
left_inv _ := by ext; | rfl | /-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f := ⟨f.hom, f.comm⟩
map_add' _ _ := rfl
map_smul' _ _ := rfl
invFun f := ⟨(forget₂ (FGModuleCat k) (ModuleCat k)).map f.hom, f.comm⟩
left_inv _ := by ext; | Mathlib.RepresentationTheory.FdRep.134_0.ADbOgJGW1JDvdmK | /-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
X Y : FdRep k G
x✝ : X ⟶ Y
⊢ AddHom.toFun
{
toAddHom :=
{ toFun := fun f => Action.Hom.mk f.hom,
map_add' :=
(_ :
∀ (x x_1 : (forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y),
(fun f => Action.Hom.mk f.hom) (x + x_1) = (fun f => Action.Hom.mk f.hom) (x + x_1)) },
map_smul' :=
(_ :
∀ (x : k) (x_1 : (forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y),
AddHom.toFun
{ toFun := fun f => Action.Hom.mk f.hom,
map_add' :=
(_ :
∀ (x x_2 : (forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y),
(fun f => Action.Hom.mk f.hom) (x + x_2) = (fun f => Action.Hom.mk f.hom) (x + x_2)) }
(x • x_1) =
AddHom.toFun
{ toFun := fun f => Action.Hom.mk f.hom,
map_add' :=
(_ :
∀ (x x_2 : (forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y),
(fun f => Action.Hom.mk f.hom) (x + x_2) = (fun f => Action.Hom.mk f.hom) (x + x_2)) }
(x • x_1)) }.toAddHom
((fun f => Action.Hom.mk ((forget₂ (FGModuleCat k) (ModuleCat k)).map f.hom)) x✝) =
x✝ | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : Module k V := by
change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : FiniteDimensional k V := by
change FiniteDimensional k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V); infer_instance
/-- All hom spaces are finite dimensional. -/
instance (V W : FdRep k G) : FiniteDimensional k (V ⟶ W) :=
FiniteDimensional.of_injective ((forget₂ (FdRep k G) (FGModuleCat k)).mapLinearMap k)
(Functor.map_injective (forget₂ (FdRep k G) (FGModuleCat k)))
/-- The monoid homomorphism corresponding to the action of `G` onto `V : FdRep k G`. -/
def ρ (V : FdRep k G) : G →* V →ₗ[k] V :=
Action.ρ V
#align fdRep.ρ FdRep.ρ
/-- The underlying `LinearEquiv` of an isomorphism of representations. -/
def isoToLinearEquiv {V W : FdRep k G} (i : V ≅ W) : V ≃ₗ[k] W :=
FGModuleCat.isoToLinearEquiv ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)
#align fdRep.iso_to_linear_equiv FdRep.isoToLinearEquiv
theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]
exact (i.hom.comm g).symm
#align fdRep.iso.conj_ρ FdRep.Iso.conj_ρ
/-- Lift an unbundled representation to `FdRep`. -/
@[simps ρ]
def of {V : Type u} [AddCommGroup V] [Module k V] [FiniteDimensional k V]
(ρ : Representation k G V) : FdRep k G :=
⟨FGModuleCat.of k V, ρ⟩
#align fdRep.of FdRep.of
instance : HasForget₂ (FdRep k G) (Rep k G) where
forget₂ := (forget₂ (FGModuleCat k) (ModuleCat k)).mapAction (MonCat.of G)
theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ := by
ext g v; rfl
#align fdRep.forget₂_ρ FdRep.forget₂_ρ
-- Verify that the monoidal structure is available.
example : MonoidalCategory (FdRep k G) := by infer_instance
example : MonoidalPreadditive (FdRep k G) := by infer_instance
example : MonoidalLinear k (FdRep k G) := by infer_instance
open FiniteDimensional
open scoped Classical
-- We need to provide this instance explicitely as otherwise `finrank_hom_simple_simple` gives a
-- deterministic timeout.
instance : HasKernels (FdRep k G) := by infer_instance
-- Verify that Schur's lemma applies out of the box.
theorem finrank_hom_simple_simple [IsAlgClosed k] (V W : FdRep k G) [Simple V] [Simple W] :
finrank k (V ⟶ W) = if Nonempty (V ≅ W) then 1 else 0 :=
CategoryTheory.finrank_hom_simple_simple k V W
#align fdRep.finrank_hom_simple_simple FdRep.finrank_hom_simple_simple
/-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f := ⟨f.hom, f.comm⟩
map_add' _ _ := rfl
map_smul' _ _ := rfl
invFun f := ⟨(forget₂ (FGModuleCat k) (ModuleCat k)).map f.hom, f.comm⟩
left_inv _ := by ext; rfl
right_inv _ := by | ext | /-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f := ⟨f.hom, f.comm⟩
map_add' _ _ := rfl
map_smul' _ _ := rfl
invFun f := ⟨(forget₂ (FGModuleCat k) (ModuleCat k)).map f.hom, f.comm⟩
left_inv _ := by ext; rfl
right_inv _ := by | Mathlib.RepresentationTheory.FdRep.134_0.ADbOgJGW1JDvdmK | /-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f | Mathlib_RepresentationTheory_FdRep |
case h.w
k G : Type u
inst✝¹ : Field k
inst✝ : Monoid G
X Y : FdRep k G
x✝¹ : X ⟶ Y
x✝ : (forget (FGModuleCat k)).obj X.V
⊢ (AddHom.toFun
{
toAddHom :=
{ toFun := fun f => Action.Hom.mk f.hom,
map_add' :=
(_ :
∀ (x x_1 : (forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y),
(fun f => Action.Hom.mk f.hom) (x + x_1) = (fun f => Action.Hom.mk f.hom) (x + x_1)) },
map_smul' :=
(_ :
∀ (x : k) (x_1 : (forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y),
AddHom.toFun
{ toFun := fun f => Action.Hom.mk f.hom,
map_add' :=
(_ :
∀ (x x_2 : (forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y),
(fun f => Action.Hom.mk f.hom) (x + x_2) = (fun f => Action.Hom.mk f.hom) (x + x_2)) }
(x • x_1) =
AddHom.toFun
{ toFun := fun f => Action.Hom.mk f.hom,
map_add' :=
(_ :
∀ (x x_2 : (forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y),
(fun f => Action.Hom.mk f.hom) (x + x_2) = (fun f => Action.Hom.mk f.hom) (x + x_2)) }
(x • x_1)) }.toAddHom
((fun f => Action.Hom.mk ((forget₂ (FGModuleCat k) (ModuleCat k)).map f.hom)) x✝¹)).hom
x✝ =
x✝¹.hom x✝ | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : Module k V := by
change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : FiniteDimensional k V := by
change FiniteDimensional k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V); infer_instance
/-- All hom spaces are finite dimensional. -/
instance (V W : FdRep k G) : FiniteDimensional k (V ⟶ W) :=
FiniteDimensional.of_injective ((forget₂ (FdRep k G) (FGModuleCat k)).mapLinearMap k)
(Functor.map_injective (forget₂ (FdRep k G) (FGModuleCat k)))
/-- The monoid homomorphism corresponding to the action of `G` onto `V : FdRep k G`. -/
def ρ (V : FdRep k G) : G →* V →ₗ[k] V :=
Action.ρ V
#align fdRep.ρ FdRep.ρ
/-- The underlying `LinearEquiv` of an isomorphism of representations. -/
def isoToLinearEquiv {V W : FdRep k G} (i : V ≅ W) : V ≃ₗ[k] W :=
FGModuleCat.isoToLinearEquiv ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)
#align fdRep.iso_to_linear_equiv FdRep.isoToLinearEquiv
theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]
exact (i.hom.comm g).symm
#align fdRep.iso.conj_ρ FdRep.Iso.conj_ρ
/-- Lift an unbundled representation to `FdRep`. -/
@[simps ρ]
def of {V : Type u} [AddCommGroup V] [Module k V] [FiniteDimensional k V]
(ρ : Representation k G V) : FdRep k G :=
⟨FGModuleCat.of k V, ρ⟩
#align fdRep.of FdRep.of
instance : HasForget₂ (FdRep k G) (Rep k G) where
forget₂ := (forget₂ (FGModuleCat k) (ModuleCat k)).mapAction (MonCat.of G)
theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ := by
ext g v; rfl
#align fdRep.forget₂_ρ FdRep.forget₂_ρ
-- Verify that the monoidal structure is available.
example : MonoidalCategory (FdRep k G) := by infer_instance
example : MonoidalPreadditive (FdRep k G) := by infer_instance
example : MonoidalLinear k (FdRep k G) := by infer_instance
open FiniteDimensional
open scoped Classical
-- We need to provide this instance explicitely as otherwise `finrank_hom_simple_simple` gives a
-- deterministic timeout.
instance : HasKernels (FdRep k G) := by infer_instance
-- Verify that Schur's lemma applies out of the box.
theorem finrank_hom_simple_simple [IsAlgClosed k] (V W : FdRep k G) [Simple V] [Simple W] :
finrank k (V ⟶ W) = if Nonempty (V ≅ W) then 1 else 0 :=
CategoryTheory.finrank_hom_simple_simple k V W
#align fdRep.finrank_hom_simple_simple FdRep.finrank_hom_simple_simple
/-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f := ⟨f.hom, f.comm⟩
map_add' _ _ := rfl
map_smul' _ _ := rfl
invFun f := ⟨(forget₂ (FGModuleCat k) (ModuleCat k)).map f.hom, f.comm⟩
left_inv _ := by ext; rfl
right_inv _ := by ext; | rfl | /-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f := ⟨f.hom, f.comm⟩
map_add' _ _ := rfl
map_smul' _ _ := rfl
invFun f := ⟨(forget₂ (FGModuleCat k) (ModuleCat k)).map f.hom, f.comm⟩
left_inv _ := by ext; rfl
right_inv _ := by ext; | Mathlib.RepresentationTheory.FdRep.134_0.ADbOgJGW1JDvdmK | /-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Group G
⊢ RightRigidCategory (FdRep k G) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : Module k V := by
change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : FiniteDimensional k V := by
change FiniteDimensional k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V); infer_instance
/-- All hom spaces are finite dimensional. -/
instance (V W : FdRep k G) : FiniteDimensional k (V ⟶ W) :=
FiniteDimensional.of_injective ((forget₂ (FdRep k G) (FGModuleCat k)).mapLinearMap k)
(Functor.map_injective (forget₂ (FdRep k G) (FGModuleCat k)))
/-- The monoid homomorphism corresponding to the action of `G` onto `V : FdRep k G`. -/
def ρ (V : FdRep k G) : G →* V →ₗ[k] V :=
Action.ρ V
#align fdRep.ρ FdRep.ρ
/-- The underlying `LinearEquiv` of an isomorphism of representations. -/
def isoToLinearEquiv {V W : FdRep k G} (i : V ≅ W) : V ≃ₗ[k] W :=
FGModuleCat.isoToLinearEquiv ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)
#align fdRep.iso_to_linear_equiv FdRep.isoToLinearEquiv
theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]
exact (i.hom.comm g).symm
#align fdRep.iso.conj_ρ FdRep.Iso.conj_ρ
/-- Lift an unbundled representation to `FdRep`. -/
@[simps ρ]
def of {V : Type u} [AddCommGroup V] [Module k V] [FiniteDimensional k V]
(ρ : Representation k G V) : FdRep k G :=
⟨FGModuleCat.of k V, ρ⟩
#align fdRep.of FdRep.of
instance : HasForget₂ (FdRep k G) (Rep k G) where
forget₂ := (forget₂ (FGModuleCat k) (ModuleCat k)).mapAction (MonCat.of G)
theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ := by
ext g v; rfl
#align fdRep.forget₂_ρ FdRep.forget₂_ρ
-- Verify that the monoidal structure is available.
example : MonoidalCategory (FdRep k G) := by infer_instance
example : MonoidalPreadditive (FdRep k G) := by infer_instance
example : MonoidalLinear k (FdRep k G) := by infer_instance
open FiniteDimensional
open scoped Classical
-- We need to provide this instance explicitely as otherwise `finrank_hom_simple_simple` gives a
-- deterministic timeout.
instance : HasKernels (FdRep k G) := by infer_instance
-- Verify that Schur's lemma applies out of the box.
theorem finrank_hom_simple_simple [IsAlgClosed k] (V W : FdRep k G) [Simple V] [Simple W] :
finrank k (V ⟶ W) = if Nonempty (V ≅ W) then 1 else 0 :=
CategoryTheory.finrank_hom_simple_simple k V W
#align fdRep.finrank_hom_simple_simple FdRep.finrank_hom_simple_simple
/-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f := ⟨f.hom, f.comm⟩
map_add' _ _ := rfl
map_smul' _ _ := rfl
invFun f := ⟨(forget₂ (FGModuleCat k) (ModuleCat k)).map f.hom, f.comm⟩
left_inv _ := by ext; rfl
right_inv _ := by ext; rfl
#align fdRep.forget₂_hom_linear_equiv FdRep.forget₂HomLinearEquiv
end FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Group G]
-- Verify that the right rigid structure is available when the monoid is a group.
noncomputable instance : RightRigidCategory (FdRep k G) := by
| change RightRigidCategory (Action (FGModuleCat k) (GroupCat.of G)) | noncomputable instance : RightRigidCategory (FdRep k G) := by
| Mathlib.RepresentationTheory.FdRep.153_0.ADbOgJGW1JDvdmK | noncomputable instance : RightRigidCategory (FdRep k G) | Mathlib_RepresentationTheory_FdRep |
k G : Type u
inst✝¹ : Field k
inst✝ : Group G
⊢ RightRigidCategory (Action (FGModuleCat k) ((forget₂ GroupCat MonCat).obj (GroupCat.of G))) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : Module k V := by
change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : FiniteDimensional k V := by
change FiniteDimensional k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V); infer_instance
/-- All hom spaces are finite dimensional. -/
instance (V W : FdRep k G) : FiniteDimensional k (V ⟶ W) :=
FiniteDimensional.of_injective ((forget₂ (FdRep k G) (FGModuleCat k)).mapLinearMap k)
(Functor.map_injective (forget₂ (FdRep k G) (FGModuleCat k)))
/-- The monoid homomorphism corresponding to the action of `G` onto `V : FdRep k G`. -/
def ρ (V : FdRep k G) : G →* V →ₗ[k] V :=
Action.ρ V
#align fdRep.ρ FdRep.ρ
/-- The underlying `LinearEquiv` of an isomorphism of representations. -/
def isoToLinearEquiv {V W : FdRep k G} (i : V ≅ W) : V ≃ₗ[k] W :=
FGModuleCat.isoToLinearEquiv ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)
#align fdRep.iso_to_linear_equiv FdRep.isoToLinearEquiv
theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]
exact (i.hom.comm g).symm
#align fdRep.iso.conj_ρ FdRep.Iso.conj_ρ
/-- Lift an unbundled representation to `FdRep`. -/
@[simps ρ]
def of {V : Type u} [AddCommGroup V] [Module k V] [FiniteDimensional k V]
(ρ : Representation k G V) : FdRep k G :=
⟨FGModuleCat.of k V, ρ⟩
#align fdRep.of FdRep.of
instance : HasForget₂ (FdRep k G) (Rep k G) where
forget₂ := (forget₂ (FGModuleCat k) (ModuleCat k)).mapAction (MonCat.of G)
theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ := by
ext g v; rfl
#align fdRep.forget₂_ρ FdRep.forget₂_ρ
-- Verify that the monoidal structure is available.
