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|---|---|---|---|---|---|---|---|---|---|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/EulerProduct.lean | DirichletCharacter.LSeries_eulerProduct | [94, 1] | [99, 87] | refine Tendsto.congr (fun n β¦ Finset.prod_congr rfl fun p hp β¦ ?_) <|
eulerProduct_of_completelyMultiplicative (toFun_on_nat_map_one Ο) (toFun_on_nat_map_mul Ο) <|
LSeriesSummable_of_one_lt_re Ο hs | N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
β’ Tendsto (fun n => β p β n.primesBelow, (1 - Ο βp * βp ^ (-s))β»ΒΉ) atTop (π (L (fun n => Ο βn) s)) | N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
n p : β
hp : p β n.primesBelow
β’ (1 - term (fun n => Ο βn) s p)β»ΒΉ = (1 - Ο βp * βp ^ (-s))β»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
β’ Tendsto (fun n => β p β n.primesBelow, (1 - Ο βp * βp ^ (-s))β»ΒΉ) atTop (π (L (fun n => Ο βn) s))
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/EulerProduct.lean | DirichletCharacter.LSeries_eulerProduct | [94, 1] | [99, 87] | rw [term_of_ne_zero (prime_of_mem_primesBelow hp).ne_zero, cpow_neg, div_eq_mul_inv] | N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
n p : β
hp : p β n.primesBelow
β’ (1 - term (fun n => Ο βn) s p)β»ΒΉ = (1 - Ο βp * βp ^ (-s))β»ΒΉ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
n p : β
hp : p β n.primesBelow
β’ (1 - term (fun n => Ο βn) s p)β»ΒΉ = (1 - Ο βp * βp ^ (-s))β»ΒΉ
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/EulerProduct.lean | DirichletCharacter.LSeries_eulerProduct' | [102, 1] | [110, 61] | rw [LSeries] | N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
β’ cexp (β' (p : Primes), -(1 - Ο ββp * ββp ^ (-s)).log) = L (fun n => Ο βn) s | N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
β’ cexp (β' (p : Primes), -(1 - Ο ββp * ββp ^ (-s)).log) = β' (n : β), term (fun n => Ο βn) s n | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
β’ cexp (β' (p : Primes), -(1 - Ο ββp * ββp ^ (-s)).log) = L (fun n => Ο βn) s
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/EulerProduct.lean | DirichletCharacter.LSeries_eulerProduct' | [102, 1] | [110, 61] | convert exp_sum_primes_log_eq_tsum (f := dirichletSummandHom Ο <| ne_zero_of_one_lt_re hs) <|
summable_dirichletSummand Ο hs | N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
β’ cexp (β' (p : Primes), -(1 - Ο ββp * ββp ^ (-s)).log) = β' (n : β), term (fun n => Ο βn) s n | case h.e'_3.h.e'_5.h.h.e
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ : β
β’ term (fun n => Ο βn) s = β(dirichletSummandHom Ο β―) | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
β’ cexp (β' (p : Primes), -(1 - Ο ββp * ββp ^ (-s)).log) = β' (n : β), term (fun n => Ο βn) s n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/EulerProduct.lean | DirichletCharacter.LSeries_eulerProduct' | [102, 1] | [110, 61] | ext n | case h.e'_3.h.e'_5.h.h.e
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ : β
β’ term (fun n => Ο βn) s = β(dirichletSummandHom Ο β―) | case h.e'_3.h.e'_5.h.h.e.h
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ n : β
β’ term (fun n => Ο βn) s n = (dirichletSummandHom Ο β―) n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_5.h.h.e
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ : β
β’ term (fun n => Ο βn) s = β(dirichletSummandHom Ο β―)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/EulerProduct.lean | DirichletCharacter.LSeries_eulerProduct' | [102, 1] | [110, 61] | rcases eq_or_ne n 0 with rfl | hn | case h.e'_3.h.e'_5.h.h.e.h
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ n : β
β’ term (fun n => Ο βn) s n = (dirichletSummandHom Ο β―) n | case h.e'_3.h.e'_5.h.h.e.h.inl
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ : β
β’ term (fun n => Ο βn) s 0 = (dirichletSummandHom Ο β―) 0
case h.e'_3.h.e'_5.h.h.e.h.inr
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ n : β
hn : n β 0
β’ term (fun n => Ο βn) s n = (dirichletSummandHom Ο β―) n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_5.h.h.e.h
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ n : β
β’ term (fun n => Ο βn) s n = (dirichletSummandHom Ο β―) n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/EulerProduct.lean | DirichletCharacter.LSeries_eulerProduct' | [102, 1] | [110, 61] | simp only [term_zero, map_zero] | case h.e'_3.h.e'_5.h.h.e.h.inl
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ : β
β’ term (fun n => Ο βn) s 0 = (dirichletSummandHom Ο β―) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_5.h.h.e.h.inl
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ : β
β’ term (fun n => Ο βn) s 0 = (dirichletSummandHom Ο β―) 0
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/EulerProduct.lean | DirichletCharacter.LSeries_eulerProduct' | [102, 1] | [110, 61] | simp [hn, dirichletSummandHom, div_eq_mul_inv, cpow_neg] | case h.e'_3.h.e'_5.h.h.e.h.inr
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ n : β
hn : n β 0
β’ term (fun n => Ο βn) s n = (dirichletSummandHom Ο β―) n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_5.h.h.e.h.inr
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ n : β
hn : n β 0
β’ term (fun n => Ο βn) s n = (dirichletSummandHom Ο β―) n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/EulerProduct.lean | ArithmeticFunction.LSeries_zeta_eulerProduct' | [117, 1] | [120, 62] | convert modOne_eq_one (R := β) βΈ LSeries_eulerProduct' (1 : DirichletCharacter β 1) hs using 7 | s : β
hs : 1 < s.re
β’ cexp (β' (p : Primes), -(1 - ββp ^ (-s)).log) = L 1 s | case h.e'_2.h.e'_1.h.e'_5.h.h.e'_3.h.e'_1.h.e'_6
s : β
hs : 1 < s.re
xβ : Primes
β’ ββxβ ^ (-s) = 1 ββxβ * ββxβ ^ (-s) | Please generate a tactic in lean4 to solve the state.
STATE:
s : β
hs : 1 < s.re
β’ cexp (β' (p : Primes), -(1 - ββp ^ (-s)).log) = L 1 s
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/EulerProduct.lean | ArithmeticFunction.LSeries_zeta_eulerProduct' | [117, 1] | [120, 62] | rw [MulChar.one_apply <| isUnit_of_subsingleton _, one_mul] | case h.e'_2.h.e'_1.h.e'_5.h.h.e'_3.h.e'_1.h.e'_6
s : β
hs : 1 < s.re
xβ : Primes
β’ ββxβ ^ (-s) = 1 ββxβ * ββxβ ^ (-s) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_1.h.e'_5.h.h.e'_3.h.e'_1.h.e'_6
s : β
hs : 1 < s.re
xβ : Primes
β’ ββxβ ^ (-s) = 1 ββxβ * ββxβ ^ (-s)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | LSeriesSummable.mul_bounded | [30, 1] | [39, 61] | refine Summable.of_norm <| (hs.const_smul c).norm.of_nonneg_of_le (fun _ β¦ norm_nonneg _) fun n β¦ ?_ | f g : β β β
c : β
s : β
hs : LSeriesSummable f s
hg : β (n : β), βg nβ β€ c
β’ LSeriesSummable (f * g) s | f g : β β β
c : β
s : β
hs : LSeriesSummable f s
hg : β (n : β), βg nβ β€ c
n : β
β’ βLSeries.term (f * g) s nβ β€ βc β’ LSeries.term f s nβ | Please generate a tactic in lean4 to solve the state.