example : MonoidalCategory (FdRep k G) := by infer_instance
example : MonoidalPreadditive (FdRep k G) := by infer_instance
example : MonoidalLinear k (FdRep k G) := by infer_instance
open FiniteDimensional
open scoped Classical
-- We need to provide this instance explicitely as otherwise `finrank_hom_simple_simple` gives a
-- deterministic timeout.
instance : HasKernels (FdRep k G) := by infer_instance
-- Verify that Schur's lemma applies out of the box.
theorem finrank_hom_simple_simple [IsAlgClosed k] (V W : FdRep k G) [Simple V] [Simple W] :
finrank k (V ⟶ W) = if Nonempty (V ≅ W) then 1 else 0 :=
CategoryTheory.finrank_hom_simple_simple k V W
#align fdRep.finrank_hom_simple_simple FdRep.finrank_hom_simple_simple
/-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f := ⟨f.hom, f.comm⟩
map_add' _ _ := rfl
map_smul' _ _ := rfl
invFun f := ⟨(forget₂ (FGModuleCat k) (ModuleCat k)).map f.hom, f.comm⟩
left_inv _ := by ext; rfl
right_inv _ := by ext; rfl
#align fdRep.forget₂_hom_linear_equiv FdRep.forget₂HomLinearEquiv
end FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Group G]
-- Verify that the right rigid structure is available when the monoid is a group.
noncomputable instance : RightRigidCategory (FdRep k G) := by
change RightRigidCategory (Action (FGModuleCat k) (GroupCat.of G)); | infer_instance | noncomputable instance : RightRigidCategory (FdRep k G) := by
change RightRigidCategory (Action (FGModuleCat k) (GroupCat.of G)); | Mathlib.RepresentationTheory.FdRep.153_0.ADbOgJGW1JDvdmK | noncomputable instance : RightRigidCategory (FdRep k G) | Mathlib_RepresentationTheory_FdRep |
k G V : Type u
inst✝⁴ : Field k
inst✝³ : Group G
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : FiniteDimensional k V
ρV : Representation k G V
W : FdRep k G
⊢ of (dual ρV) ⊗ W ≅ of (linHom ρV (ρ W)) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : Module k V := by
change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : FiniteDimensional k V := by
change FiniteDimensional k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V); infer_instance
/-- All hom spaces are finite dimensional. -/
instance (V W : FdRep k G) : FiniteDimensional k (V ⟶ W) :=
FiniteDimensional.of_injective ((forget₂ (FdRep k G) (FGModuleCat k)).mapLinearMap k)
(Functor.map_injective (forget₂ (FdRep k G) (FGModuleCat k)))
/-- The monoid homomorphism corresponding to the action of `G` onto `V : FdRep k G`. -/
def ρ (V : FdRep k G) : G →* V →ₗ[k] V :=
Action.ρ V
#align fdRep.ρ FdRep.ρ
/-- The underlying `LinearEquiv` of an isomorphism of representations. -/
def isoToLinearEquiv {V W : FdRep k G} (i : V ≅ W) : V ≃ₗ[k] W :=
FGModuleCat.isoToLinearEquiv ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)
#align fdRep.iso_to_linear_equiv FdRep.isoToLinearEquiv
theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]
exact (i.hom.comm g).symm
#align fdRep.iso.conj_ρ FdRep.Iso.conj_ρ
/-- Lift an unbundled representation to `FdRep`. -/
@[simps ρ]
def of {V : Type u} [AddCommGroup V] [Module k V] [FiniteDimensional k V]
(ρ : Representation k G V) : FdRep k G :=
⟨FGModuleCat.of k V, ρ⟩
#align fdRep.of FdRep.of
instance : HasForget₂ (FdRep k G) (Rep k G) where
forget₂ := (forget₂ (FGModuleCat k) (ModuleCat k)).mapAction (MonCat.of G)
theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ := by
ext g v; rfl
#align fdRep.forget₂_ρ FdRep.forget₂_ρ
-- Verify that the monoidal structure is available.
example : MonoidalCategory (FdRep k G) := by infer_instance
example : MonoidalPreadditive (FdRep k G) := by infer_instance
example : MonoidalLinear k (FdRep k G) := by infer_instance
open FiniteDimensional
open scoped Classical
-- We need to provide this instance explicitely as otherwise `finrank_hom_simple_simple` gives a
-- deterministic timeout.
instance : HasKernels (FdRep k G) := by infer_instance
-- Verify that Schur's lemma applies out of the box.
theorem finrank_hom_simple_simple [IsAlgClosed k] (V W : FdRep k G) [Simple V] [Simple W] :
finrank k (V ⟶ W) = if Nonempty (V ≅ W) then 1 else 0 :=
CategoryTheory.finrank_hom_simple_simple k V W
#align fdRep.finrank_hom_simple_simple FdRep.finrank_hom_simple_simple
/-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f := ⟨f.hom, f.comm⟩
map_add' _ _ := rfl
map_smul' _ _ := rfl
invFun f := ⟨(forget₂ (FGModuleCat k) (ModuleCat k)).map f.hom, f.comm⟩
left_inv _ := by ext; rfl
right_inv _ := by ext; rfl
#align fdRep.forget₂_hom_linear_equiv FdRep.forget₂HomLinearEquiv
end FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Group G]
-- Verify that the right rigid structure is available when the monoid is a group.
noncomputable instance : RightRigidCategory (FdRep k G) := by
change RightRigidCategory (Action (FGModuleCat k) (GroupCat.of G)); infer_instance
end FdRep
namespace FdRep
-- The variables in this section are slightly weird, living half in `Representation` and half in
-- `FdRep`. When we have a better API for general monoidal closed and rigid categories and these
-- structures on `FdRep`, we should remove the dependancy of statements about `FdRep` on
-- `Representation.linHom` and `Representation.dual`. The isomorphism `dualTensorIsoLinHom`
-- below should then just be obtained from general results about rigid categories.
open Representation
variable {k G V : Type u} [Field k] [Group G]
variable [AddCommGroup V] [Module k V]
variable [FiniteDimensional k V]
variable (ρV : Representation k G V) (W : FdRep k G)
open scoped MonoidalCategory
/-- Auxiliary definition for `FdRep.dualTensorIsoLinHom`. -/
noncomputable def dualTensorIsoLinHomAux :
(FdRep.of ρV.dual ⊗ W).V ≅ (FdRep.of (linHom ρV W.ρ)).V :=
-- Porting note: had to make all types explicit arguments
@LinearEquiv.toFGModuleCatIso k _ (FdRep.of ρV.dual ⊗ W).V (V →ₗ[k] W)
_ _ _ _ _ _ (dualTensorHomEquiv k V W)
#align fdRep.dual_tensor_iso_lin_hom_aux FdRep.dualTensorIsoLinHomAux
/-- When `V` and `W` are finite dimensional representations of a group `G`, the isomorphism
`dualTensorHomEquiv k V W` of vector spaces induces an isomorphism of representations. -/
noncomputable def dualTensorIsoLinHom : FdRep.of ρV.dual ⊗ W ≅ FdRep.of (linHom ρV W.ρ) := by
| refine Action.mkIso (dualTensorIsoLinHomAux ρV W) ?_ | /-- When `V` and `W` are finite dimensional representations of a group `G`, the isomorphism
`dualTensorHomEquiv k V W` of vector spaces induces an isomorphism of representations. -/
noncomputable def dualTensorIsoLinHom : FdRep.of ρV.dual ⊗ W ≅ FdRep.of (linHom ρV W.ρ) := by
| Mathlib.RepresentationTheory.FdRep.182_0.ADbOgJGW1JDvdmK | /-- When `V` and `W` are finite dimensional representations of a group `G`, the isomorphism
`dualTensorHomEquiv k V W` of vector spaces induces an isomorphism of representations. -/
noncomputable def dualTensorIsoLinHom : FdRep.of ρV.dual ⊗ W ≅ FdRep.of (linHom ρV W.ρ) | Mathlib_RepresentationTheory_FdRep |
k G V : Type u
inst✝⁴ : Field k
inst✝³ : Group G
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : FiniteDimensional k V
ρV : Representation k G V
W : FdRep k G
⊢ ∀ (g : ↑(MonCat.of G)),
(of (dual ρV) ⊗ W).ρ g ≫ (dualTensorIsoLinHomAux ρV W).hom =
(dualTensorIsoLinHomAux ρV W).hom ≫ (of (linHom ρV (ρ W))).ρ g | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
/-!
# `FdRep k G` is the category of finite dimensional `k`-linear representations of `G`.
If `V : FdRep k G`, there is a coercion that allows you to treat `V` as a type,
and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances.
Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.
Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,
you can construct the bundled representation as `Rep.of ρ`.
We verify that `FdRep k G` is a `k`-linear monoidal category, and rigid when `G` is a group.
`FdRep k G` has all finite limits.
## TODO
* `FdRep k G ≌ FullSubcategory (FiniteDimensional k)`
* Upgrade the right rigid structure to a rigid structure
(this just needs to be done for `FGModuleCat`).
* `FdRep k G` has all finite colimits.
* `FdRep k G` is abelian.
* `FdRep k G ≌ FGModuleCat (MonoidAlgebra k G)`.
-/
suppress_compilation
universe u
open CategoryTheory
open CategoryTheory.Limits
set_option linter.uppercaseLean3 false -- `fdRep`
/-- The category of finite dimensional `k`-linear representations of a monoid `G`. -/
abbrev FdRep (k G : Type u) [Field k] [Monoid G] :=
Action (FGModuleCat.{u} k) (MonCat.of G)
#align fdRep FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Monoid G]
-- Porting note: `@[derive]` didn't work for `FdRep`. Add the 4 instances here.
instance : LargeCategory (FdRep k G) := inferInstance
instance : ConcreteCategory (FdRep k G) := inferInstance
instance : Preadditive (FdRep k G) := inferInstance
instance : HasFiniteLimits (FdRep k G) := inferInstance
instance : Linear k (FdRep k G) := by infer_instance
instance : CoeSort (FdRep k G) (Type u) :=
ConcreteCategory.hasCoeToSort _
instance (V : FdRep k G) : AddCommGroup V := by
change AddCommGroup ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : Module k V := by
change Module k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj; infer_instance
instance (V : FdRep k G) : FiniteDimensional k V := by
change FiniteDimensional k ((forget₂ (FdRep k G) (FGModuleCat k)).obj V); infer_instance
/-- All hom spaces are finite dimensional. -/
instance (V W : FdRep k G) : FiniteDimensional k (V ⟶ W) :=
FiniteDimensional.of_injective ((forget₂ (FdRep k G) (FGModuleCat k)).mapLinearMap k)
(Functor.map_injective (forget₂ (FdRep k G) (FGModuleCat k)))
/-- The monoid homomorphism corresponding to the action of `G` onto `V : FdRep k G`. -/
def ρ (V : FdRep k G) : G →* V →ₗ[k] V :=
Action.ρ V
#align fdRep.ρ FdRep.ρ
/-- The underlying `LinearEquiv` of an isomorphism of representations. -/
def isoToLinearEquiv {V W : FdRep k G} (i : V ≅ W) : V ≃ₗ[k] W :=
FGModuleCat.isoToLinearEquiv ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)
#align fdRep.iso_to_linear_equiv FdRep.isoToLinearEquiv
theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]
exact (i.hom.comm g).symm
#align fdRep.iso.conj_ρ FdRep.Iso.conj_ρ
/-- Lift an unbundled representation to `FdRep`. -/
@[simps ρ]
def of {V : Type u} [AddCommGroup V] [Module k V] [FiniteDimensional k V]
(ρ : Representation k G V) : FdRep k G :=
⟨FGModuleCat.of k V, ρ⟩
#align fdRep.of FdRep.of
instance : HasForget₂ (FdRep k G) (Rep k G) where
forget₂ := (forget₂ (FGModuleCat k) (ModuleCat k)).mapAction (MonCat.of G)
theorem forget₂_ρ (V : FdRep k G) : ((forget₂ (FdRep k G) (Rep k G)).obj V).ρ = V.ρ := by
ext g v; rfl
#align fdRep.forget₂_ρ FdRep.forget₂_ρ
-- Verify that the monoidal structure is available.
example : MonoidalCategory (FdRep k G) := by infer_instance
example : MonoidalPreadditive (FdRep k G) := by infer_instance
example : MonoidalLinear k (FdRep k G) := by infer_instance
open FiniteDimensional
open scoped Classical
-- We need to provide this instance explicitely as otherwise `finrank_hom_simple_simple` gives a
-- deterministic timeout.
instance : HasKernels (FdRep k G) := by infer_instance
-- Verify that Schur's lemma applies out of the box.
theorem finrank_hom_simple_simple [IsAlgClosed k] (V W : FdRep k G) [Simple V] [Simple W] :
finrank k (V ⟶ W) = if Nonempty (V ≅ W) then 1 else 0 :=
CategoryTheory.finrank_hom_simple_simple k V W
#align fdRep.finrank_hom_simple_simple FdRep.finrank_hom_simple_simple
/-- The forgetful functor to `Rep k G` preserves hom-sets and their vector space structure. -/
def forget₂HomLinearEquiv (X Y : FdRep k G) :
((forget₂ (FdRep k G) (Rep k G)).obj X ⟶ (forget₂ (FdRep k G) (Rep k G)).obj Y) ≃ₗ[k] X ⟶ Y
where
toFun f := ⟨f.hom, f.comm⟩
map_add' _ _ := rfl
map_smul' _ _ := rfl
invFun f := ⟨(forget₂ (FGModuleCat k) (ModuleCat k)).map f.hom, f.comm⟩
left_inv _ := by ext; rfl
right_inv _ := by ext; rfl
#align fdRep.forget₂_hom_linear_equiv FdRep.forget₂HomLinearEquiv
end FdRep
namespace FdRep
variable {k G : Type u} [Field k] [Group G]
-- Verify that the right rigid structure is available when the monoid is a group.
noncomputable instance : RightRigidCategory (FdRep k G) := by
change RightRigidCategory (Action (FGModuleCat k) (GroupCat.of G)); infer_instance
end FdRep
namespace FdRep
-- The variables in this section are slightly weird, living half in `Representation` and half in
-- `FdRep`. When we have a better API for general monoidal closed and rigid categories and these
-- structures on `FdRep`, we should remove the dependancy of statements about `FdRep` on
-- `Representation.linHom` and `Representation.dual`. The isomorphism `dualTensorIsoLinHom`
-- below should then just be obtained from general results about rigid categories.