STATE:
f g : β β β
c : β
s : β
hs : LSeriesSummable f s
hg : β (n : β), βg nβ β€ c
β’ LSeriesSummable (f * g) s
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | LSeriesSummable.mul_bounded | [30, 1] | [39, 61] | rw [Complex.real_smul, β LSeries.term_smul_apply, mul_comm] | f g : β β β
c : β
s : β
hs : LSeriesSummable f s
hg : β (n : β), βg nβ β€ c
n : β
β’ βLSeries.term (f * g) s nβ β€ βc β’ LSeries.term f s nβ | f g : β β β
c : β
s : β
hs : LSeriesSummable f s
hg : β (n : β), βg nβ β€ c
n : β
β’ βLSeries.term (g * f) s nβ β€ βLSeries.term (βc β’ f) s nβ | Please generate a tactic in lean4 to solve the state.
STATE:
f g : β β β
c : β
s : β
hs : LSeriesSummable f s
hg : β (n : β), βg nβ β€ c
n : β
β’ βLSeries.term (f * g) s nβ β€ βc β’ LSeries.term f s nβ
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | LSeriesSummable.mul_bounded | [30, 1] | [39, 61] | refine LSeries.norm_term_le s ?_ | f g : β β β
c : β
s : β
hs : LSeriesSummable f s
hg : β (n : β), βg nβ β€ c
n : β
β’ βLSeries.term (g * f) s nβ β€ βLSeries.term (βc β’ f) s nβ | f g : β β β
c : β
s : β
hs : LSeriesSummable f s
hg : β (n : β), βg nβ β€ c
n : β
β’ β(g * f) nβ β€ β(βc β’ f) nβ | Please generate a tactic in lean4 to solve the state.
STATE:
f g : β β β
c : β
s : β
hs : LSeriesSummable f s
hg : β (n : β), βg nβ β€ c
n : β
β’ βLSeries.term (g * f) s nβ β€ βLSeries.term (βc β’ f) s nβ
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | LSeriesSummable.mul_bounded | [30, 1] | [39, 61] | have hc : β(c : β)β = c := by
simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self, (norm_nonneg _).trans (hg 0)] | f g : β β β
c : β
s : β
hs : LSeriesSummable f s
hg : β (n : β), βg nβ β€ c
n : β
β’ β(g * f) nβ β€ β(βc β’ f) nβ | f g : β β β
c : β
s : β
hs : LSeriesSummable f s
hg : β (n : β), βg nβ β€ c
n : β
hc : ββcβ = c
β’ β(g * f) nβ β€ β(βc β’ f) nβ | Please generate a tactic in lean4 to solve the state.
STATE:
f g : β β β
c : β
s : β
hs : LSeriesSummable f s
hg : β (n : β), βg nβ β€ c
n : β
β’ β(g * f) nβ β€ β(βc β’ f) nβ
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | LSeriesSummable.mul_bounded | [30, 1] | [39, 61] | simpa only [Pi.mul_apply, norm_mul, Pi.smul_apply, smul_eq_mul, hc]
using mul_le_mul_of_nonneg_right (hg n) <| norm_nonneg _ | f g : β β β
c : β
s : β
hs : LSeriesSummable f s
hg : β (n : β), βg nβ β€ c
n : β
hc : ββcβ = c
β’ β(g * f) nβ β€ β(βc β’ f) nβ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f g : β β β
c : β
s : β
hs : LSeriesSummable f s
hg : β (n : β), βg nβ β€ c
n : β
hc : ββcβ = c
β’ β(g * f) nβ β€ β(βc β’ f) nβ
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | LSeriesSummable.mul_bounded | [30, 1] | [39, 61] | simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self, (norm_nonneg _).trans (hg 0)] | f g : β β β
c : β
s : β
hs : LSeriesSummable f s
hg : β (n : β), βg nβ β€ c
n : β
β’ ββcβ = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f g : β β β
c : β
s : β
hs : LSeriesSummable f s
hg : β (n : β), βg nβ β€ c
n : β
β’ ββcβ = c
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | LSeriesSummable.mul_moebius | [42, 1] | [46, 36] | refine hf.mul_bounded (c := 1) fun n β¦ ?_ | f : β β β
s : β
hf : LSeriesSummable f s
β’ LSeriesSummable (f * fun n => β(ΞΌ n)) s | f : β β β
s : β
hf : LSeriesSummable f s
n : β
β’ ββ(ΞΌ n)β β€ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
s : β
hf : LSeriesSummable f s
β’ LSeriesSummable (f * fun n => β(ΞΌ n)) s
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | LSeriesSummable.mul_moebius | [42, 1] | [46, 36] | simp only [Complex.norm_int] | f : β β β
s : β
hf : LSeriesSummable f s
n : β
β’ ββ(ΞΌ n)β β€ 1 | f : β β β
s : β
hf : LSeriesSummable f s
n : β
β’ |β(ΞΌ n)| β€ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
s : β
hf : LSeriesSummable f s
n : β
β’ ββ(ΞΌ n)β β€ 1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | LSeriesSummable.mul_moebius | [42, 1] | [46, 36] | exact_mod_cast abs_moebius_le_one | f : β β β
s : β
hf : LSeriesSummable f s
n : β
β’ |β(ΞΌ n)| β€ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
s : β
hf : LSeriesSummable f s
n : β
β’ |β(ΞΌ n)| β€ 1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | LSeries.mul_convolution_distrib | [51, 1] | [59, 28] | ext n | R : Type u_1
instβ : CommSemiring R
Ο : β β R
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
f g : β β R
β’ Ο * (f β g) = Ο * f β (Ο * g) | case h
R : Type u_1
instβ : CommSemiring R
Ο : β β R
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
f g : β β R
n : β
β’ (Ο * (f β g)) n = (Ο * f β (Ο * g)) n | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : CommSemiring R
Ο : β β R
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
f g : β β R
β’ Ο * (f β g) = Ο * f β (Ο * g)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | LSeries.mul_convolution_distrib | [51, 1] | [59, 28] | simp only [Pi.mul_apply, LSeries.convolution_def, Finset.mul_sum] | case h
R : Type u_1
instβ : CommSemiring R
Ο : β β R
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
f g : β β R
n : β
β’ (Ο * (f β g)) n = (Ο * f β (Ο * g)) n | case h
R : Type u_1
instβ : CommSemiring R
Ο : β β R
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
f g : β β R
n : β
β’ β i β n.divisorsAntidiagonal, Ο n * (f i.1 * g i.2) = β x β n.divisorsAntidiagonal, Ο x.1 * f x.1 * (Ο x.2 * g x.2) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
instβ : CommSemiring R
Ο : β β R
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
f g : β β R
n : β
β’ (Ο * (f β g)) n = (Ο * f β (Ο * g)) n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | LSeries.mul_convolution_distrib | [51, 1] | [59, 28] | refine Finset.sum_congr rfl fun p hp β¦ ?_ | case h
R : Type u_1
instβ : CommSemiring R
Ο : β β R
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
f g : β β R
n : β
β’ β i β n.divisorsAntidiagonal, Ο n * (f i.1 * g i.2) = β x β n.divisorsAntidiagonal, Ο x.1 * f x.1 * (Ο x.2 * g x.2) | case h
R : Type u_1
instβ : CommSemiring R
Ο : β β R
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
f g : β β R
n : β
p : β Γ β
hp : p β n.divisorsAntidiagonal
β’ Ο n * (f p.1 * g p.2) = Ο p.1 * f p.1 * (Ο p.2 * g p.2) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
instβ : CommSemiring R
Ο : β β R