open Representation
variable {k G V : Type u} [Field k] [Group G]
variable [AddCommGroup V] [Module k V]
variable [FiniteDimensional k V]
variable (ρV : Representation k G V) (W : FdRep k G)
open scoped MonoidalCategory
/-- Auxiliary definition for `FdRep.dualTensorIsoLinHom`. -/
noncomputable def dualTensorIsoLinHomAux :
(FdRep.of ρV.dual ⊗ W).V ≅ (FdRep.of (linHom ρV W.ρ)).V :=
-- Porting note: had to make all types explicit arguments
@LinearEquiv.toFGModuleCatIso k _ (FdRep.of ρV.dual ⊗ W).V (V →ₗ[k] W)
_ _ _ _ _ _ (dualTensorHomEquiv k V W)
#align fdRep.dual_tensor_iso_lin_hom_aux FdRep.dualTensorIsoLinHomAux
/-- When `V` and `W` are finite dimensional representations of a group `G`, the isomorphism
`dualTensorHomEquiv k V W` of vector spaces induces an isomorphism of representations. -/
noncomputable def dualTensorIsoLinHom : FdRep.of ρV.dual ⊗ W ≅ FdRep.of (linHom ρV W.ρ) := by
refine Action.mkIso (dualTensorIsoLinHomAux ρV W) ?_
| convert dualTensorHom_comm ρV W.ρ | /-- When `V` and `W` are finite dimensional representations of a group `G`, the isomorphism
`dualTensorHomEquiv k V W` of vector spaces induces an isomorphism of representations. -/
noncomputable def dualTensorIsoLinHom : FdRep.of ρV.dual ⊗ W ≅ FdRep.of (linHom ρV W.ρ) := by
refine Action.mkIso (dualTensorIsoLinHomAux ρV W) ?_
| Mathlib.RepresentationTheory.FdRep.182_0.ADbOgJGW1JDvdmK | /-- When `V` and `W` are finite dimensional representations of a group `G`, the isomorphism
`dualTensorHomEquiv k V W` of vector spaces induces an isomorphism of representations. -/
noncomputable def dualTensorIsoLinHom : FdRep.of ρV.dual ⊗ W ≅ FdRep.of (linHom ρV W.ρ) | Mathlib_RepresentationTheory_FdRep |
α : Type u_1
s : Set α
inst✝² : Preorder α
inst✝¹ : SupSet α
inst✝ : Inhabited ↑s
t : Set ↑s
h' : Set.Nonempty t
h'' : BddAbove t
h : sSup (Subtype.val '' t) ∈ s
⊢ sSup (Subtype.val '' t) = ↑(sSup t) | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by | simp [dif_pos, h, h', h''] | theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by | Mathlib.Order.CompleteLatticeIntervals.57_0.e28Rmw8JX0zQo3b | theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝² : Preorder α
inst✝¹ : SupSet α
inst✝ : Inhabited ↑s
⊢ sSup ∅ = default | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
| simp [sSup] | theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
| Mathlib.Order.CompleteLatticeIntervals.62_0.e28Rmw8JX0zQo3b | theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝² : Preorder α
inst✝¹ : SupSet α
inst✝ : Inhabited ↑s
t : Set ↑s
ht : ¬BddAbove t
⊢ sSup t = default | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
| simp [sSup, ht] | theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
| Mathlib.Order.CompleteLatticeIntervals.66_0.e28Rmw8JX0zQo3b | theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝² : Preorder α
inst✝¹ : InfSet α
inst✝ : Inhabited ↑s
t : Set ↑s
h' : Set.Nonempty t
h'' : BddBelow t
h : sInf (Subtype.val '' t) ∈ s
⊢ sInf (Subtype.val '' t) = ↑(sInf t) | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by | simp [dif_pos, h, h', h''] | theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by | Mathlib.Order.CompleteLatticeIntervals.97_0.e28Rmw8JX0zQo3b | theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝² : Preorder α
inst✝¹ : InfSet α
inst✝ : Inhabited ↑s
⊢ sInf ∅ = default | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
| simp [sInf] | theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
| Mathlib.Order.CompleteLatticeIntervals.102_0.e28Rmw8JX0zQo3b | theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝² : Preorder α
inst✝¹ : InfSet α
inst✝ : Inhabited ↑s
t : Set ↑s
ht : ¬BddBelow t
⊢ sInf t = default | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
| simp [sInf, ht] | theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
| Mathlib.Order.CompleteLatticeIntervals.106_0.e28Rmw8JX0zQo3b | theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSet s
src✝¹ : Lattice ↑s := DistribLattice.toLattice
src✝ : LinearOrder ↑s := inferInstance
⊢ ∀ (s_1 : Set ↑s) (a : ↑s), BddAbove s_1 → a ∈ s_1 → a ≤ sSup s_1 | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
| rintro t c h_bdd hct | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
| Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSet s
src✝¹ : Lattice ↑s := DistribLattice.toLattice
src✝ : LinearOrder ↑s := inferInstance
t : Set ↑s
c : ↑s
h_bdd : BddAbove t
hct : c ∈ t
⊢ c ≤ sSup t | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
| rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)] | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
| Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSet s
src✝¹ : Lattice ↑s := DistribLattice.toLattice
src✝ : LinearOrder ↑s := inferInstance
t : Set ↑s
c : ↑s
h_bdd : BddAbove t
hct : c ∈ t
⊢ ↑c ≤ sSup (Subtype.val '' t) | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
| exact (Subtype.mono_coe _).le_csSup_image hct h_bdd | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
| Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSet s
src✝¹ : Lattice ↑s := DistribLattice.toLattice
src✝ : LinearOrder ↑s := inferInstance
⊢ ∀ (s_1 : Set ↑s) (a : ↑s), Set.Nonempty s_1 → a ∈ upperBounds s_1 → sSup s_1 ≤ a | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
| rintro t B ht hB | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
| Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSet s
src✝¹ : Lattice ↑s := DistribLattice.toLattice
src✝ : LinearOrder ↑s := inferInstance
t : Set ↑s
B : ↑s
ht : Set.Nonempty t
hB : B ∈ upperBounds t
⊢ sSup t ≤ B | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
| rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)] | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
| Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSet s
src✝¹ : Lattice ↑s := DistribLattice.toLattice
src✝ : LinearOrder ↑s := inferInstance
t : Set ↑s
B : ↑s
ht : Set.Nonempty t
hB : B ∈ upperBounds t
⊢ sSup (Subtype.val '' t) ≤ ↑B | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
| exact (Subtype.mono_coe s).csSup_image_le ht hB | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
| Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSet s
src✝¹ : Lattice ↑s := DistribLattice.toLattice
src✝ : LinearOrder ↑s := inferInstance
⊢ ∀ (s_1 : Set ↑s) (a : ↑s), BddBelow s_1 → a ∈ s_1 → sInf s_1 ≤ a | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
| rintro t c h_bdd hct | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
| Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSet s
src✝¹ : Lattice ↑s := DistribLattice.toLattice
src✝ : LinearOrder ↑s := inferInstance
t : Set ↑s
c : ↑s
h_bdd : BddBelow t
hct : c ∈ t
⊢ sInf t ≤ c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
| rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)] | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
| Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSet s
src✝¹ : Lattice ↑s := DistribLattice.toLattice
src✝ : LinearOrder ↑s := inferInstance
t : Set ↑s
c : ↑s
h_bdd : BddBelow t
hct : c ∈ t
⊢ sInf (Subtype.val '' t) ≤ ↑c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
| exact (Subtype.mono_coe s).csInf_image_le hct h_bdd | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
| Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSet s
src✝¹ : Lattice ↑s := DistribLattice.toLattice
src✝ : LinearOrder ↑s := inferInstance
⊢ ∀ (s_1 : Set ↑s) (a : ↑s), Set.Nonempty s_1 → a ∈ lowerBounds s_1 → a ≤ sInf s_1 | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
| intro t B ht hB | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
| Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSet s
src✝¹ : Lattice ↑s := DistribLattice.toLattice
src✝ : LinearOrder ↑s := inferInstance
t : Set ↑s
B : ↑s
ht : Set.Nonempty t
hB : B ∈ lowerBounds t
⊢ B ≤ sInf t | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
| rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)] | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
| Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSet s
src✝¹ : Lattice ↑s := DistribLattice.toLattice
src✝ : LinearOrder ↑s := inferInstance
t : Set ↑s
B : ↑s
ht : Set.Nonempty t
hB : B ∈ lowerBounds t
⊢ ↑B ≤ sInf (Subtype.val '' t) | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
| exact (Subtype.mono_coe s).le_csInf_image ht hB | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
| Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSet s
src✝¹ : Lattice ↑s := DistribLattice.toLattice
src✝ : LinearOrder ↑s := inferInstance
t : Set ↑s
ht : ¬BddAbove t
⊢ sSup t = sSup ∅ | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by | simp [ht] | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by | Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, Set.Nonempty t → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, Set.Nonempty t → BddBelow t → sInf (Subtype.val '' t) ∈ s
src✝³ : SupSet ↑s := subsetSupSet s
src✝² : InfSet ↑s := subsetInfSet s
src✝¹ : Lattice ↑s := DistribLattice.toLattice
src✝ : LinearOrder ↑s := inferInstance
t : Set ↑s
ht : ¬BddBelow t
⊢ sInf t = sInf ∅ | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by | simp [ht] | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by | Mathlib.Order.CompleteLatticeIntervals.120_0.e28Rmw8JX0zQo3b | /-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s✝ : Set α
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hs : OrdConnected s
t : Set ↑s
ht : Set.Nonempty t
h_bdd : BddAbove t
⊢ sSup (Subtype.val '' t) ∈ s | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
| obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht | /-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
| Mathlib.Order.CompleteLatticeIntervals.150_0.e28Rmw8JX0zQo3b | /-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s | Mathlib_Order_CompleteLatticeIntervals |
case intro
α : Type u_1
s✝ : Set α
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hs : OrdConnected s
t : Set ↑s
h_bdd : BddAbove t
c : ↑s
hct : c ∈ t
⊢ sSup (Subtype.val '' t) ∈ s | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
| obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd | /-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
| Mathlib.Order.CompleteLatticeIntervals.150_0.e28Rmw8JX0zQo3b | /-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s | Mathlib_Order_CompleteLatticeIntervals |
case intro.intro
α : Type u_1
s✝ : Set α
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hs : OrdConnected s
t : Set ↑s
c : ↑s
hct : c ∈ t
B : ↑s
hB : B ∈ upperBounds t
⊢ sSup (Subtype.val '' t) ∈ s | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
| refine' hs.out c.2 B.2 ⟨_, _⟩ | /-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
| Mathlib.Order.CompleteLatticeIntervals.150_0.e28Rmw8JX0zQo3b | /-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s | Mathlib_Order_CompleteLatticeIntervals |
case intro.intro.refine'_1
α : Type u_1
s✝ : Set α
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hs : OrdConnected s
t : Set ↑s
c : ↑s
hct : c ∈ t
B : ↑s
hB : B ∈ upperBounds t
⊢ ↑c ≤ sSup (Subtype.val '' t) | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· | exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩ | /-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· | Mathlib.Order.CompleteLatticeIntervals.150_0.e28Rmw8JX0zQo3b | /-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s | Mathlib_Order_CompleteLatticeIntervals |
case intro.intro.refine'_2
α : Type u_1
s✝ : Set α
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hs : OrdConnected s
t : Set ↑s
c : ↑s
hct : c ∈ t
B : ↑s
hB : B ∈ upperBounds t
⊢ sSup (Subtype.val '' t) ≤ ↑B | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· | exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB | /-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· | Mathlib.Order.CompleteLatticeIntervals.150_0.e28Rmw8JX0zQo3b | /-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s✝ : Set α
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hs : OrdConnected s
t : Set ↑s
ht : Set.Nonempty t
h_bdd : BddBelow t
⊢ sInf (Subtype.val '' t) ∈ s | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
| obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht | /-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
| Mathlib.Order.CompleteLatticeIntervals.161_0.e28Rmw8JX0zQo3b | /-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s | Mathlib_Order_CompleteLatticeIntervals |
case intro
α : Type u_1
s✝ : Set α
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hs : OrdConnected s
t : Set ↑s
h_bdd : BddBelow t
c : ↑s
hct : c ∈ t
⊢ sInf (Subtype.val '' t) ∈ s | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
| obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd | /-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
| Mathlib.Order.CompleteLatticeIntervals.161_0.e28Rmw8JX0zQo3b | /-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s | Mathlib_Order_CompleteLatticeIntervals |
case intro.intro
α : Type u_1
s✝ : Set α
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hs : OrdConnected s
t : Set ↑s
c : ↑s
hct : c ∈ t
B : ↑s
hB : B ∈ lowerBounds t
⊢ sInf (Subtype.val '' t) ∈ s | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
| refine' hs.out B.2 c.2 ⟨_, _⟩ | /-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
| Mathlib.Order.CompleteLatticeIntervals.161_0.e28Rmw8JX0zQo3b | /-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s | Mathlib_Order_CompleteLatticeIntervals |
case intro.intro.refine'_1
α : Type u_1
s✝ : Set α
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hs : OrdConnected s
t : Set ↑s
c : ↑s
hct : c ∈ t
B : ↑s
hB : B ∈ lowerBounds t
⊢ ↑B ≤ sInf (Subtype.val '' t) | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· | exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB | /-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· | Mathlib.Order.CompleteLatticeIntervals.161_0.e28Rmw8JX0zQo3b | /-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s | Mathlib_Order_CompleteLatticeIntervals |
case intro.intro.refine'_2
α : Type u_1
s✝ : Set α
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hs : OrdConnected s
t : Set ↑s
c : ↑s
hct : c ∈ t
B : ↑s
hB : B ∈ lowerBounds t
⊢ sInf (Subtype.val '' t) ≤ ↑c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· | exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩ | /-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· | Mathlib.Order.CompleteLatticeIntervals.161_0.e28Rmw8JX0zQo3b | /-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
hS : ¬S = ∅
⊢ sSup (Subtype.val '' S) ∈ Icc a b | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
| rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
| Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
hS : Set.Nonempty S
⊢ sSup (Subtype.val '' S) ∈ Icc a b | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
| refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩ | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
| Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
hS : Set.Nonempty S
⊢ a ≤ sSup (Subtype.val '' S) | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
| obtain ⟨c, hc⟩ := hS | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
| Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case intro
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
⊢ a ≤ sSup (Subtype.val '' S) | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
| exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩) | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
| Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
⊢ c ≤ sSup S | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
| by_cases hS : S = ∅ | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
| Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case pos
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
hS : S = ∅
⊢ c ≤ sSup S | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> | simp only [hS, dite_true, dite_false] | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case neg
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
hS : ¬S = ∅
⊢ c ≤ sSup S | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> | simp only [hS, dite_true, dite_false] | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case pos
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
hS : S = ∅
⊢ c ≤ { val := a, property := (_ : a ≤ a ∧ a ≤ b) } | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· | simp [hS] at hc | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case neg
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
hS : ¬S = ∅
⊢ c ≤ { val := sSup (Subtype.val '' S), property := (_ : a ≤ sSup (Subtype.val '' S) ∧ sSup (Subtype.val '' S) ≤ b) } | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· | exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩ | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : ∀ b_1 ∈ S, b_1 ≤ c
⊢ sSup S ≤ c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
| by_cases hS : S = ∅ | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
| Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case pos
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : ∀ b_1 ∈ S, b_1 ≤ c
hS : S = ∅
⊢ sSup S ≤ c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> | simp only [hS, dite_true, dite_false] | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case neg
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : ∀ b_1 ∈ S, b_1 ≤ c
hS : ¬S = ∅
⊢ sSup S ≤ c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> | simp only [hS, dite_true, dite_false] | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case pos
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : ∀ b_1 ∈ S, b_1 ≤ c
hS : S = ∅
⊢ { val := a, property := (_ : a ≤ a ∧ a ≤ b) } ≤ c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· | exact c.2.1 | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case neg
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : ∀ b_1 ∈ S, b_1 ≤ c
hS : ¬S = ∅
⊢ { val := sSup (Subtype.val '' S), property := (_ : a ≤ sSup (Subtype.val '' S) ∧ sSup (Subtype.val '' S) ≤ b) } ≤ c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· | exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h) | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
hS : ¬S = ∅
⊢ sInf (Subtype.val '' S) ∈ Icc a b | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
| rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
| Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
hS : Set.Nonempty S
⊢ sInf (Subtype.val '' S) ∈ Icc a b | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
| refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩ | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
| Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
hS : Set.Nonempty S
⊢ sInf (Subtype.val '' S) ≤ b | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
| obtain ⟨c, hc⟩ := hS | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
| Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case intro
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
⊢ sInf (Subtype.val '' S) ≤ b | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
obtain ⟨c, hc⟩ := hS
| exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2 | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
obtain ⟨c, hc⟩ := hS
| Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
⊢ sInf S ≤ c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
obtain ⟨c, hc⟩ := hS
exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2⟩
sInf_le S c hc := by
| by_cases hS : S = ∅ | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
obtain ⟨c, hc⟩ := hS
exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2⟩
sInf_le S c hc := by
| Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case pos
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
hS : S = ∅
⊢ sInf S ≤ c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
obtain ⟨c, hc⟩ := hS
exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2⟩
sInf_le S c hc := by
by_cases hS : S = ∅ <;> | simp only [hS, dite_true, dite_false] | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
obtain ⟨c, hc⟩ := hS
exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2⟩
sInf_le S c hc := by
by_cases hS : S = ∅ <;> | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case neg
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
hS : ¬S = ∅
⊢ sInf S ≤ c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
obtain ⟨c, hc⟩ := hS
exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2⟩
sInf_le S c hc := by
by_cases hS : S = ∅ <;> | simp only [hS, dite_true, dite_false] | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
obtain ⟨c, hc⟩ := hS
exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2⟩
sInf_le S c hc := by
by_cases hS : S = ∅ <;> | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case pos
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
hS : S = ∅
⊢ { val := b, property := (_ : a ≤ b ∧ b ≤ b) } ≤ c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
obtain ⟨c, hc⟩ := hS
exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2⟩
sInf_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· | simp [hS] at hc | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
obtain ⟨c, hc⟩ := hS
exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2⟩
sInf_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case neg
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : c ∈ S
hS : ¬S = ∅
⊢ { val := sInf (Subtype.val '' S), property := (_ : a ≤ sInf (Subtype.val '' S) ∧ sInf (Subtype.val '' S) ≤ b) } ≤ c | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
obtain ⟨c, hc⟩ := hS
exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2⟩
sInf_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· | exact csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩ | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
obtain ⟨c, hc⟩ := hS
exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2⟩
sInf_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : ∀ b_1 ∈ S, c ≤ b_1
⊢ c ≤ sInf S | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
obtain ⟨c, hc⟩ := hS
exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2⟩
sInf_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩
le_sInf S c hc := by
| by_cases hS : S = ∅ | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
obtain ⟨c, hc⟩ := hS
exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2⟩
sInf_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩
le_sInf S c hc := by
| Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case pos
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : ∀ b_1 ∈ S, c ≤ b_1
hS : S = ∅
⊢ c ≤ sInf S | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
obtain ⟨c, hc⟩ := hS
exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2⟩
sInf_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩
le_sInf S c hc := by
by_cases hS : S = ∅ <;> | simp only [hS, dite_true, dite_false] | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
obtain ⟨c, hc⟩ := hS
exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2⟩
sInf_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩
le_sInf S c hc := by
by_cases hS : S = ∅ <;> | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case neg
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : ∀ b_1 ∈ S, c ≤ b_1
hS : ¬S = ∅
⊢ c ≤ sInf S | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
obtain ⟨c, hc⟩ := hS
exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2⟩
sInf_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩
le_sInf S c hc := by
by_cases hS : S = ∅ <;> | simp only [hS, dite_true, dite_false] | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
obtain ⟨c, hc⟩ := hS
exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2⟩
sInf_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩
le_sInf S c hc := by
by_cases hS : S = ∅ <;> | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case pos
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : ∀ b_1 ∈ S, c ≤ b_1
hS : S = ∅
⊢ c ≤ { val := b, property := (_ : a ≤ b ∧ b ≤ b) } | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
obtain ⟨c, hc⟩ := hS
exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2⟩
sInf_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩
le_sInf S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· | exact c.2.2 | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
obtain ⟨c, hc⟩ := hS
exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2⟩
sInf_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩
le_sInf S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
case neg
α : Type u_1
s : Set α
inst✝ : ConditionallyCompleteLattice α
a b : α
h : a ≤ b
src✝ : BoundedOrder ↑(Icc a b) := Icc.boundedOrder h
S : Set ↑(Icc a b)
c : ↑(Icc a b)
hc : ∀ b_1 ∈ S, c ≤ b_1
hS : ¬S = ∅
⊢ c ≤ { val := sInf (Subtype.val '' S), property := (_ : a ≤ sInf (Subtype.val '' S) ∧ sInf (Subtype.val '' S) ≤ b) } | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Data.Set.Intervals.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-! # Subtypes of conditionally complete linear orders
In this file we give conditions on a subset of a conditionally complete linear order, to ensure that
the subtype is itself conditionally complete.
We check that an `OrdConnected` set satisfies these conditions.
## TODO
Add appropriate instances for all `Set.Ixx`. This requires a refactor that will allow different
default values for `sSup` and `sInf`.
-/
open Classical
open Set
variable {α : Type*} (s : Set α)
section SupSet
variable [Preorder α] [SupSet α]
/-- `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Sup subsetSupSet
attribute [local instance] subsetSupSet
@[simp]
theorem subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl
#align subset_Sup_def subset_sSup_def
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Sup_of_within subset_sSup_of_within
theorem subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup]
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht]
end SupSet
section InfSet
variable [Preorder α] [InfSet α]
/-- `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. -/
noncomputable def subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
#align subset_has_Inf subsetInfSet
attribute [local instance] subsetInfSet
@[simp]
theorem subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl
#align subset_Inf_def subset_sInf_def
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [dif_pos, h, h', h'']
#align subset_Inf_of_within subset_sInf_of_within
theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht]
end InfSet
section OrdConnected
variable [ConditionallyCompleteLinearOrder α]
attribute [local instance] subsetSupSet
attribute [local instance] subsetInfSet
/-- For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. -/
@[reducible]
noncomputable def subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet s, subsetInfSet s, DistribLattice.toLattice, (inferInstance : LinearOrder s) with
le_csSup := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ⟨c, hct⟩ h_bdd (h_Sup ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe _).le_csSup_image hct h_bdd
csSup_le := by
rintro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sSup_of_within s ht ⟨B, hB⟩ (h_Sup ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).csSup_image_le ht hB
le_csInf := by
intro t B ht hB
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ht ⟨B, hB⟩ (h_Inf ht ⟨B, hB⟩)]
exact (Subtype.mono_coe s).le_csInf_image ht hB
csInf_le := by
rintro t c h_bdd hct
rw [← Subtype.coe_le_coe, ← subset_sInf_of_within s ⟨c, hct⟩ h_bdd (h_Inf ⟨c, hct⟩ h_bdd)]
exact (Subtype.mono_coe s).csInf_image_le hct h_bdd
csSup_of_not_bddAbove := fun t ht ↦ by simp [ht]
csInf_of_not_bddBelow := fun t ht ↦ by simp [ht] }
#align subset_conditionally_complete_linear_order subsetConditionallyCompleteLinearOrder
/-- The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. -/
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine' hs.out c.2 B.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csSup_image hct ⟨B, hB⟩
· exact (Subtype.mono_coe s).csSup_image_le ⟨c, hct⟩ hB
#align Sup_within_of_ord_connected sSup_within_of_ordConnected
/-- The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. -/
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine' hs.out B.2 c.2 ⟨_, _⟩
· exact (Subtype.mono_coe s).le_csInf_image ⟨c, hct⟩ hB
· exact (Subtype.mono_coe s).csInf_image_le hct ⟨B, hB⟩
#align Inf_within_of_ord_connected sInf_within_of_ordConnected
/-- A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. -/
noncomputable instance ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h)
#align ord_connected_subset_conditionally_complete_linear_order ordConnectedSubsetConditionallyCompleteLinearOrder
end OrdConnected
section Icc
/-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
obtain ⟨c, hc⟩ := hS
exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2⟩
sInf_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩
le_sInf S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.2
· | exact le_csInf ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h) | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ := Set.Icc.boundedOrder h
sSup S := if hS : S = ∅ then ⟨a, le_rfl, h⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨_, csSup_le (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.2)⟩
obtain ⟨c, hc⟩ := hS
exact c.2.1.trans (le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩)⟩
le_sSup S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact le_csSup ⟨b, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.2⟩ ⟨c, hc, rfl⟩
sSup_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.1
· exact csSup_le ((Set.nonempty_iff_ne_empty.mpr hS).image (↑))
(fun _ ⟨d, h, hd⟩ ↦ hd ▸ hc d h)
sInf S := if hS : S = ∅ then ⟨b, h, le_rfl⟩ else ⟨sInf ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hS
refine' ⟨le_csInf (hS.image (↑)) (fun _ ⟨c, _, hc⟩ ↦ hc ▸ c.2.1), _⟩
obtain ⟨c, hc⟩ := hS
exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2⟩
sInf_le S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· simp [hS] at hc
· exact csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩
le_sInf S c hc := by
by_cases hS : S = ∅ <;> simp only [hS, dite_true, dite_false]
· exact c.2.2
· | Mathlib.Order.CompleteLatticeIntervals.185_0.e28Rmw8JX0zQo3b | /-- Complete lattice structure on `Set.Icc` -/
noncomputable def Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} (h : a ≤ b) : CompleteLattice (Set.Icc a b) where
__ | Mathlib_Order_CompleteLatticeIntervals |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ c ∈ perpBisector p₁ p₂ ↔ inner ((Equiv.pointReflection c) p₁ -ᵥ p₂) (p₂ -ᵥ p₁) = 0 | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
/-!
# Perpendicular bisector of a segment
We define `AffineSubspace.perpBisector p₁ p₂` to be the perpendicular bisector of the segment
`[p₁, p₂]`, as a bundled affine subspace. We also prove that a point belongs to the perpendicular
bisector if and only if it is equidistant from `p₁` and `p₂`, as well as a few linear equations that
define this subspace.
## Keywords
euclidean geometry, perpendicular, perpendicular bisector, line segment bisector, equidistant
-/
set_option autoImplicit true
open Set
open scoped BigOperators RealInnerProductSpace
variable [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
variable [NormedAddTorsor V P]
noncomputable section
namespace AffineSubspace
variable {c c₁ c₂ p₁ p₂ : P}
/-- Perpendicular bisector of a segment in a Euclidean affine space. -/
def perpBisector (p₁ p₂ : P) : AffineSubspace ℝ P :=
.comap ((AffineEquiv.vaddConst ℝ (midpoint ℝ p₁ p₂)).symm : P →ᵃ[ℝ] V) <|
(LinearMap.ker (innerₛₗ ℝ (p₂ -ᵥ p₁))).toAffineSubspace
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `p₂ -ᵥ p₁` is orthogonal to
`c -ᵥ midpoint ℝ p₁ p₂`. -/
theorem mem_perpBisector_iff_inner_eq_zero' :
c ∈ perpBisector p₁ p₂ ↔ ⟪p₂ -ᵥ p₁, c -ᵥ midpoint ℝ p₁ p₂⟫ = 0 :=
Iff.rfl
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `c -ᵥ midpoint ℝ p₁ p₂` is
orthogonal to `p₂ -ᵥ p₁`. -/
theorem mem_perpBisector_iff_inner_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ midpoint ℝ p₁ p₂, p₂ -ᵥ p₁⟫ = 0 :=
inner_eq_zero_symm
theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 := by
| rw [mem_perpBisector_iff_inner_eq_zero, Equiv.pointReflection_apply,
vsub_midpoint, invOf_eq_inv, ← smul_add, real_inner_smul_left, vadd_vsub_assoc] | theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 := by
| Mathlib.Geometry.Euclidean.PerpBisector.54_0.WKtplj3xHYGfYbJ | theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ 2⁻¹ * inner (c -ᵥ p₁ + (c -ᵥ p₂)) (p₂ -ᵥ p₁) = 0 ↔ inner (c -ᵥ p₁ + (c -ᵥ p₂)) (p₂ -ᵥ p₁) = 0 | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
/-!
# Perpendicular bisector of a segment
We define `AffineSubspace.perpBisector p₁ p₂` to be the perpendicular bisector of the segment
`[p₁, p₂]`, as a bundled affine subspace. We also prove that a point belongs to the perpendicular
bisector if and only if it is equidistant from `p₁` and `p₂`, as well as a few linear equations that
define this subspace.
## Keywords
euclidean geometry, perpendicular, perpendicular bisector, line segment bisector, equidistant
-/
set_option autoImplicit true
open Set
open scoped BigOperators RealInnerProductSpace
variable [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
variable [NormedAddTorsor V P]
noncomputable section
namespace AffineSubspace
variable {c c₁ c₂ p₁ p₂ : P}
/-- Perpendicular bisector of a segment in a Euclidean affine space. -/
def perpBisector (p₁ p₂ : P) : AffineSubspace ℝ P :=
.comap ((AffineEquiv.vaddConst ℝ (midpoint ℝ p₁ p₂)).symm : P →ᵃ[ℝ] V) <|
(LinearMap.ker (innerₛₗ ℝ (p₂ -ᵥ p₁))).toAffineSubspace
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `p₂ -ᵥ p₁` is orthogonal to
`c -ᵥ midpoint ℝ p₁ p₂`. -/
theorem mem_perpBisector_iff_inner_eq_zero' :
c ∈ perpBisector p₁ p₂ ↔ ⟪p₂ -ᵥ p₁, c -ᵥ midpoint ℝ p₁ p₂⟫ = 0 :=
Iff.rfl
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `c -ᵥ midpoint ℝ p₁ p₂` is
orthogonal to `p₂ -ᵥ p₁`. -/
theorem mem_perpBisector_iff_inner_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ midpoint ℝ p₁ p₂, p₂ -ᵥ p₁⟫ = 0 :=
inner_eq_zero_symm
theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, Equiv.pointReflection_apply,
vsub_midpoint, invOf_eq_inv, ← smul_add, real_inner_smul_left, vadd_vsub_assoc]
| simp | theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, Equiv.pointReflection_apply,
vsub_midpoint, invOf_eq_inv, ← smul_add, real_inner_smul_left, vadd_vsub_assoc]
| Mathlib.Geometry.Euclidean.PerpBisector.54_0.WKtplj3xHYGfYbJ | theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ c ∈ perpBisector p₁ ((Equiv.pointReflection p₂) p₁) ↔ inner (c -ᵥ p₂) (p₁ -ᵥ p₂) = 0 | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
/-!