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
f g : β β R
n : β
β’ β i β n.divisorsAntidiagonal, Ο n * (f i.1 * g i.2) = β x β n.divisorsAntidiagonal, Ο x.1 * f x.1 * (Ο x.2 * g x.2)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | LSeries.mul_convolution_distrib | [51, 1] | [59, 28] | rw [(Nat.mem_divisorsAntidiagonal.mp hp).1.symm, hΟ] | case h
R : Type u_1
instβ : CommSemiring R
Ο : β β R
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
f g : β β R
n : β
p : β Γ β
hp : p β n.divisorsAntidiagonal
β’ Ο n * (f p.1 * g p.2) = Ο p.1 * f p.1 * (Ο p.2 * g p.2) | case h
R : Type u_1
instβ : CommSemiring R
Ο : β β R
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
f g : β β R
n : β
p : β Γ β
hp : p β n.divisorsAntidiagonal
β’ Ο p.1 * Ο p.2 * (f p.1 * g p.2) = Ο p.1 * f p.1 * (Ο p.2 * g p.2) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
instβ : CommSemiring R
Ο : β β R
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
f g : β β R
n : β
p : β Γ β
hp : p β n.divisorsAntidiagonal
β’ Ο n * (f p.1 * g p.2) = Ο p.1 * f p.1 * (Ο p.2 * g p.2)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | LSeries.mul_convolution_distrib | [51, 1] | [59, 28] | exact mul_mul_mul_comm .. | case h
R : Type u_1
instβ : CommSemiring R
Ο : β β R
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
f g : β β R
n : β
p : β Γ β
hp : p β n.divisorsAntidiagonal
β’ Ο p.1 * Ο p.2 * (f p.1 * g p.2) = Ο p.1 * f p.1 * (Ο p.2 * g p.2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
instβ : CommSemiring R
Ο : β β R
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
f g : β β R
n : β
p : β Γ β
hp : p β n.divisorsAntidiagonal
β’ Ο p.1 * Ο p.2 * (f p.1 * g p.2) = Ο p.1 * f p.1 * (Ο p.2 * g p.2)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | LSeries.convolution_mul_moebius | [62, 1] | [72, 68] | nth_rewrite 1 [β mul_one Ο] | Ο : β β β
hβ : Ο 1 = 1
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
this : (1 β fun x => β(ΞΌ x)) = Ξ΄
β’ Ο β (Ο * fun n => β(ΞΌ n)) = Ξ΄ | Ο : β β β
hβ : Ο 1 = 1
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
this : (1 β fun x => β(ΞΌ x)) = Ξ΄
β’ Ο * 1 β (Ο * fun n => β(ΞΌ n)) = Ξ΄ | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : β β β
hβ : Ο 1 = 1
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
this : (1 β fun x => β(ΞΌ x)) = Ξ΄
β’ Ο β (Ο * fun n => β(ΞΌ n)) = Ξ΄
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | LSeries.convolution_mul_moebius | [62, 1] | [72, 68] | simp only [β mul_convolution_distrib hΟ 1 βΞΌ, this, mul_delta hβ] | Ο : β β β
hβ : Ο 1 = 1
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
this : (1 β fun x => β(ΞΌ x)) = Ξ΄
β’ Ο * 1 β (Ο * fun n => β(ΞΌ n)) = Ξ΄ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : β β β
hβ : Ο 1 = 1
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
this : (1 β fun x => β(ΞΌ x)) = Ξ΄
β’ Ο * 1 β (Ο * fun n => β(ΞΌ n)) = Ξ΄
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | LSeries.convolution_mul_moebius | [62, 1] | [72, 68] | rw [one_convolution_eq_zeta_convolution, β one_eq_delta] | Ο : β β β
hβ : Ο 1 = 1
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
β’ (1 β fun x => β(ΞΌ x)) = Ξ΄ | Ο : β β β
hβ : Ο 1 = 1
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
β’ ((fun x => β(ΞΆ x)) β fun x => β(ΞΌ x)) = fun n => 1 n | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : β β β
hβ : Ο 1 = 1
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
β’ (1 β fun x => β(ΞΌ x)) = Ξ΄
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | LSeries.convolution_mul_moebius | [62, 1] | [72, 68] | change β(ΞΆ : ArithmeticFunction β) β β(ΞΌ : ArithmeticFunction β) = β(1 : ArithmeticFunction β) | Ο : β β β
hβ : Ο 1 = 1
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
β’ ((fun x => β(ΞΆ x)) β fun x => β(ΞΌ x)) = fun n => 1 n | Ο : β β β
hβ : Ο 1 = 1
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
β’ ββΞΆ β ββΞΌ = β1 | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : β β β
hβ : Ο 1 = 1
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
β’ ((fun x => β(ΞΆ x)) β fun x => β(ΞΌ x)) = fun n => 1 n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | LSeries.convolution_mul_moebius | [62, 1] | [72, 68] | simp only [coe_mul, coe_zeta_mul_coe_moebius] | Ο : β β β
hβ : Ο 1 = 1
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
β’ ββΞΆ β ββΞΌ = β1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : β β β
hβ : Ο 1 = 1
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
β’ ββΞΆ β ββΞΌ = β1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | LSeries.mul_mu_eq_one | [75, 1] | [80, 23] | rw [β LSeries_convolution' hs ?_, convolution_mul_moebius hβ hΟ, LSeries_delta, Pi.one_apply] | Ο : β β β
hβ : Ο 1 = 1
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
s : β
hs : LSeriesSummable Ο s
β’ L Ο s * L (Ο * fun n => β(ΞΌ n)) s = 1 | Ο : β β β
hβ : Ο 1 = 1
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
s : β
hs : LSeriesSummable Ο s
β’ LSeriesSummable (Ο * fun n => β(ΞΌ n)) s | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : β β β
hβ : Ο 1 = 1
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
s : β
hs : LSeriesSummable Ο s
β’ L Ο s * L (Ο * fun n => β(ΞΌ n)) s = 1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | LSeries.mul_mu_eq_one | [75, 1] | [80, 23] | exact hs.mul_moebius | Ο : β β β
hβ : Ο 1 = 1
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
s : β
hs : LSeriesSummable Ο s
β’ LSeriesSummable (Ο * fun n => β(ΞΌ n)) s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : β β β
hβ : Ο 1 = 1
hΟ : β (m n : β), Ο (m * n) = Ο m * Ο n
s : β
hs : LSeriesSummable Ο s
β’ LSeriesSummable (Ο * fun n => β(ΞΌ n)) s
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | DirichletCharacter.toFun_on_nat_map_one | [92, 1] | [93, 32] | simp only [cast_one, map_one] | N : β
Ο : DirichletCharacter β N
β’ (fun n => Ο βn) 1 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
β’ (fun n => Ο βn) 1 = 1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/DirichletLSeries.lean | DirichletCharacter.toFun_on_nat_map_mul | [95, 1] | [97, 32] | simp only [cast_mul, map_mul] | N : β
Ο : DirichletCharacter β N
m n : β
β’ (fun n => Ο βn) (m * n) = (fun n => Ο βn) m * (fun n => Ο βn) n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
m n : β
β’ (fun n => Ο βn) (m * n) = (fun n => Ο βn) m * (fun n => Ο βn) n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | DirichletCharacter.LSeries_eulerProduct' | [42, 1] | [51, 61] | rw [LSeries] | N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
β’ cexp (β' (p : Primes), -(1 - Ο ββp * ββp ^ (-s)).log) = L (fun n => Ο βn) s | N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
β’ cexp (β' (p : Primes), -(1 - Ο ββp * ββp ^ (-s)).log) = β' (n : β), term (fun n => Ο βn) s n | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
β’ cexp (β' (p : Primes), -(1 - Ο ββp * ββp ^ (-s)).log) = L (fun n => Ο βn) s
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | DirichletCharacter.LSeries_eulerProduct' | [42, 1] | [51, 61] | convert exp_sum_primes_log_eq_tsum (f := dirichletSummandHom Ο <| ne_zero_of_one_lt_re hs) <|
summable_dirichletSummand Ο hs | N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
β’ cexp (β' (p : Primes), -(1 - Ο ββp * ββp ^ (-s)).log) = β' (n : β), term (fun n => Ο βn) s n | case h.e'_3.h.e'_5.h.h.e
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ : β
β’ term (fun n => Ο βn) s = β(dirichletSummandHom Ο β―) | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
β’ cexp (β' (p : Primes), -(1 - Ο ββp * ββp ^ (-s)).log) = β' (n : β), term (fun n => Ο βn) s n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | DirichletCharacter.LSeries_eulerProduct' | [42, 1] | [51, 61] | ext n | case h.e'_3.h.e'_5.h.h.e
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ : β
β’ term (fun n => Ο βn) s = β(dirichletSummandHom Ο β―) | case h.e'_3.h.e'_5.h.h.e.h
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ n : β
β’ term (fun n => Ο βn) s n = (dirichletSummandHom Ο β―) n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_5.h.h.e
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ : β
β’ term (fun n => Ο βn) s = β(dirichletSummandHom Ο β―)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | DirichletCharacter.LSeries_eulerProduct' | [42, 1] | [51, 61] | rcases eq_or_ne n 0 with rfl | hn | case h.e'_3.h.e'_5.h.h.e.h
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ n : β
β’ term (fun n => Ο βn) s n = (dirichletSummandHom Ο β―) n | case h.e'_3.h.e'_5.h.h.e.h.inl
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ : β
β’ term (fun n => Ο βn) s 0 = (dirichletSummandHom Ο β―) 0
case h.e'_3.h.e'_5.h.h.e.h.inr
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ n : β
hn : n β 0
β’ term (fun n => Ο βn) s n = (dirichletSummandHom Ο β―) n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_5.h.h.e.h
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ n : β
β’ term (fun n => Ο βn) s n = (dirichletSummandHom Ο β―) n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | DirichletCharacter.LSeries_eulerProduct' | [42, 1] | [51, 61] | simp only [term_zero, map_zero] | case h.e'_3.h.e'_5.h.h.e.h.inl
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ : β
β’ term (fun n => Ο βn) s 0 = (dirichletSummandHom Ο β―) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_5.h.h.e.h.inl
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ : β
β’ term (fun n => Ο βn) s 0 = (dirichletSummandHom Ο β―) 0
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | DirichletCharacter.LSeries_eulerProduct' | [42, 1] | [51, 61] | simp [hn, dirichletSummandHom, div_eq_mul_inv, cpow_neg] | case h.e'_3.h.e'_5.h.h.e.h.inr
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ n : β
hn : n β 0
β’ term (fun n => Ο βn) s n = (dirichletSummandHom Ο β―) n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_5.h.h.e.h.inr
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
xβ n : β
hn : n β 0
β’ term (fun n => Ο βn) s n = (dirichletSummandHom Ο β―) n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | ArithmeticFunction.LSeries_zeta_eulerProduct' | [56, 1] | [59, 62] | convert modOne_eq_one (R := β) βΈ LSeries_eulerProduct' Οβ hs using 7 | s : β
hs : 1 < s.re
β’ cexp (β' (p : Primes), -(1 - ββp ^ (-s)).log) = L 1 s | case h.e'_2.h.e'_1.h.e'_5.h.h.e'_3.h.e'_1.h.e'_6
s : β
hs : 1 < s.re
xβ : Primes
β’ ββxβ ^ (-s) = 1 ββxβ * ββxβ ^ (-s) | Please generate a tactic in lean4 to solve the state.
STATE:
s : β
hs : 1 < s.re
β’ cexp (β' (p : Primes), -(1 - ββp ^ (-s)).log) = L 1 s
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | ArithmeticFunction.LSeries_zeta_eulerProduct' | [56, 1] | [59, 62] | rw [MulChar.one_apply <| isUnit_of_subsingleton _, one_mul] | case h.e'_2.h.e'_1.h.e'_5.h.h.e'_3.h.e'_1.h.e'_6
s : β
hs : 1 < s.re
xβ : Primes
β’ ββxβ ^ (-s) = 1 ββxβ * ββxβ ^ (-s) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_1.h.e'_5.h.h.e'_3.h.e'_1.h.e'_6
s : β
hs : 1 < s.re
xβ : Primes
β’ ββxβ ^ (-s) = 1 ββxβ * ββxβ ^ (-s)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | summable_neg_log_one_sub_char_mul_prime_cpow | [69, 1] | [79, 41] | have (p : Nat.Primes) : βΟ p * (p : β) ^ (-s)β β€ (p : β) ^ (-s).re := by
rw [norm_mul, norm_natCast_cpow_of_re_ne_zero _ <| re_neg_ne_zero_of_one_lt_re hs]
calc βΟ pβ * (p : β) ^ (-s).re
_ β€ 1 * (p : β) ^ (-s.re) := by gcongr; exact DirichletCharacter.norm_le_one Ο _
_ = _ := one_mul _ | N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
β’ Summable fun p => -(1 - Ο ββp * ββp ^ (-s)).log | N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
this : β (p : Nat.Primes), βΟ ββp * ββp ^ (-s)β β€ ββp ^ (-s).re
β’ Summable fun p => -(1 - Ο ββp * ββp ^ (-s)).log | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
β’ Summable fun p => -(1 - Ο ββp * ββp ^ (-s)).log
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | summable_neg_log_one_sub_char_mul_prime_cpow | [69, 1] | [79, 41] | refine (Nat.Primes.summable_rpow.mpr ?_).of_nonneg_of_le (fun _ β¦ norm_nonneg _) this
|>.of_norm.neg_clog_one_sub | N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
this : β (p : Nat.Primes), βΟ ββp * ββp ^ (-s)β β€ ββp ^ (-s).re
β’ Summable fun p => -(1 - Ο ββp * ββp ^ (-s)).log | N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
this : β (p : Nat.Primes), βΟ ββp * ββp ^ (-s)β β€ ββp ^ (-s).re
β’ (-s).re < -1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
this : β (p : Nat.Primes), βΟ ββp * ββp ^ (-s)β β€ ββp ^ (-s).re