# Perpendicular bisector of a segment
We define `AffineSubspace.perpBisector p₁ p₂` to be the perpendicular bisector of the segment
`[p₁, p₂]`, as a bundled affine subspace. We also prove that a point belongs to the perpendicular
bisector if and only if it is equidistant from `p₁` and `p₂`, as well as a few linear equations that
define this subspace.
## Keywords
euclidean geometry, perpendicular, perpendicular bisector, line segment bisector, equidistant
-/
set_option autoImplicit true
open Set
open scoped BigOperators RealInnerProductSpace
variable [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
variable [NormedAddTorsor V P]
noncomputable section
namespace AffineSubspace
variable {c c₁ c₂ p₁ p₂ : P}
/-- Perpendicular bisector of a segment in a Euclidean affine space. -/
def perpBisector (p₁ p₂ : P) : AffineSubspace ℝ P :=
.comap ((AffineEquiv.vaddConst ℝ (midpoint ℝ p₁ p₂)).symm : P →ᵃ[ℝ] V) <|
(LinearMap.ker (innerₛₗ ℝ (p₂ -ᵥ p₁))).toAffineSubspace
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `p₂ -ᵥ p₁` is orthogonal to
`c -ᵥ midpoint ℝ p₁ p₂`. -/
theorem mem_perpBisector_iff_inner_eq_zero' :
c ∈ perpBisector p₁ p₂ ↔ ⟪p₂ -ᵥ p₁, c -ᵥ midpoint ℝ p₁ p₂⟫ = 0 :=
Iff.rfl
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `c -ᵥ midpoint ℝ p₁ p₂` is
orthogonal to `p₂ -ᵥ p₁`. -/
theorem mem_perpBisector_iff_inner_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ midpoint ℝ p₁ p₂, p₂ -ᵥ p₁⟫ = 0 :=
inner_eq_zero_symm
theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, Equiv.pointReflection_apply,
vsub_midpoint, invOf_eq_inv, ← smul_add, real_inner_smul_left, vadd_vsub_assoc]
simp
theorem mem_perpBisector_pointReflection_iff_inner_eq_zero :
c ∈ perpBisector p₁ (Equiv.pointReflection p₂ p₁) ↔ ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ = 0 := by
| rw [mem_perpBisector_iff_inner_eq_zero, midpoint_pointReflection_right,
Equiv.pointReflection_apply, vadd_vsub_assoc, inner_add_right, add_self_eq_zero,
← neg_eq_zero, ← inner_neg_right, neg_vsub_eq_vsub_rev] | theorem mem_perpBisector_pointReflection_iff_inner_eq_zero :
c ∈ perpBisector p₁ (Equiv.pointReflection p₂ p₁) ↔ ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ = 0 := by
| Mathlib.Geometry.Euclidean.PerpBisector.60_0.WKtplj3xHYGfYbJ | theorem mem_perpBisector_pointReflection_iff_inner_eq_zero :
c ∈ perpBisector p₁ (Equiv.pointReflection p₂ p₁) ↔ ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ = 0 | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁✝ p₂✝ p₁ p₂ : P
⊢ midpoint ℝ p₁ p₂ ∈ perpBisector p₁ p₂ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
/-!
# Perpendicular bisector of a segment
We define `AffineSubspace.perpBisector p₁ p₂` to be the perpendicular bisector of the segment
`[p₁, p₂]`, as a bundled affine subspace. We also prove that a point belongs to the perpendicular
bisector if and only if it is equidistant from `p₁` and `p₂`, as well as a few linear equations that
define this subspace.
## Keywords
euclidean geometry, perpendicular, perpendicular bisector, line segment bisector, equidistant
-/
set_option autoImplicit true
open Set
open scoped BigOperators RealInnerProductSpace
variable [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
variable [NormedAddTorsor V P]
noncomputable section
namespace AffineSubspace
variable {c c₁ c₂ p₁ p₂ : P}
/-- Perpendicular bisector of a segment in a Euclidean affine space. -/
def perpBisector (p₁ p₂ : P) : AffineSubspace ℝ P :=
.comap ((AffineEquiv.vaddConst ℝ (midpoint ℝ p₁ p₂)).symm : P →ᵃ[ℝ] V) <|
(LinearMap.ker (innerₛₗ ℝ (p₂ -ᵥ p₁))).toAffineSubspace
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `p₂ -ᵥ p₁` is orthogonal to
`c -ᵥ midpoint ℝ p₁ p₂`. -/
theorem mem_perpBisector_iff_inner_eq_zero' :
c ∈ perpBisector p₁ p₂ ↔ ⟪p₂ -ᵥ p₁, c -ᵥ midpoint ℝ p₁ p₂⟫ = 0 :=
Iff.rfl
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `c -ᵥ midpoint ℝ p₁ p₂` is
orthogonal to `p₂ -ᵥ p₁`. -/
theorem mem_perpBisector_iff_inner_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ midpoint ℝ p₁ p₂, p₂ -ᵥ p₁⟫ = 0 :=
inner_eq_zero_symm
theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, Equiv.pointReflection_apply,
vsub_midpoint, invOf_eq_inv, ← smul_add, real_inner_smul_left, vadd_vsub_assoc]
simp
theorem mem_perpBisector_pointReflection_iff_inner_eq_zero :
c ∈ perpBisector p₁ (Equiv.pointReflection p₂ p₁) ↔ ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, midpoint_pointReflection_right,
Equiv.pointReflection_apply, vadd_vsub_assoc, inner_add_right, add_self_eq_zero,
← neg_eq_zero, ← inner_neg_right, neg_vsub_eq_vsub_rev]
theorem midpoint_mem_perpBisector (p₁ p₂ : P) :
midpoint ℝ p₁ p₂ ∈ perpBisector p₁ p₂ := by
| simp [mem_perpBisector_iff_inner_eq_zero] | theorem midpoint_mem_perpBisector (p₁ p₂ : P) :
midpoint ℝ p₁ p₂ ∈ perpBisector p₁ p₂ := by
| Mathlib.Geometry.Euclidean.PerpBisector.66_0.WKtplj3xHYGfYbJ | theorem midpoint_mem_perpBisector (p₁ p₂ : P) :
midpoint ℝ p₁ p₂ ∈ perpBisector p₁ p₂ | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁✝ p₂✝ p₁ p₂ : P
⊢ direction (perpBisector p₁ p₂) = (Submodule.span ℝ {p₂ -ᵥ p₁})ᗮ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
/-!
# Perpendicular bisector of a segment
We define `AffineSubspace.perpBisector p₁ p₂` to be the perpendicular bisector of the segment
`[p₁, p₂]`, as a bundled affine subspace. We also prove that a point belongs to the perpendicular
bisector if and only if it is equidistant from `p₁` and `p₂`, as well as a few linear equations that
define this subspace.
## Keywords
euclidean geometry, perpendicular, perpendicular bisector, line segment bisector, equidistant
-/
set_option autoImplicit true
open Set
open scoped BigOperators RealInnerProductSpace
variable [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
variable [NormedAddTorsor V P]
noncomputable section
namespace AffineSubspace
variable {c c₁ c₂ p₁ p₂ : P}
/-- Perpendicular bisector of a segment in a Euclidean affine space. -/
def perpBisector (p₁ p₂ : P) : AffineSubspace ℝ P :=
.comap ((AffineEquiv.vaddConst ℝ (midpoint ℝ p₁ p₂)).symm : P →ᵃ[ℝ] V) <|
(LinearMap.ker (innerₛₗ ℝ (p₂ -ᵥ p₁))).toAffineSubspace
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `p₂ -ᵥ p₁` is orthogonal to
`c -ᵥ midpoint ℝ p₁ p₂`. -/
theorem mem_perpBisector_iff_inner_eq_zero' :
c ∈ perpBisector p₁ p₂ ↔ ⟪p₂ -ᵥ p₁, c -ᵥ midpoint ℝ p₁ p₂⟫ = 0 :=
Iff.rfl
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `c -ᵥ midpoint ℝ p₁ p₂` is
orthogonal to `p₂ -ᵥ p₁`. -/
theorem mem_perpBisector_iff_inner_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ midpoint ℝ p₁ p₂, p₂ -ᵥ p₁⟫ = 0 :=
inner_eq_zero_symm
theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, Equiv.pointReflection_apply,
vsub_midpoint, invOf_eq_inv, ← smul_add, real_inner_smul_left, vadd_vsub_assoc]
simp
theorem mem_perpBisector_pointReflection_iff_inner_eq_zero :
c ∈ perpBisector p₁ (Equiv.pointReflection p₂ p₁) ↔ ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, midpoint_pointReflection_right,
Equiv.pointReflection_apply, vadd_vsub_assoc, inner_add_right, add_self_eq_zero,
← neg_eq_zero, ← inner_neg_right, neg_vsub_eq_vsub_rev]
theorem midpoint_mem_perpBisector (p₁ p₂ : P) :
midpoint ℝ p₁ p₂ ∈ perpBisector p₁ p₂ := by
simp [mem_perpBisector_iff_inner_eq_zero]
theorem perpBisector_nonempty : (perpBisector p₁ p₂ : Set P).Nonempty :=
⟨_, midpoint_mem_perpBisector _ _⟩
@[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ := by
| erw [perpBisector, comap_symm, map_direction, Submodule.map_id,
Submodule.toAffineSubspace_direction] | @[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ := by
| Mathlib.Geometry.Euclidean.PerpBisector.73_0.WKtplj3xHYGfYbJ | @[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁✝ p₂✝ p₁ p₂ : P
⊢ LinearMap.ker ((innerₛₗ ℝ) (p₂ -ᵥ p₁)) = (Submodule.span ℝ {p₂ -ᵥ p₁})ᗮ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
/-!
# Perpendicular bisector of a segment
We define `AffineSubspace.perpBisector p₁ p₂` to be the perpendicular bisector of the segment
`[p₁, p₂]`, as a bundled affine subspace. We also prove that a point belongs to the perpendicular
bisector if and only if it is equidistant from `p₁` and `p₂`, as well as a few linear equations that
define this subspace.
## Keywords
euclidean geometry, perpendicular, perpendicular bisector, line segment bisector, equidistant
-/
set_option autoImplicit true
open Set
open scoped BigOperators RealInnerProductSpace
variable [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
variable [NormedAddTorsor V P]
noncomputable section
namespace AffineSubspace
variable {c c₁ c₂ p₁ p₂ : P}
/-- Perpendicular bisector of a segment in a Euclidean affine space. -/
def perpBisector (p₁ p₂ : P) : AffineSubspace ℝ P :=
.comap ((AffineEquiv.vaddConst ℝ (midpoint ℝ p₁ p₂)).symm : P →ᵃ[ℝ] V) <|
(LinearMap.ker (innerₛₗ ℝ (p₂ -ᵥ p₁))).toAffineSubspace
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `p₂ -ᵥ p₁` is orthogonal to
`c -ᵥ midpoint ℝ p₁ p₂`. -/
theorem mem_perpBisector_iff_inner_eq_zero' :
c ∈ perpBisector p₁ p₂ ↔ ⟪p₂ -ᵥ p₁, c -ᵥ midpoint ℝ p₁ p₂⟫ = 0 :=
Iff.rfl
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `c -ᵥ midpoint ℝ p₁ p₂` is
orthogonal to `p₂ -ᵥ p₁`. -/
theorem mem_perpBisector_iff_inner_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ midpoint ℝ p₁ p₂, p₂ -ᵥ p₁⟫ = 0 :=
inner_eq_zero_symm
theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, Equiv.pointReflection_apply,
vsub_midpoint, invOf_eq_inv, ← smul_add, real_inner_smul_left, vadd_vsub_assoc]
simp
theorem mem_perpBisector_pointReflection_iff_inner_eq_zero :
c ∈ perpBisector p₁ (Equiv.pointReflection p₂ p₁) ↔ ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, midpoint_pointReflection_right,
Equiv.pointReflection_apply, vadd_vsub_assoc, inner_add_right, add_self_eq_zero,
← neg_eq_zero, ← inner_neg_right, neg_vsub_eq_vsub_rev]
theorem midpoint_mem_perpBisector (p₁ p₂ : P) :
midpoint ℝ p₁ p₂ ∈ perpBisector p₁ p₂ := by
simp [mem_perpBisector_iff_inner_eq_zero]
theorem perpBisector_nonempty : (perpBisector p₁ p₂ : Set P).Nonempty :=
⟨_, midpoint_mem_perpBisector _ _⟩
@[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ := by
erw [perpBisector, comap_symm, map_direction, Submodule.map_id,
Submodule.toAffineSubspace_direction]
| ext x | @[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ := by
erw [perpBisector, comap_symm, map_direction, Submodule.map_id,
Submodule.toAffineSubspace_direction]
| Mathlib.Geometry.Euclidean.PerpBisector.73_0.WKtplj3xHYGfYbJ | @[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ | Mathlib_Geometry_Euclidean_PerpBisector |
case h
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁✝ p₂✝ p₁ p₂ : P
x : V
⊢ x ∈ LinearMap.ker ((innerₛₗ ℝ) (p₂ -ᵥ p₁)) ↔ x ∈ (Submodule.span ℝ {p₂ -ᵥ p₁})ᗮ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
/-!
# Perpendicular bisector of a segment
We define `AffineSubspace.perpBisector p₁ p₂` to be the perpendicular bisector of the segment
`[p₁, p₂]`, as a bundled affine subspace. We also prove that a point belongs to the perpendicular
bisector if and only if it is equidistant from `p₁` and `p₂`, as well as a few linear equations that
define this subspace.