β’ Summable fun p => -(1 - Ο ββp * ββp ^ (-s)).log
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | summable_neg_log_one_sub_char_mul_prime_cpow | [69, 1] | [79, 41] | simp only [neg_re, neg_lt_neg_iff, hs] | N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
this : β (p : Nat.Primes), βΟ ββp * ββp ^ (-s)β β€ ββp ^ (-s).re
β’ (-s).re < -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
this : β (p : Nat.Primes), βΟ ββp * ββp ^ (-s)β β€ ββp ^ (-s).re
β’ (-s).re < -1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | summable_neg_log_one_sub_char_mul_prime_cpow | [69, 1] | [79, 41] | rw [norm_mul, norm_natCast_cpow_of_re_ne_zero _ <| re_neg_ne_zero_of_one_lt_re hs] | N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
p : Nat.Primes
β’ βΟ ββp * ββp ^ (-s)β β€ ββp ^ (-s).re | N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
p : Nat.Primes
β’ βΟ ββpβ * ββp ^ (-s).re β€ ββp ^ (-s).re | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
p : Nat.Primes
β’ βΟ ββp * ββp ^ (-s)β β€ ββp ^ (-s).re
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | summable_neg_log_one_sub_char_mul_prime_cpow | [69, 1] | [79, 41] | calc βΟ pβ * (p : β) ^ (-s).re
_ β€ 1 * (p : β) ^ (-s.re) := by gcongr; exact DirichletCharacter.norm_le_one Ο _
_ = _ := one_mul _ | N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
p : Nat.Primes
β’ βΟ ββpβ * ββp ^ (-s).re β€ ββp ^ (-s).re | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
p : Nat.Primes
β’ βΟ ββpβ * ββp ^ (-s).re β€ ββp ^ (-s).re
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | summable_neg_log_one_sub_char_mul_prime_cpow | [69, 1] | [79, 41] | gcongr | N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
p : Nat.Primes
β’ βΟ ββpβ * ββp ^ (-s).re β€ 1 * ββp ^ (-s.re) | case h
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
p : Nat.Primes
β’ βΟ ββpβ β€ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
p : Nat.Primes
β’ βΟ ββpβ * ββp ^ (-s).re β€ 1 * ββp ^ (-s.re)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | summable_neg_log_one_sub_char_mul_prime_cpow | [69, 1] | [79, 41] | exact DirichletCharacter.norm_le_one Ο _ | case h
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
p : Nat.Primes
β’ βΟ ββpβ β€ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
N : β
Ο : DirichletCharacter β N
s : β
hs : 1 < s.re
p : Nat.Primes
β’ βΟ ββpβ β€ 1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg' | [81, 1] | [104, 22] | have hacβ : β(a : β)β < 1 := by
simp only [norm_eq_abs, abs_ofReal, _root_.abs_of_nonneg haβ, haβ] | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
β’ 0 β€ 3 * (-(1 - βa).log).re + 4 * (-(1 - βa * z).log).re + (-(1 - βa * z ^ 2).log).re | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
β’ 0 β€ 3 * (-(1 - βa).log).re + 4 * (-(1 - βa * z).log).re + (-(1 - βa * z ^ 2).log).re | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
β’ 0 β€ 3 * (-(1 - βa).log).re + 4 * (-(1 - βa * z).log).re + (-(1 - βa * z ^ 2).log).re
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg' | [81, 1] | [104, 22] | have hacβ : βa * zβ < 1 := by rwa [norm_mul, hz, mul_one] | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
β’ 0 β€ 3 * (-(1 - βa).log).re + 4 * (-(1 - βa * z).log).re + (-(1 - βa * z ^ 2).log).re | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
β’ 0 β€ 3 * (-(1 - βa).log).re + 4 * (-(1 - βa * z).log).re + (-(1 - βa * z ^ 2).log).re | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
β’ 0 β€ 3 * (-(1 - βa).log).re + 4 * (-(1 - βa * z).log).re + (-(1 - βa * z ^ 2).log).re
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg' | [81, 1] | [104, 22] | have hacβ : βa * z ^ 2β < 1 := by rwa [norm_mul, norm_pow, hz, one_pow, mul_one] | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
β’ 0 β€ 3 * (-(1 - βa).log).re + 4 * (-(1 - βa * z).log).re + (-(1 - βa * z ^ 2).log).re | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
β’ 0 β€ 3 * (-(1 - βa).log).re + 4 * (-(1 - βa * z).log).re + (-(1 - βa * z ^ 2).log).re | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
β’ 0 β€ 3 * (-(1 - βa).log).re + 4 * (-(1 - βa * z).log).re + (-(1 - βa * z ^ 2).log).re
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg' | [81, 1] | [104, 22] | have Hβ := (hasSum_re <| hasSum_taylorSeries_neg_log hacβ).mul_left 3 | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
β’ 0 β€ 3 * (-(1 - βa).log).re + 4 * (-(1 - βa * z).log).re + (-(1 - βa * z ^ 2).log).re | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
Hβ : HasSum (fun i => 3 * (βa ^ i / βi).re) (3 * (-(1 - βa).log).re)
β’ 0 β€ 3 * (-(1 - βa).log).re + 4 * (-(1 - βa * z).log).re + (-(1 - βa * z ^ 2).log).re | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
β’ 0 β€ 3 * (-(1 - βa).log).re + 4 * (-(1 - βa * z).log).re + (-(1 - βa * z ^ 2).log).re
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg' | [81, 1] | [104, 22] | have Hβ := (hasSum_re <| hasSum_taylorSeries_neg_log hacβ).mul_left 4 | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
Hβ : HasSum (fun i => 3 * (βa ^ i / βi).re) (3 * (-(1 - βa).log).re)
β’ 0 β€ 3 * (-(1 - βa).log).re + 4 * (-(1 - βa * z).log).re + (-(1 - βa * z ^ 2).log).re | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
Hβ : HasSum (fun i => 3 * (βa ^ i / βi).re) (3 * (-(1 - βa).log).re)
Hβ : HasSum (fun i => 4 * ((βa * z) ^ i / βi).re) (4 * (-(1 - βa * z).log).re)
β’ 0 β€ 3 * (-(1 - βa).log).re + 4 * (-(1 - βa * z).log).re + (-(1 - βa * z ^ 2).log).re | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
Hβ : HasSum (fun i => 3 * (βa ^ i / βi).re) (3 * (-(1 - βa).log).re)
β’ 0 β€ 3 * (-(1 - βa).log).re + 4 * (-(1 - βa * z).log).re + (-(1 - βa * z ^ 2).log).re
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg' | [81, 1] | [104, 22] | have Hβ := hasSum_re <| hasSum_taylorSeries_neg_log hacβ | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
Hβ : HasSum (fun i => 3 * (βa ^ i / βi).re) (3 * (-(1 - βa).log).re)
Hβ : HasSum (fun i => 4 * ((βa * z) ^ i / βi).re) (4 * (-(1 - βa * z).log).re)
β’ 0 β€ 3 * (-(1 - βa).log).re + 4 * (-(1 - βa * z).log).re + (-(1 - βa * z ^ 2).log).re | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
Hβ : HasSum (fun i => 3 * (βa ^ i / βi).re) (3 * (-(1 - βa).log).re)
Hβ : HasSum (fun i => 4 * ((βa * z) ^ i / βi).re) (4 * (-(1 - βa * z).log).re)