## Keywords
euclidean geometry, perpendicular, perpendicular bisector, line segment bisector, equidistant
-/
set_option autoImplicit true
open Set
open scoped BigOperators RealInnerProductSpace
variable [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
variable [NormedAddTorsor V P]
noncomputable section
namespace AffineSubspace
variable {c c₁ c₂ p₁ p₂ : P}
/-- Perpendicular bisector of a segment in a Euclidean affine space. -/
def perpBisector (p₁ p₂ : P) : AffineSubspace ℝ P :=
.comap ((AffineEquiv.vaddConst ℝ (midpoint ℝ p₁ p₂)).symm : P →ᵃ[ℝ] V) <|
(LinearMap.ker (innerₛₗ ℝ (p₂ -ᵥ p₁))).toAffineSubspace
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `p₂ -ᵥ p₁` is orthogonal to
`c -ᵥ midpoint ℝ p₁ p₂`. -/
theorem mem_perpBisector_iff_inner_eq_zero' :
c ∈ perpBisector p₁ p₂ ↔ ⟪p₂ -ᵥ p₁, c -ᵥ midpoint ℝ p₁ p₂⟫ = 0 :=
Iff.rfl
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `c -ᵥ midpoint ℝ p₁ p₂` is
orthogonal to `p₂ -ᵥ p₁`. -/
theorem mem_perpBisector_iff_inner_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ midpoint ℝ p₁ p₂, p₂ -ᵥ p₁⟫ = 0 :=
inner_eq_zero_symm
theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, Equiv.pointReflection_apply,
vsub_midpoint, invOf_eq_inv, ← smul_add, real_inner_smul_left, vadd_vsub_assoc]
simp
theorem mem_perpBisector_pointReflection_iff_inner_eq_zero :
c ∈ perpBisector p₁ (Equiv.pointReflection p₂ p₁) ↔ ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, midpoint_pointReflection_right,
Equiv.pointReflection_apply, vadd_vsub_assoc, inner_add_right, add_self_eq_zero,
← neg_eq_zero, ← inner_neg_right, neg_vsub_eq_vsub_rev]
theorem midpoint_mem_perpBisector (p₁ p₂ : P) :
midpoint ℝ p₁ p₂ ∈ perpBisector p₁ p₂ := by
simp [mem_perpBisector_iff_inner_eq_zero]
theorem perpBisector_nonempty : (perpBisector p₁ p₂ : Set P).Nonempty :=
⟨_, midpoint_mem_perpBisector _ _⟩
@[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ := by
erw [perpBisector, comap_symm, map_direction, Submodule.map_id,
Submodule.toAffineSubspace_direction]
ext x
| exact Submodule.mem_orthogonal_singleton_iff_inner_right.symm | @[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ := by
erw [perpBisector, comap_symm, map_direction, Submodule.map_id,
Submodule.toAffineSubspace_direction]
ext x
| Mathlib.Geometry.Euclidean.PerpBisector.73_0.WKtplj3xHYGfYbJ | @[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ c ∈ perpBisector p₁ p₂ ↔ inner (c -ᵥ p₁) (p₂ -ᵥ p₁) = inner (c -ᵥ p₂) (p₁ -ᵥ p₂) | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
/-!
# Perpendicular bisector of a segment
We define `AffineSubspace.perpBisector p₁ p₂` to be the perpendicular bisector of the segment
`[p₁, p₂]`, as a bundled affine subspace. We also prove that a point belongs to the perpendicular
bisector if and only if it is equidistant from `p₁` and `p₂`, as well as a few linear equations that
define this subspace.
## Keywords
euclidean geometry, perpendicular, perpendicular bisector, line segment bisector, equidistant
-/
set_option autoImplicit true
open Set
open scoped BigOperators RealInnerProductSpace
variable [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
variable [NormedAddTorsor V P]
noncomputable section
namespace AffineSubspace
variable {c c₁ c₂ p₁ p₂ : P}
/-- Perpendicular bisector of a segment in a Euclidean affine space. -/
def perpBisector (p₁ p₂ : P) : AffineSubspace ℝ P :=
.comap ((AffineEquiv.vaddConst ℝ (midpoint ℝ p₁ p₂)).symm : P →ᵃ[ℝ] V) <|
(LinearMap.ker (innerₛₗ ℝ (p₂ -ᵥ p₁))).toAffineSubspace
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `p₂ -ᵥ p₁` is orthogonal to
`c -ᵥ midpoint ℝ p₁ p₂`. -/
theorem mem_perpBisector_iff_inner_eq_zero' :
c ∈ perpBisector p₁ p₂ ↔ ⟪p₂ -ᵥ p₁, c -ᵥ midpoint ℝ p₁ p₂⟫ = 0 :=
Iff.rfl
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `c -ᵥ midpoint ℝ p₁ p₂` is
orthogonal to `p₂ -ᵥ p₁`. -/
theorem mem_perpBisector_iff_inner_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ midpoint ℝ p₁ p₂, p₂ -ᵥ p₁⟫ = 0 :=
inner_eq_zero_symm
theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, Equiv.pointReflection_apply,
vsub_midpoint, invOf_eq_inv, ← smul_add, real_inner_smul_left, vadd_vsub_assoc]
simp
theorem mem_perpBisector_pointReflection_iff_inner_eq_zero :
c ∈ perpBisector p₁ (Equiv.pointReflection p₂ p₁) ↔ ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, midpoint_pointReflection_right,
Equiv.pointReflection_apply, vadd_vsub_assoc, inner_add_right, add_self_eq_zero,
← neg_eq_zero, ← inner_neg_right, neg_vsub_eq_vsub_rev]
theorem midpoint_mem_perpBisector (p₁ p₂ : P) :
midpoint ℝ p₁ p₂ ∈ perpBisector p₁ p₂ := by
simp [mem_perpBisector_iff_inner_eq_zero]
theorem perpBisector_nonempty : (perpBisector p₁ p₂ : Set P).Nonempty :=
⟨_, midpoint_mem_perpBisector _ _⟩
@[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ := by
erw [perpBisector, comap_symm, map_direction, Submodule.map_id,
Submodule.toAffineSubspace_direction]
ext x
exact Submodule.mem_orthogonal_singleton_iff_inner_right.symm
theorem mem_perpBisector_iff_inner_eq_inner :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ := by
| rw [Iff.comm, mem_perpBisector_iff_inner_eq_zero, ← add_neg_eq_zero, ← inner_neg_right,
neg_vsub_eq_vsub_rev, ← inner_add_left, vsub_midpoint, invOf_eq_inv, ← smul_add,
real_inner_smul_left] | theorem mem_perpBisector_iff_inner_eq_inner :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ := by
| Mathlib.Geometry.Euclidean.PerpBisector.81_0.WKtplj3xHYGfYbJ | theorem mem_perpBisector_iff_inner_eq_inner :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ inner (c -ᵥ p₁ + (c -ᵥ p₂)) (p₂ -ᵥ p₁) = 0 ↔ 2⁻¹ * inner (c -ᵥ p₁ + (c -ᵥ p₂)) (p₂ -ᵥ p₁) = 0 | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
/-!
# Perpendicular bisector of a segment
We define `AffineSubspace.perpBisector p₁ p₂` to be the perpendicular bisector of the segment
`[p₁, p₂]`, as a bundled affine subspace. We also prove that a point belongs to the perpendicular
bisector if and only if it is equidistant from `p₁` and `p₂`, as well as a few linear equations that
define this subspace.
## Keywords
euclidean geometry, perpendicular, perpendicular bisector, line segment bisector, equidistant
-/
set_option autoImplicit true
open Set
open scoped BigOperators RealInnerProductSpace
variable [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
variable [NormedAddTorsor V P]
noncomputable section
namespace AffineSubspace
variable {c c₁ c₂ p₁ p₂ : P}
/-- Perpendicular bisector of a segment in a Euclidean affine space. -/
def perpBisector (p₁ p₂ : P) : AffineSubspace ℝ P :=
.comap ((AffineEquiv.vaddConst ℝ (midpoint ℝ p₁ p₂)).symm : P →ᵃ[ℝ] V) <|
(LinearMap.ker (innerₛₗ ℝ (p₂ -ᵥ p₁))).toAffineSubspace
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `p₂ -ᵥ p₁` is orthogonal to
`c -ᵥ midpoint ℝ p₁ p₂`. -/
theorem mem_perpBisector_iff_inner_eq_zero' :
c ∈ perpBisector p₁ p₂ ↔ ⟪p₂ -ᵥ p₁, c -ᵥ midpoint ℝ p₁ p₂⟫ = 0 :=
Iff.rfl
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `c -ᵥ midpoint ℝ p₁ p₂` is
orthogonal to `p₂ -ᵥ p₁`. -/
theorem mem_perpBisector_iff_inner_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ midpoint ℝ p₁ p₂, p₂ -ᵥ p₁⟫ = 0 :=
inner_eq_zero_symm
theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, Equiv.pointReflection_apply,
vsub_midpoint, invOf_eq_inv, ← smul_add, real_inner_smul_left, vadd_vsub_assoc]
simp
theorem mem_perpBisector_pointReflection_iff_inner_eq_zero :
c ∈ perpBisector p₁ (Equiv.pointReflection p₂ p₁) ↔ ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, midpoint_pointReflection_right,
Equiv.pointReflection_apply, vadd_vsub_assoc, inner_add_right, add_self_eq_zero,
← neg_eq_zero, ← inner_neg_right, neg_vsub_eq_vsub_rev]
theorem midpoint_mem_perpBisector (p₁ p₂ : P) :
midpoint ℝ p₁ p₂ ∈ perpBisector p₁ p₂ := by
simp [mem_perpBisector_iff_inner_eq_zero]
theorem perpBisector_nonempty : (perpBisector p₁ p₂ : Set P).Nonempty :=
⟨_, midpoint_mem_perpBisector _ _⟩
@[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ := by
erw [perpBisector, comap_symm, map_direction, Submodule.map_id,
Submodule.toAffineSubspace_direction]
ext x
exact Submodule.mem_orthogonal_singleton_iff_inner_right.symm
theorem mem_perpBisector_iff_inner_eq_inner :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ := by
rw [Iff.comm, mem_perpBisector_iff_inner_eq_zero, ← add_neg_eq_zero, ← inner_neg_right,
neg_vsub_eq_vsub_rev, ← inner_add_left, vsub_midpoint, invOf_eq_inv, ← smul_add,
real_inner_smul_left]; | simp | theorem mem_perpBisector_iff_inner_eq_inner :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ := by
rw [Iff.comm, mem_perpBisector_iff_inner_eq_zero, ← add_neg_eq_zero, ← inner_neg_right,
neg_vsub_eq_vsub_rev, ← inner_add_left, vsub_midpoint, invOf_eq_inv, ← smul_add,
real_inner_smul_left]; | Mathlib.Geometry.Euclidean.PerpBisector.81_0.WKtplj3xHYGfYbJ | theorem mem_perpBisector_iff_inner_eq_inner :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ c ∈ perpBisector p₁ p₂ ↔ inner (c -ᵥ p₁) (p₂ -ᵥ p₁) = dist p₁ p₂ ^ 2 / 2 | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
/-!
# Perpendicular bisector of a segment
We define `AffineSubspace.perpBisector p₁ p₂` to be the perpendicular bisector of the segment
`[p₁, p₂]`, as a bundled affine subspace. We also prove that a point belongs to the perpendicular
bisector if and only if it is equidistant from `p₁` and `p₂`, as well as a few linear equations that
define this subspace.
## Keywords
euclidean geometry, perpendicular, perpendicular bisector, line segment bisector, equidistant
-/
set_option autoImplicit true
open Set
open scoped BigOperators RealInnerProductSpace
variable [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
variable [NormedAddTorsor V P]
noncomputable section
namespace AffineSubspace
variable {c c₁ c₂ p₁ p₂ : P}
/-- Perpendicular bisector of a segment in a Euclidean affine space. -/
def perpBisector (p₁ p₂ : P) : AffineSubspace ℝ P :=
.comap ((AffineEquiv.vaddConst ℝ (midpoint ℝ p₁ p₂)).symm : P →ᵃ[ℝ] V) <|
(LinearMap.ker (innerₛₗ ℝ (p₂ -ᵥ p₁))).toAffineSubspace
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `p₂ -ᵥ p₁` is orthogonal to
`c -ᵥ midpoint ℝ p₁ p₂`. -/
theorem mem_perpBisector_iff_inner_eq_zero' :
c ∈ perpBisector p₁ p₂ ↔ ⟪p₂ -ᵥ p₁, c -ᵥ midpoint ℝ p₁ p₂⟫ = 0 :=
Iff.rfl
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `c -ᵥ midpoint ℝ p₁ p₂` is
orthogonal to `p₂ -ᵥ p₁`. -/
theorem mem_perpBisector_iff_inner_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ midpoint ℝ p₁ p₂, p₂ -ᵥ p₁⟫ = 0 :=
inner_eq_zero_symm
theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, Equiv.pointReflection_apply,
vsub_midpoint, invOf_eq_inv, ← smul_add, real_inner_smul_left, vadd_vsub_assoc]
simp
theorem mem_perpBisector_pointReflection_iff_inner_eq_zero :
c ∈ perpBisector p₁ (Equiv.pointReflection p₂ p₁) ↔ ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, midpoint_pointReflection_right,
Equiv.pointReflection_apply, vadd_vsub_assoc, inner_add_right, add_self_eq_zero,
← neg_eq_zero, ← inner_neg_right, neg_vsub_eq_vsub_rev]
theorem midpoint_mem_perpBisector (p₁ p₂ : P) :
midpoint ℝ p₁ p₂ ∈ perpBisector p₁ p₂ := by
simp [mem_perpBisector_iff_inner_eq_zero]
theorem perpBisector_nonempty : (perpBisector p₁ p₂ : Set P).Nonempty :=
⟨_, midpoint_mem_perpBisector _ _⟩
@[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ := by
erw [perpBisector, comap_symm, map_direction, Submodule.map_id,
Submodule.toAffineSubspace_direction]
ext x
exact Submodule.mem_orthogonal_singleton_iff_inner_right.symm
theorem mem_perpBisector_iff_inner_eq_inner :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ := by
rw [Iff.comm, mem_perpBisector_iff_inner_eq_zero, ← add_neg_eq_zero, ← inner_neg_right,
neg_vsub_eq_vsub_rev, ← inner_add_left, vsub_midpoint, invOf_eq_inv, ← smul_add,
real_inner_smul_left]; simp
theorem mem_perpBisector_iff_inner_eq :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = (dist p₁ p₂) ^ 2 / 2 := by
| rw [mem_perpBisector_iff_inner_eq_zero, ← vsub_sub_vsub_cancel_right _ _ p₁, inner_sub_left,
sub_eq_zero, midpoint_vsub_left, invOf_eq_inv, real_inner_smul_left, real_inner_self_eq_norm_sq,
dist_eq_norm_vsub' V, div_eq_inv_mul] | theorem mem_perpBisector_iff_inner_eq :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = (dist p₁ p₂) ^ 2 / 2 := by
| Mathlib.Geometry.Euclidean.PerpBisector.87_0.WKtplj3xHYGfYbJ | theorem mem_perpBisector_iff_inner_eq :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = (dist p₁ p₂) ^ 2 / 2 | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ c ∈ perpBisector p₁ p₂ ↔ dist c p₁ = dist c p₂ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
/-!
# Perpendicular bisector of a segment
We define `AffineSubspace.perpBisector p₁ p₂` to be the perpendicular bisector of the segment
`[p₁, p₂]`, as a bundled affine subspace. We also prove that a point belongs to the perpendicular
bisector if and only if it is equidistant from `p₁` and `p₂`, as well as a few linear equations that
define this subspace.