Hβ : HasSum (fun x => ((βa * z ^ 2) ^ x / βx).re) (-(1 - βa * z ^ 2).log).re
β’ 0 β€ 3 * (-(1 - βa).log).re + 4 * (-(1 - βa * z).log).re + (-(1 - βa * z ^ 2).log).re | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
Hβ : HasSum (fun i => 3 * (βa ^ i / βi).re) (3 * (-(1 - βa).log).re)
Hβ : HasSum (fun i => 4 * ((βa * z) ^ i / βi).re) (4 * (-(1 - βa * z).log).re)
β’ 0 β€ 3 * (-(1 - βa).log).re + 4 * (-(1 - βa * z).log).re + (-(1 - βa * z ^ 2).log).re
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg' | [81, 1] | [104, 22] | rw [β ((Hβ.add Hβ).add Hβ).tsum_eq] | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
Hβ : HasSum (fun i => 3 * (βa ^ i / βi).re) (3 * (-(1 - βa).log).re)
Hβ : HasSum (fun i => 4 * ((βa * z) ^ i / βi).re) (4 * (-(1 - βa * z).log).re)
Hβ : HasSum (fun x => ((βa * z ^ 2) ^ x / βx).re) (-(1 - βa * z ^ 2).log).re
β’ 0 β€ 3 * (-(1 - βa).log).re + 4 * (-(1 - βa * z).log).re + (-(1 - βa * z ^ 2).log).re | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
Hβ : HasSum (fun i => 3 * (βa ^ i / βi).re) (3 * (-(1 - βa).log).re)
Hβ : HasSum (fun i => 4 * ((βa * z) ^ i / βi).re) (4 * (-(1 - βa * z).log).re)
Hβ : HasSum (fun x => ((βa * z ^ 2) ^ x / βx).re) (-(1 - βa * z ^ 2).log).re
β’ 0 β€ β' (b : β), (3 * (βa ^ b / βb).re + 4 * ((βa * z) ^ b / βb).re + ((βa * z ^ 2) ^ b / βb).re) | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
Hβ : HasSum (fun i => 3 * (βa ^ i / βi).re) (3 * (-(1 - βa).log).re)
Hβ : HasSum (fun i => 4 * ((βa * z) ^ i / βi).re) (4 * (-(1 - βa * z).log).re)
Hβ : HasSum (fun x => ((βa * z ^ 2) ^ x / βx).re) (-(1 - βa * z ^ 2).log).re
β’ 0 β€ 3 * (-(1 - βa).log).re + 4 * (-(1 - βa * z).log).re + (-(1 - βa * z ^ 2).log).re
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg' | [81, 1] | [104, 22] | clear Hβ Hβ Hβ | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
Hβ : HasSum (fun i => 3 * (βa ^ i / βi).re) (3 * (-(1 - βa).log).re)
Hβ : HasSum (fun i => 4 * ((βa * z) ^ i / βi).re) (4 * (-(1 - βa * z).log).re)
Hβ : HasSum (fun x => ((βa * z ^ 2) ^ x / βx).re) (-(1 - βa * z ^ 2).log).re
β’ 0 β€ β' (b : β), (3 * (βa ^ b / βb).re + 4 * ((βa * z) ^ b / βb).re + ((βa * z ^ 2) ^ b / βb).re) | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
β’ 0 β€ β' (b : β), (3 * (βa ^ b / βb).re + 4 * ((βa * z) ^ b / βb).re + ((βa * z ^ 2) ^ b / βb).re) | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
Hβ : HasSum (fun i => 3 * (βa ^ i / βi).re) (3 * (-(1 - βa).log).re)
Hβ : HasSum (fun i => 4 * ((βa * z) ^ i / βi).re) (4 * (-(1 - βa * z).log).re)
Hβ : HasSum (fun x => ((βa * z ^ 2) ^ x / βx).re) (-(1 - βa * z ^ 2).log).re
β’ 0 β€ β' (b : β), (3 * (βa ^ b / βb).re + 4 * ((βa * z) ^ b / βb).re + ((βa * z ^ 2) ^ b / βb).re)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg' | [81, 1] | [104, 22] | refine tsum_nonneg fun n β¦ ?_ | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
β’ 0 β€ β' (b : β), (3 * (βa ^ b / βb).re + 4 * ((βa * z) ^ b / βb).re + ((βa * z ^ 2) ^ b / βb).re) | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
n : β
β’ 0 β€ 3 * (βa ^ n / βn).re + 4 * ((βa * z) ^ n / βn).re + ((βa * z ^ 2) ^ n / βn).re | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
β’ 0 β€ β' (b : β), (3 * (βa ^ b / βb).re + 4 * ((βa * z) ^ b / βb).re + ((βa * z ^ 2) ^ b / βb).re)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg' | [81, 1] | [104, 22] | simp only [mul_pow, β ofReal_pow, div_natCast_re, ofReal_re, mul_re, ofReal_im, zero_mul,
sub_zero] | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
n : β
β’ 0 β€ 3 * (βa ^ n / βn).re + 4 * ((βa * z) ^ n / βn).re + ((βa * z ^ 2) ^ n / βn).re | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
n : β
β’ 0 β€ 3 * (a ^ n / βn) + 4 * (a ^ n * (z ^ n).re / βn) + a ^ n * ((z ^ 2) ^ n).re / βn | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
n : β
β’ 0 β€ 3 * (βa ^ n / βn).re + 4 * ((βa * z) ^ n / βn).re + ((βa * z ^ 2) ^ n / βn).re
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg' | [81, 1] | [104, 22] | rcases n.eq_zero_or_pos with rfl | hn | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
n : β
β’ 0 β€ 3 * (a ^ n / βn) + 4 * (a ^ n * (z ^ n).re / βn) + a ^ n * ((z ^ 2) ^ n).re / βn | case inl
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
β’ 0 β€ 3 * (a ^ 0 / β0) + 4 * (a ^ 0 * (z ^ 0).re / β0) + a ^ 0 * ((z ^ 2) ^ 0).re / β0
case inr
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
n : β
hn : n > 0
β’ 0 β€ 3 * (a ^ n / βn) + 4 * (a ^ n * (z ^ n).re / βn) + a ^ n * ((z ^ 2) ^ n).re / βn | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
n : β
β’ 0 β€ 3 * (a ^ n / βn) + 4 * (a ^ n * (z ^ n).re / βn) + a ^ n * ((z ^ 2) ^ n).re / βn
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg' | [81, 1] | [104, 22] | field_simp | case inr
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
n : β
hn : n > 0
β’ 0 β€ 3 * (a ^ n / βn) + 4 * (a ^ n * (z ^ n).re / βn) + a ^ n * ((z ^ 2) ^ n).re / βn | case inr
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
n : β
hn : n > 0
β’ 0 β€ (3 * a ^ n + 4 * (a ^ n * (z ^ n).re) + a ^ n * ((z ^ 2) ^ n).re) / βn | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
n : β
hn : n > 0
β’ 0 β€ 3 * (a ^ n / βn) + 4 * (a ^ n * (z ^ n).re / βn) + a ^ n * ((z ^ 2) ^ n).re / βn
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg' | [81, 1] | [104, 22] | refine div_nonneg ?_ n.cast_nonneg | case inr
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
n : β
hn : n > 0
β’ 0 β€ (3 * a ^ n + 4 * (a ^ n * (z ^ n).re) + a ^ n * ((z ^ 2) ^ n).re) / βn | case inr
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
n : β
hn : n > 0
β’ 0 β€ 3 * a ^ n + 4 * (a ^ n * (z ^ n).re) + a ^ n * ((z ^ 2) ^ n).re | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
n : β
hn : n > 0
β’ 0 β€ (3 * a ^ n + 4 * (a ^ n * (z ^ n).re) + a ^ n * ((z ^ 2) ^ n).re) / βn
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg' | [81, 1] | [104, 22] | rw [β pow_mul, pow_mul', sq, mul_re, β sq, β sq, β sq_abs_sub_sq_re, β norm_eq_abs, norm_pow, hz] | case inr
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
n : β
hn : n > 0
β’ 0 β€ 3 * a ^ n + 4 * (a ^ n * (z ^ n).re) + a ^ n * ((z ^ 2) ^ n).re | case inr
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
n : β
hn : n > 0
β’ 0 β€ 3 * a ^ n + 4 * (a ^ n * (z ^ n).re) + a ^ n * ((z ^ n).re ^ 2 - ((1 ^ n) ^ 2 - (z ^ n).re ^ 2)) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
n : β
hn : n > 0
β’ 0 β€ 3 * a ^ n + 4 * (a ^ n * (z ^ n).re) + a ^ n * ((z ^ 2) ^ n).re
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg' | [81, 1] | [104, 22] | calc
0 β€ 2 * a ^ n * ((z ^ n).re + 1) ^ 2 := by positivity
_ = _ := by ring | case inr
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
n : β
hn : n > 0
β’ 0 β€ 3 * a ^ n + 4 * (a ^ n * (z ^ n).re) + a ^ n * ((z ^ n).re ^ 2 - ((1 ^ n) ^ 2 - (z ^ n).re ^ 2)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
n : β
hn : n > 0
β’ 0 β€ 3 * a ^ n + 4 * (a ^ n * (z ^ n).re) + a ^ n * ((z ^ n).re ^ 2 - ((1 ^ n) ^ 2 - (z ^ n).re ^ 2))
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg' | [81, 1] | [104, 22] | simp only [norm_eq_abs, abs_ofReal, _root_.abs_of_nonneg haβ, haβ] | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
β’ ββaβ < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
β’ ββaβ < 1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg' | [81, 1] | [104, 22] | rwa [norm_mul, hz, mul_one] | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
β’ ββa * zβ < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
β’ ββa * zβ < 1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg' | [81, 1] | [104, 22] | rwa [norm_mul, norm_pow, hz, one_pow, mul_one] | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
β’ ββa * z ^ 2β < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
β’ ββa * z ^ 2β < 1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg' | [81, 1] | [104, 22] | simp | case inl
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
β’ 0 β€ 3 * (a ^ 0 / β0) + 4 * (a ^ 0 * (z ^ 0).re / β0) + a ^ 0 * ((z ^ 2) ^ 0).re / β0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
β’ 0 β€ 3 * (a ^ 0 / β0) + 4 * (a ^ 0 * (z ^ 0).re / β0) + a ^ 0 * ((z ^ 2) ^ 0).re / β0
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg' | [81, 1] | [104, 22] | positivity | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
n : β
hn : n > 0
β’ 0 β€ 2 * a ^ n * ((z ^ n).re + 1) ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
n : β
hn : n > 0
β’ 0 β€ 2 * a ^ n * ((z ^ n).re + 1) ^ 2
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg' | [81, 1] | [104, 22] | ring | a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
n : β
hn : n > 0
β’ 2 * a ^ n * ((z ^ n).re + 1) ^ 2 =
3 * a ^ n + 4 * (a ^ n * (z ^ n).re) + a ^ n * ((z ^ n).re ^ 2 - ((1 ^ n) ^ 2 - (z ^ n).re ^ 2)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
haβ : 0 β€ a
haβ : a < 1
z : β
hz : βzβ = 1
hacβ : ββaβ < 1
hacβ : ββa * zβ < 1
hacβ : ββa * z ^ 2β < 1
n : β
hn : n > 0
β’ 2 * a ^ n * ((z ^ n).re + 1) ^ 2 =
3 * a ^ n + 4 * (a ^ n * (z ^ n).re) + a ^ n * ((z ^ n).re ^ 2 - ((1 ^ n) ^ 2 - (z ^ n).re ^ 2))
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg_dirichlet | [106, 1] | [135, 37] | by_cases hn' : IsUnit (n : ZMod N) | N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
β’ 0 β€
3 * (-(1 - 1 βn * βn ^ (-βx)).log).re + 4 * (-(1 - Ο βn * βn ^ (-(βx + I * βy))).log).re +
(-(1 - Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy))).log).re | case pos
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
β’ 0 β€
3 * (-(1 - 1 βn * βn ^ (-βx)).log).re + 4 * (-(1 - Ο βn * βn ^ (-(βx + I * βy))).log).re +
(-(1 - Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy))).log).re
case neg
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : Β¬IsUnit βn
β’ 0 β€
3 * (-(1 - 1 βn * βn ^ (-βx)).log).re + 4 * (-(1 - Ο βn * βn ^ (-(βx + I * βy))).log).re +
(-(1 - Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy))).log).re | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
β’ 0 β€
3 * (-(1 - 1 βn * βn ^ (-βx)).log).re + 4 * (-(1 - Ο βn * βn ^ (-(βx + I * βy))).log).re +
(-(1 - Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy))).log).re
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg_dirichlet | [106, 1] | [135, 37] | have haβ : 0 β€ (n : β) ^ (-x) := Real.rpow_nonneg n.cast_nonneg _ | case pos
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
β’ 0 β€
3 * (-(1 - 1 βn * βn ^ (-βx)).log).re + 4 * (-(1 - Ο βn * βn ^ (-(βx + I * βy))).log).re +
(-(1 - Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy))).log).re | case pos
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
β’ 0 β€
3 * (-(1 - 1 βn * βn ^ (-βx)).log).re + 4 * (-(1 - Ο βn * βn ^ (-(βx + I * βy))).log).re +
(-(1 - Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy))).log).re | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
β’ 0 β€
3 * (-(1 - 1 βn * βn ^ (-βx)).log).re + 4 * (-(1 - Ο βn * βn ^ (-(βx + I * βy))).log).re +
(-(1 - Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy))).log).re
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg_dirichlet | [106, 1] | [135, 37] | have haβ : (n : β) ^ (-x) < 1 := by
simpa only [Real.rpow_lt_one_iff n.cast_nonneg, Nat.cast_eq_zero, Nat.one_lt_cast,
Left.neg_neg_iff, Nat.cast_lt_one, Left.neg_pos_iff]
using Or.inr <| Or.inl β¨hn, zero_lt_one.trans hxβ© | case pos
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
β’ 0 β€
3 * (-(1 - 1 βn * βn ^ (-βx)).log).re + 4 * (-(1 - Ο βn * βn ^ (-(βx + I * βy))).log).re +
(-(1 - Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy))).log).re | case pos
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
β’ 0 β€
3 * (-(1 - 1 βn * βn ^ (-βx)).log).re + 4 * (-(1 - Ο βn * βn ^ (-(βx + I * βy))).log).re +
(-(1 - Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy))).log).re | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
β’ 0 β€
3 * (-(1 - 1 βn * βn ^ (-βx)).log).re + 4 * (-(1 - Ο βn * βn ^ (-(βx + I * βy))).log).re +
(-(1 - Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy))).log).re
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg_dirichlet | [106, 1] | [135, 37] | have hz : βΟ n * (n : β) ^ (-(I * y))β = 1 := by
rw [norm_mul, β hn'.unit_spec, DirichletCharacter.unit_norm_eq_one Ο hn'.unit, one_mul,
norm_eq_abs, abs_cpow_of_imp fun h β¦ False.elim <| by linarith [Nat.cast_eq_zero.mp h, hn]]
simp | case pos
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
β’ 0 β€