## Keywords
euclidean geometry, perpendicular, perpendicular bisector, line segment bisector, equidistant
-/
set_option autoImplicit true
open Set
open scoped BigOperators RealInnerProductSpace
variable [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
variable [NormedAddTorsor V P]
noncomputable section
namespace AffineSubspace
variable {c c₁ c₂ p₁ p₂ : P}
/-- Perpendicular bisector of a segment in a Euclidean affine space. -/
def perpBisector (p₁ p₂ : P) : AffineSubspace ℝ P :=
.comap ((AffineEquiv.vaddConst ℝ (midpoint ℝ p₁ p₂)).symm : P →ᵃ[ℝ] V) <|
(LinearMap.ker (innerₛₗ ℝ (p₂ -ᵥ p₁))).toAffineSubspace
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `p₂ -ᵥ p₁` is orthogonal to
`c -ᵥ midpoint ℝ p₁ p₂`. -/
theorem mem_perpBisector_iff_inner_eq_zero' :
c ∈ perpBisector p₁ p₂ ↔ ⟪p₂ -ᵥ p₁, c -ᵥ midpoint ℝ p₁ p₂⟫ = 0 :=
Iff.rfl
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `c -ᵥ midpoint ℝ p₁ p₂` is
orthogonal to `p₂ -ᵥ p₁`. -/
theorem mem_perpBisector_iff_inner_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ midpoint ℝ p₁ p₂, p₂ -ᵥ p₁⟫ = 0 :=
inner_eq_zero_symm
theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, Equiv.pointReflection_apply,
vsub_midpoint, invOf_eq_inv, ← smul_add, real_inner_smul_left, vadd_vsub_assoc]
simp
theorem mem_perpBisector_pointReflection_iff_inner_eq_zero :
c ∈ perpBisector p₁ (Equiv.pointReflection p₂ p₁) ↔ ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, midpoint_pointReflection_right,
Equiv.pointReflection_apply, vadd_vsub_assoc, inner_add_right, add_self_eq_zero,
← neg_eq_zero, ← inner_neg_right, neg_vsub_eq_vsub_rev]
theorem midpoint_mem_perpBisector (p₁ p₂ : P) :
midpoint ℝ p₁ p₂ ∈ perpBisector p₁ p₂ := by
simp [mem_perpBisector_iff_inner_eq_zero]
theorem perpBisector_nonempty : (perpBisector p₁ p₂ : Set P).Nonempty :=
⟨_, midpoint_mem_perpBisector _ _⟩
@[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ := by
erw [perpBisector, comap_symm, map_direction, Submodule.map_id,
Submodule.toAffineSubspace_direction]
ext x
exact Submodule.mem_orthogonal_singleton_iff_inner_right.symm
theorem mem_perpBisector_iff_inner_eq_inner :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ := by
rw [Iff.comm, mem_perpBisector_iff_inner_eq_zero, ← add_neg_eq_zero, ← inner_neg_right,
neg_vsub_eq_vsub_rev, ← inner_add_left, vsub_midpoint, invOf_eq_inv, ← smul_add,
real_inner_smul_left]; simp
theorem mem_perpBisector_iff_inner_eq :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = (dist p₁ p₂) ^ 2 / 2 := by
rw [mem_perpBisector_iff_inner_eq_zero, ← vsub_sub_vsub_cancel_right _ _ p₁, inner_sub_left,
sub_eq_zero, midpoint_vsub_left, invOf_eq_inv, real_inner_smul_left, real_inner_self_eq_norm_sq,
dist_eq_norm_vsub' V, div_eq_inv_mul]
theorem mem_perpBisector_iff_dist_eq : c ∈ perpBisector p₁ p₂ ↔ dist c p₁ = dist c p₂ := by
| rw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, ← real_inner_add_sub_eq_zero_iff,
vsub_sub_vsub_cancel_left, inner_add_left, add_eq_zero_iff_eq_neg, ← inner_neg_right,
neg_vsub_eq_vsub_rev, mem_perpBisector_iff_inner_eq_inner] | theorem mem_perpBisector_iff_dist_eq : c ∈ perpBisector p₁ p₂ ↔ dist c p₁ = dist c p₂ := by
| Mathlib.Geometry.Euclidean.PerpBisector.93_0.WKtplj3xHYGfYbJ | theorem mem_perpBisector_iff_dist_eq : c ∈ perpBisector p₁ p₂ ↔ dist c p₁ = dist c p₂ | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ c ∈ perpBisector p₁ p₂ ↔ dist p₁ c = dist p₂ c | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
/-!
# Perpendicular bisector of a segment
We define `AffineSubspace.perpBisector p₁ p₂` to be the perpendicular bisector of the segment
`[p₁, p₂]`, as a bundled affine subspace. We also prove that a point belongs to the perpendicular
bisector if and only if it is equidistant from `p₁` and `p₂`, as well as a few linear equations that
define this subspace.
## Keywords
euclidean geometry, perpendicular, perpendicular bisector, line segment bisector, equidistant
-/
set_option autoImplicit true
open Set
open scoped BigOperators RealInnerProductSpace
variable [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
variable [NormedAddTorsor V P]
noncomputable section
namespace AffineSubspace
variable {c c₁ c₂ p₁ p₂ : P}
/-- Perpendicular bisector of a segment in a Euclidean affine space. -/
def perpBisector (p₁ p₂ : P) : AffineSubspace ℝ P :=
.comap ((AffineEquiv.vaddConst ℝ (midpoint ℝ p₁ p₂)).symm : P →ᵃ[ℝ] V) <|
(LinearMap.ker (innerₛₗ ℝ (p₂ -ᵥ p₁))).toAffineSubspace
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `p₂ -ᵥ p₁` is orthogonal to
`c -ᵥ midpoint ℝ p₁ p₂`. -/
theorem mem_perpBisector_iff_inner_eq_zero' :
c ∈ perpBisector p₁ p₂ ↔ ⟪p₂ -ᵥ p₁, c -ᵥ midpoint ℝ p₁ p₂⟫ = 0 :=
Iff.rfl
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `c -ᵥ midpoint ℝ p₁ p₂` is
orthogonal to `p₂ -ᵥ p₁`. -/
theorem mem_perpBisector_iff_inner_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ midpoint ℝ p₁ p₂, p₂ -ᵥ p₁⟫ = 0 :=
inner_eq_zero_symm
theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, Equiv.pointReflection_apply,
vsub_midpoint, invOf_eq_inv, ← smul_add, real_inner_smul_left, vadd_vsub_assoc]
simp
theorem mem_perpBisector_pointReflection_iff_inner_eq_zero :
c ∈ perpBisector p₁ (Equiv.pointReflection p₂ p₁) ↔ ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, midpoint_pointReflection_right,
Equiv.pointReflection_apply, vadd_vsub_assoc, inner_add_right, add_self_eq_zero,
← neg_eq_zero, ← inner_neg_right, neg_vsub_eq_vsub_rev]
theorem midpoint_mem_perpBisector (p₁ p₂ : P) :
midpoint ℝ p₁ p₂ ∈ perpBisector p₁ p₂ := by
simp [mem_perpBisector_iff_inner_eq_zero]
theorem perpBisector_nonempty : (perpBisector p₁ p₂ : Set P).Nonempty :=
⟨_, midpoint_mem_perpBisector _ _⟩
@[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ := by
erw [perpBisector, comap_symm, map_direction, Submodule.map_id,
Submodule.toAffineSubspace_direction]
ext x
exact Submodule.mem_orthogonal_singleton_iff_inner_right.symm
theorem mem_perpBisector_iff_inner_eq_inner :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ := by
rw [Iff.comm, mem_perpBisector_iff_inner_eq_zero, ← add_neg_eq_zero, ← inner_neg_right,
neg_vsub_eq_vsub_rev, ← inner_add_left, vsub_midpoint, invOf_eq_inv, ← smul_add,
real_inner_smul_left]; simp
theorem mem_perpBisector_iff_inner_eq :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = (dist p₁ p₂) ^ 2 / 2 := by
rw [mem_perpBisector_iff_inner_eq_zero, ← vsub_sub_vsub_cancel_right _ _ p₁, inner_sub_left,
sub_eq_zero, midpoint_vsub_left, invOf_eq_inv, real_inner_smul_left, real_inner_self_eq_norm_sq,
dist_eq_norm_vsub' V, div_eq_inv_mul]
theorem mem_perpBisector_iff_dist_eq : c ∈ perpBisector p₁ p₂ ↔ dist c p₁ = dist c p₂ := by
rw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, ← real_inner_add_sub_eq_zero_iff,
vsub_sub_vsub_cancel_left, inner_add_left, add_eq_zero_iff_eq_neg, ← inner_neg_right,
neg_vsub_eq_vsub_rev, mem_perpBisector_iff_inner_eq_inner]
theorem mem_perpBisector_iff_dist_eq' : c ∈ perpBisector p₁ p₂ ↔ dist p₁ c = dist p₂ c := by
| simp only [mem_perpBisector_iff_dist_eq, dist_comm] | theorem mem_perpBisector_iff_dist_eq' : c ∈ perpBisector p₁ p₂ ↔ dist p₁ c = dist p₂ c := by
| Mathlib.Geometry.Euclidean.PerpBisector.98_0.WKtplj3xHYGfYbJ | theorem mem_perpBisector_iff_dist_eq' : c ∈ perpBisector p₁ p₂ ↔ dist p₁ c = dist p₂ c | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁✝ p₂✝ p₁ p₂ : P
⊢ perpBisector p₁ p₂ = perpBisector p₂ p₁ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
/-!
# Perpendicular bisector of a segment
We define `AffineSubspace.perpBisector p₁ p₂` to be the perpendicular bisector of the segment
`[p₁, p₂]`, as a bundled affine subspace. We also prove that a point belongs to the perpendicular
bisector if and only if it is equidistant from `p₁` and `p₂`, as well as a few linear equations that
define this subspace.
## Keywords
euclidean geometry, perpendicular, perpendicular bisector, line segment bisector, equidistant
-/
set_option autoImplicit true
open Set
open scoped BigOperators RealInnerProductSpace
variable [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
variable [NormedAddTorsor V P]
noncomputable section
namespace AffineSubspace
variable {c c₁ c₂ p₁ p₂ : P}
/-- Perpendicular bisector of a segment in a Euclidean affine space. -/
def perpBisector (p₁ p₂ : P) : AffineSubspace ℝ P :=
.comap ((AffineEquiv.vaddConst ℝ (midpoint ℝ p₁ p₂)).symm : P →ᵃ[ℝ] V) <|
(LinearMap.ker (innerₛₗ ℝ (p₂ -ᵥ p₁))).toAffineSubspace
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `p₂ -ᵥ p₁` is orthogonal to
`c -ᵥ midpoint ℝ p₁ p₂`. -/
theorem mem_perpBisector_iff_inner_eq_zero' :
c ∈ perpBisector p₁ p₂ ↔ ⟪p₂ -ᵥ p₁, c -ᵥ midpoint ℝ p₁ p₂⟫ = 0 :=
Iff.rfl
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `c -ᵥ midpoint ℝ p₁ p₂` is
orthogonal to `p₂ -ᵥ p₁`. -/
theorem mem_perpBisector_iff_inner_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ midpoint ℝ p₁ p₂, p₂ -ᵥ p₁⟫ = 0 :=
inner_eq_zero_symm
theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, Equiv.pointReflection_apply,
vsub_midpoint, invOf_eq_inv, ← smul_add, real_inner_smul_left, vadd_vsub_assoc]
simp
theorem mem_perpBisector_pointReflection_iff_inner_eq_zero :
c ∈ perpBisector p₁ (Equiv.pointReflection p₂ p₁) ↔ ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, midpoint_pointReflection_right,
Equiv.pointReflection_apply, vadd_vsub_assoc, inner_add_right, add_self_eq_zero,
← neg_eq_zero, ← inner_neg_right, neg_vsub_eq_vsub_rev]
theorem midpoint_mem_perpBisector (p₁ p₂ : P) :
midpoint ℝ p₁ p₂ ∈ perpBisector p₁ p₂ := by
simp [mem_perpBisector_iff_inner_eq_zero]
theorem perpBisector_nonempty : (perpBisector p₁ p₂ : Set P).Nonempty :=
⟨_, midpoint_mem_perpBisector _ _⟩
@[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ := by
erw [perpBisector, comap_symm, map_direction, Submodule.map_id,
Submodule.toAffineSubspace_direction]
ext x
exact Submodule.mem_orthogonal_singleton_iff_inner_right.symm
theorem mem_perpBisector_iff_inner_eq_inner :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ := by
rw [Iff.comm, mem_perpBisector_iff_inner_eq_zero, ← add_neg_eq_zero, ← inner_neg_right,
neg_vsub_eq_vsub_rev, ← inner_add_left, vsub_midpoint, invOf_eq_inv, ← smul_add,
real_inner_smul_left]; simp
theorem mem_perpBisector_iff_inner_eq :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = (dist p₁ p₂) ^ 2 / 2 := by
rw [mem_perpBisector_iff_inner_eq_zero, ← vsub_sub_vsub_cancel_right _ _ p₁, inner_sub_left,
sub_eq_zero, midpoint_vsub_left, invOf_eq_inv, real_inner_smul_left, real_inner_self_eq_norm_sq,
dist_eq_norm_vsub' V, div_eq_inv_mul]
theorem mem_perpBisector_iff_dist_eq : c ∈ perpBisector p₁ p₂ ↔ dist c p₁ = dist c p₂ := by
rw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, ← real_inner_add_sub_eq_zero_iff,
vsub_sub_vsub_cancel_left, inner_add_left, add_eq_zero_iff_eq_neg, ← inner_neg_right,
neg_vsub_eq_vsub_rev, mem_perpBisector_iff_inner_eq_inner]
theorem mem_perpBisector_iff_dist_eq' : c ∈ perpBisector p₁ p₂ ↔ dist p₁ c = dist p₂ c := by
simp only [mem_perpBisector_iff_dist_eq, dist_comm]
theorem perpBisector_comm (p₁ p₂ : P) : perpBisector p₁ p₂ = perpBisector p₂ p₁ := by
| ext c | theorem perpBisector_comm (p₁ p₂ : P) : perpBisector p₁ p₂ = perpBisector p₂ p₁ := by
| Mathlib.Geometry.Euclidean.PerpBisector.101_0.WKtplj3xHYGfYbJ | theorem perpBisector_comm (p₁ p₂ : P) : perpBisector p₁ p₂ = perpBisector p₂ p₁ | Mathlib_Geometry_Euclidean_PerpBisector |
case h
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c✝ c₁ c₂ p₁✝ p₂✝ p₁ p₂ c : P
⊢ c ∈ perpBisector p₁ p₂ ↔ c ∈ perpBisector p₂ p₁ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
/-!
# Perpendicular bisector of a segment
We define `AffineSubspace.perpBisector p₁ p₂` to be the perpendicular bisector of the segment
`[p₁, p₂]`, as a bundled affine subspace. We also prove that a point belongs to the perpendicular
bisector if and only if it is equidistant from `p₁` and `p₂`, as well as a few linear equations that
define this subspace.
## Keywords
euclidean geometry, perpendicular, perpendicular bisector, line segment bisector, equidistant
-/
set_option autoImplicit true
open Set
open scoped BigOperators RealInnerProductSpace
variable [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
variable [NormedAddTorsor V P]
noncomputable section
namespace AffineSubspace
variable {c c₁ c₂ p₁ p₂ : P}
/-- Perpendicular bisector of a segment in a Euclidean affine space. -/
def perpBisector (p₁ p₂ : P) : AffineSubspace ℝ P :=
.comap ((AffineEquiv.vaddConst ℝ (midpoint ℝ p₁ p₂)).symm : P →ᵃ[ℝ] V) <|
(LinearMap.ker (innerₛₗ ℝ (p₂ -ᵥ p₁))).toAffineSubspace
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `p₂ -ᵥ p₁` is orthogonal to
`c -ᵥ midpoint ℝ p₁ p₂`. -/
theorem mem_perpBisector_iff_inner_eq_zero' :
c ∈ perpBisector p₁ p₂ ↔ ⟪p₂ -ᵥ p₁, c -ᵥ midpoint ℝ p₁ p₂⟫ = 0 :=
Iff.rfl
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `c -ᵥ midpoint ℝ p₁ p₂` is
orthogonal to `p₂ -ᵥ p₁`. -/
theorem mem_perpBisector_iff_inner_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ midpoint ℝ p₁ p₂, p₂ -ᵥ p₁⟫ = 0 :=
inner_eq_zero_symm
theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, Equiv.pointReflection_apply,
vsub_midpoint, invOf_eq_inv, ← smul_add, real_inner_smul_left, vadd_vsub_assoc]
simp
theorem mem_perpBisector_pointReflection_iff_inner_eq_zero :
c ∈ perpBisector p₁ (Equiv.pointReflection p₂ p₁) ↔ ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, midpoint_pointReflection_right,
Equiv.pointReflection_apply, vadd_vsub_assoc, inner_add_right, add_self_eq_zero,
← neg_eq_zero, ← inner_neg_right, neg_vsub_eq_vsub_rev]
theorem midpoint_mem_perpBisector (p₁ p₂ : P) :
midpoint ℝ p₁ p₂ ∈ perpBisector p₁ p₂ := by
simp [mem_perpBisector_iff_inner_eq_zero]
theorem perpBisector_nonempty : (perpBisector p₁ p₂ : Set P).Nonempty :=
⟨_, midpoint_mem_perpBisector _ _⟩
@[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ := by
erw [perpBisector, comap_symm, map_direction, Submodule.map_id,
Submodule.toAffineSubspace_direction]
ext x
exact Submodule.mem_orthogonal_singleton_iff_inner_right.symm
theorem mem_perpBisector_iff_inner_eq_inner :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ := by
rw [Iff.comm, mem_perpBisector_iff_inner_eq_zero, ← add_neg_eq_zero, ← inner_neg_right,
neg_vsub_eq_vsub_rev, ← inner_add_left, vsub_midpoint, invOf_eq_inv, ← smul_add,
real_inner_smul_left]; simp
theorem mem_perpBisector_iff_inner_eq :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = (dist p₁ p₂) ^ 2 / 2 := by
rw [mem_perpBisector_iff_inner_eq_zero, ← vsub_sub_vsub_cancel_right _ _ p₁, inner_sub_left,
sub_eq_zero, midpoint_vsub_left, invOf_eq_inv, real_inner_smul_left, real_inner_self_eq_norm_sq,
dist_eq_norm_vsub' V, div_eq_inv_mul]
theorem mem_perpBisector_iff_dist_eq : c ∈ perpBisector p₁ p₂ ↔ dist c p₁ = dist c p₂ := by
rw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, ← real_inner_add_sub_eq_zero_iff,
vsub_sub_vsub_cancel_left, inner_add_left, add_eq_zero_iff_eq_neg, ← inner_neg_right,
neg_vsub_eq_vsub_rev, mem_perpBisector_iff_inner_eq_inner]
theorem mem_perpBisector_iff_dist_eq' : c ∈ perpBisector p₁ p₂ ↔ dist p₁ c = dist p₂ c := by
simp only [mem_perpBisector_iff_dist_eq, dist_comm]
theorem perpBisector_comm (p₁ p₂ : P) : perpBisector p₁ p₂ = perpBisector p₂ p₁ := by
ext c; | simp only [mem_perpBisector_iff_dist_eq, eq_comm] | theorem perpBisector_comm (p₁ p₂ : P) : perpBisector p₁ p₂ = perpBisector p₂ p₁ := by
ext c; | Mathlib.Geometry.Euclidean.PerpBisector.101_0.WKtplj3xHYGfYbJ | theorem perpBisector_comm (p₁ p₂ : P) : perpBisector p₁ p₂ = perpBisector p₂ p₁ | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ p₂ ∈ perpBisector p₁ p₂ ↔ p₁ = p₂ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
/-!
# Perpendicular bisector of a segment
We define `AffineSubspace.perpBisector p₁ p₂` to be the perpendicular bisector of the segment
`[p₁, p₂]`, as a bundled affine subspace. We also prove that a point belongs to the perpendicular
bisector if and only if it is equidistant from `p₁` and `p₂`, as well as a few linear equations that
define this subspace.
## Keywords
euclidean geometry, perpendicular, perpendicular bisector, line segment bisector, equidistant
-/
set_option autoImplicit true
open Set
open scoped BigOperators RealInnerProductSpace
variable [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
variable [NormedAddTorsor V P]
noncomputable section
namespace AffineSubspace
variable {c c₁ c₂ p₁ p₂ : P}
/-- Perpendicular bisector of a segment in a Euclidean affine space. -/
def perpBisector (p₁ p₂ : P) : AffineSubspace ℝ P :=
.comap ((AffineEquiv.vaddConst ℝ (midpoint ℝ p₁ p₂)).symm : P →ᵃ[ℝ] V) <|
(LinearMap.ker (innerₛₗ ℝ (p₂ -ᵥ p₁))).toAffineSubspace
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `p₂ -ᵥ p₁` is orthogonal to
`c -ᵥ midpoint ℝ p₁ p₂`. -/
theorem mem_perpBisector_iff_inner_eq_zero' :
c ∈ perpBisector p₁ p₂ ↔ ⟪p₂ -ᵥ p₁, c -ᵥ midpoint ℝ p₁ p₂⟫ = 0 :=
Iff.rfl
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `c -ᵥ midpoint ℝ p₁ p₂` is
orthogonal to `p₂ -ᵥ p₁`. -/
theorem mem_perpBisector_iff_inner_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ midpoint ℝ p₁ p₂, p₂ -ᵥ p₁⟫ = 0 :=
inner_eq_zero_symm
theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, Equiv.pointReflection_apply,
vsub_midpoint, invOf_eq_inv, ← smul_add, real_inner_smul_left, vadd_vsub_assoc]
simp
theorem mem_perpBisector_pointReflection_iff_inner_eq_zero :
c ∈ perpBisector p₁ (Equiv.pointReflection p₂ p₁) ↔ ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, midpoint_pointReflection_right,
Equiv.pointReflection_apply, vadd_vsub_assoc, inner_add_right, add_self_eq_zero,
← neg_eq_zero, ← inner_neg_right, neg_vsub_eq_vsub_rev]
theorem midpoint_mem_perpBisector (p₁ p₂ : P) :
midpoint ℝ p₁ p₂ ∈ perpBisector p₁ p₂ := by
simp [mem_perpBisector_iff_inner_eq_zero]
theorem perpBisector_nonempty : (perpBisector p₁ p₂ : Set P).Nonempty :=
⟨_, midpoint_mem_perpBisector _ _⟩
@[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ := by
erw [perpBisector, comap_symm, map_direction, Submodule.map_id,
Submodule.toAffineSubspace_direction]
ext x
exact Submodule.mem_orthogonal_singleton_iff_inner_right.symm
theorem mem_perpBisector_iff_inner_eq_inner :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ := by
rw [Iff.comm, mem_perpBisector_iff_inner_eq_zero, ← add_neg_eq_zero, ← inner_neg_right,
neg_vsub_eq_vsub_rev, ← inner_add_left, vsub_midpoint, invOf_eq_inv, ← smul_add,
real_inner_smul_left]; simp
theorem mem_perpBisector_iff_inner_eq :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = (dist p₁ p₂) ^ 2 / 2 := by
rw [mem_perpBisector_iff_inner_eq_zero, ← vsub_sub_vsub_cancel_right _ _ p₁, inner_sub_left,
sub_eq_zero, midpoint_vsub_left, invOf_eq_inv, real_inner_smul_left, real_inner_self_eq_norm_sq,
dist_eq_norm_vsub' V, div_eq_inv_mul]
theorem mem_perpBisector_iff_dist_eq : c ∈ perpBisector p₁ p₂ ↔ dist c p₁ = dist c p₂ := by
rw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, ← real_inner_add_sub_eq_zero_iff,
vsub_sub_vsub_cancel_left, inner_add_left, add_eq_zero_iff_eq_neg, ← inner_neg_right,
neg_vsub_eq_vsub_rev, mem_perpBisector_iff_inner_eq_inner]
theorem mem_perpBisector_iff_dist_eq' : c ∈ perpBisector p₁ p₂ ↔ dist p₁ c = dist p₂ c := by
simp only [mem_perpBisector_iff_dist_eq, dist_comm]
theorem perpBisector_comm (p₁ p₂ : P) : perpBisector p₁ p₂ = perpBisector p₂ p₁ := by
ext c; simp only [mem_perpBisector_iff_dist_eq, eq_comm]
@[simp] theorem right_mem_perpBisector : p₂ ∈ perpBisector p₁ p₂ ↔ p₁ = p₂ := by
| simpa [mem_perpBisector_iff_inner_eq_inner] using eq_comm | @[simp] theorem right_mem_perpBisector : p₂ ∈ perpBisector p₁ p₂ ↔ p₁ = p₂ := by
| Mathlib.Geometry.Euclidean.PerpBisector.104_0.WKtplj3xHYGfYbJ | @[simp] theorem right_mem_perpBisector : p₂ ∈ perpBisector p₁ p₂ ↔ p₁ = p₂ | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ p₁ ∈ perpBisector p₁ p₂ ↔ p₁ = p₂ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
/-!
# Perpendicular bisector of a segment
We define `AffineSubspace.perpBisector p₁ p₂` to be the perpendicular bisector of the segment
`[p₁, p₂]`, as a bundled affine subspace. We also prove that a point belongs to the perpendicular
bisector if and only if it is equidistant from `p₁` and `p₂`, as well as a few linear equations that
define this subspace.
## Keywords
euclidean geometry, perpendicular, perpendicular bisector, line segment bisector, equidistant
-/
set_option autoImplicit true
open Set
open scoped BigOperators RealInnerProductSpace
variable [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
variable [NormedAddTorsor V P]
noncomputable section
namespace AffineSubspace
variable {c c₁ c₂ p₁ p₂ : P}
/-- Perpendicular bisector of a segment in a Euclidean affine space. -/
def perpBisector (p₁ p₂ : P) : AffineSubspace ℝ P :=
.comap ((AffineEquiv.vaddConst ℝ (midpoint ℝ p₁ p₂)).symm : P →ᵃ[ℝ] V) <|
(LinearMap.ker (innerₛₗ ℝ (p₂ -ᵥ p₁))).toAffineSubspace
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `p₂ -ᵥ p₁` is orthogonal to
`c -ᵥ midpoint ℝ p₁ p₂`. -/
theorem mem_perpBisector_iff_inner_eq_zero' :
c ∈ perpBisector p₁ p₂ ↔ ⟪p₂ -ᵥ p₁, c -ᵥ midpoint ℝ p₁ p₂⟫ = 0 :=
Iff.rfl
/-- A point `c` belongs the perpendicular bisector of `[p₁, p₂] iff `c -ᵥ midpoint ℝ p₁ p₂` is
orthogonal to `p₂ -ᵥ p₁`. -/
theorem mem_perpBisector_iff_inner_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ midpoint ℝ p₁ p₂, p₂ -ᵥ p₁⟫ = 0 :=
inner_eq_zero_symm
theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, Equiv.pointReflection_apply,
vsub_midpoint, invOf_eq_inv, ← smul_add, real_inner_smul_left, vadd_vsub_assoc]
simp
theorem mem_perpBisector_pointReflection_iff_inner_eq_zero :
c ∈ perpBisector p₁ (Equiv.pointReflection p₂ p₁) ↔ ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ = 0 := by
rw [mem_perpBisector_iff_inner_eq_zero, midpoint_pointReflection_right,
Equiv.pointReflection_apply, vadd_vsub_assoc, inner_add_right, add_self_eq_zero,
← neg_eq_zero, ← inner_neg_right, neg_vsub_eq_vsub_rev]
theorem midpoint_mem_perpBisector (p₁ p₂ : P) :
midpoint ℝ p₁ p₂ ∈ perpBisector p₁ p₂ := by
simp [mem_perpBisector_iff_inner_eq_zero]
theorem perpBisector_nonempty : (perpBisector p₁ p₂ : Set P).Nonempty :=
⟨_, midpoint_mem_perpBisector _ _⟩
@[simp]
theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ := by
erw [perpBisector, comap_symm, map_direction, Submodule.map_id,
Submodule.toAffineSubspace_direction]
ext x
exact Submodule.mem_orthogonal_singleton_iff_inner_right.symm
theorem mem_perpBisector_iff_inner_eq_inner :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ := by
rw [Iff.comm, mem_perpBisector_iff_inner_eq_zero, ← add_neg_eq_zero, ← inner_neg_right,
neg_vsub_eq_vsub_rev, ← inner_add_left, vsub_midpoint, invOf_eq_inv, ← smul_add,
real_inner_smul_left]; simp
theorem mem_perpBisector_iff_inner_eq :
c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = (dist p₁ p₂) ^ 2 / 2 := by
rw [mem_perpBisector_iff_inner_eq_zero, ← vsub_sub_vsub_cancel_right _ _ p₁, inner_sub_left,
sub_eq_zero, midpoint_vsub_left, invOf_eq_inv, real_inner_smul_left, real_inner_self_eq_norm_sq,
dist_eq_norm_vsub' V, div_eq_inv_mul]
theorem mem_perpBisector_iff_dist_eq : c ∈ perpBisector p₁ p₂ ↔ dist c p₁ = dist c p₂ := by
rw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, ← real_inner_add_sub_eq_zero_iff,
vsub_sub_vsub_cancel_left, inner_add_left, add_eq_zero_iff_eq_neg, ← inner_neg_right,
neg_vsub_eq_vsub_rev, mem_perpBisector_iff_inner_eq_inner]
theorem mem_perpBisector_iff_dist_eq' : c ∈ perpBisector p₁ p₂ ↔ dist p₁ c = dist p₂ c := by
simp only [mem_perpBisector_iff_dist_eq, dist_comm]
theorem perpBisector_comm (p₁ p₂ : P) : perpBisector p₁ p₂ = perpBisector p₂ p₁ := by
ext c; simp only [mem_perpBisector_iff_dist_eq, eq_comm]
@[simp] theorem right_mem_perpBisector : p₂ ∈ perpBisector p₁ p₂ ↔ p₁ = p₂ := by
simpa [mem_perpBisector_iff_inner_eq_inner] using eq_comm
@[simp] theorem left_mem_perpBisector : p₁ ∈ perpBisector p₁ p₂ ↔ p₁ = p₂ := by
| rw [perpBisector_comm, right_mem_perpBisector, eq_comm] | @[simp] theorem left_mem_perpBisector : p₁ ∈ perpBisector p₁ p₂ ↔ p₁ = p₂ := by
| Mathlib.Geometry.Euclidean.PerpBisector.107_0.WKtplj3xHYGfYbJ | @[simp] theorem left_mem_perpBisector : p₁ ∈ perpBisector p₁ p₂ ↔ p₁ = p₂ | Mathlib_Geometry_Euclidean_PerpBisector |
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