3 * (-(1 - 1 βn * βn ^ (-βx)).log).re + 4 * (-(1 - Ο βn * βn ^ (-(βx + I * βy))).log).re +
(-(1 - Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy))).log).re | case pos
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ 0 β€
3 * (-(1 - 1 βn * βn ^ (-βx)).log).re + 4 * (-(1 - Ο βn * βn ^ (-(βx + I * βy))).log).re +
(-(1 - Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy))).log).re | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
β’ 0 β€
3 * (-(1 - 1 βn * βn ^ (-βx)).log).re + 4 * (-(1 - Ο βn * βn ^ (-(βx + I * βy))).log).re +
(-(1 - Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy))).log).re
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg_dirichlet | [106, 1] | [135, 37] | rw [MulChar.one_apply hn', one_mul] | case pos
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ 0 β€
3 * (-(1 - 1 βn * βn ^ (-βx)).log).re + 4 * (-(1 - Ο βn * βn ^ (-(βx + I * βy))).log).re +
(-(1 - Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy))).log).re | case pos
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ 0 β€
3 * (-(1 - βn ^ (-βx)).log).re + 4 * (-(1 - Ο βn * βn ^ (-(βx + I * βy))).log).re +
(-(1 - Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy))).log).re | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ 0 β€
3 * (-(1 - 1 βn * βn ^ (-βx)).log).re + 4 * (-(1 - Ο βn * βn ^ (-(βx + I * βy))).log).re +
(-(1 - Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy))).log).re
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg_dirichlet | [106, 1] | [135, 37] | convert re_log_comb_nonneg' haβ haβ hz using 6 | case pos
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ 0 β€
3 * (-(1 - βn ^ (-βx)).log).re + 4 * (-(1 - Ο βn * βn ^ (-(βx + I * βy))).log).re +
(-(1 - Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy))).log).re | case h.e'_4.h.e'_5.h.e'_5.h.e'_6.h.e'_1.h.e'_3
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ (1 - βn ^ (-βx)).log = (1 - β(βn ^ (-x))).log
case h.e'_4.h.e'_5.h.e'_6.h.e'_6.h.e'_1.h.e'_3
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ (1 - Ο βn * βn ^ (-(βx + I * βy))).log = (1 - β(βn ^ (-x)) * (Ο βn * βn ^ (-(I * βy)))).log
case h.e'_4.h.e'_6.h.e'_1.h.e'_3.h.e'_1.h.e'_6
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy)) = β(βn ^ (-x)) * (Ο βn * βn ^ (-(I * βy))) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ 0 β€
3 * (-(1 - βn ^ (-βx)).log).re + 4 * (-(1 - Ο βn * βn ^ (-(βx + I * βy))).log).re +
(-(1 - Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy))).log).re
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg_dirichlet | [106, 1] | [135, 37] | simpa only [Real.rpow_lt_one_iff n.cast_nonneg, Nat.cast_eq_zero, Nat.one_lt_cast,
Left.neg_neg_iff, Nat.cast_lt_one, Left.neg_pos_iff]
using Or.inr <| Or.inl β¨hn, zero_lt_one.trans hxβ© | N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
β’ βn ^ (-x) < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
β’ βn ^ (-x) < 1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg_dirichlet | [106, 1] | [135, 37] | rw [norm_mul, β hn'.unit_spec, DirichletCharacter.unit_norm_eq_one Ο hn'.unit, one_mul,
norm_eq_abs, abs_cpow_of_imp fun h β¦ False.elim <| by linarith [Nat.cast_eq_zero.mp h, hn]] | N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
β’ βΟ βn * βn ^ (-(I * βy))β = 1 | N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
β’ Complex.abs βn ^ (-(I * βy)).re / ((βn).arg * (-(I * βy)).im).exp = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
β’ βΟ βn * βn ^ (-(I * βy))β = 1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg_dirichlet | [106, 1] | [135, 37] | simp | N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
β’ Complex.abs βn ^ (-(I * βy)).re / ((βn).arg * (-(I * βy)).im).exp = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
β’ Complex.abs βn ^ (-(I * βy)).re / ((βn).arg * (-(I * βy)).im).exp = 1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg_dirichlet | [106, 1] | [135, 37] | linarith [Nat.cast_eq_zero.mp h, hn] | N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
h : βn = 0
β’ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
h : βn = 0
β’ False
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg_dirichlet | [106, 1] | [135, 37] | congr 2 | case h.e'_4.h.e'_5.h.e'_5.h.e'_6.h.e'_1.h.e'_3
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ (1 - βn ^ (-βx)).log = (1 - β(βn ^ (-x))).log | case h.e'_4.h.e'_5.h.e'_5.h.e'_6.h.e'_1.h.e'_3.e_x.e_a
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ βn ^ (-βx) = β(βn ^ (-x)) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.e'_5.h.e'_5.h.e'_6.h.e'_1.h.e'_3
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ (1 - βn ^ (-βx)).log = (1 - β(βn ^ (-x))).log
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg_dirichlet | [106, 1] | [135, 37] | exact_mod_cast (ofReal_cpow n.cast_nonneg (-x)).symm | case h.e'_4.h.e'_5.h.e'_5.h.e'_6.h.e'_1.h.e'_3.e_x.e_a
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ βn ^ (-βx) = β(βn ^ (-x)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.e'_5.h.e'_5.h.e'_6.h.e'_1.h.e'_3.e_x.e_a
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ βn ^ (-βx) = β(βn ^ (-x))
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg_dirichlet | [106, 1] | [135, 37] | congr 2 | case h.e'_4.h.e'_5.h.e'_6.h.e'_6.h.e'_1.h.e'_3
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ (1 - Ο βn * βn ^ (-(βx + I * βy))).log = (1 - β(βn ^ (-x)) * (Ο βn * βn ^ (-(I * βy)))).log | case h.e'_4.h.e'_5.h.e'_6.h.e'_6.h.e'_1.h.e'_3.e_x.e_a
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ Ο βn * βn ^ (-(βx + I * βy)) = β(βn ^ (-x)) * (Ο βn * βn ^ (-(I * βy))) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.e'_5.h.e'_6.h.e'_6.h.e'_1.h.e'_3
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ (1 - Ο βn * βn ^ (-(βx + I * βy))).log = (1 - β(βn ^ (-x)) * (Ο βn * βn ^ (-(I * βy)))).log
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg_dirichlet | [106, 1] | [135, 37] | rw [neg_add, cpow_add _ _ <| by norm_cast; linarith, β ofReal_neg,
ofReal_cpow n.cast_nonneg (-x), ofReal_natCast] | case h.e'_4.h.e'_5.h.e'_6.h.e'_6.h.e'_1.h.e'_3.e_x.e_a
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ Ο βn * βn ^ (-(βx + I * βy)) = β(βn ^ (-x)) * (Ο βn * βn ^ (-(I * βy))) | case h.e'_4.h.e'_5.h.e'_6.h.e'_6.h.e'_1.h.e'_3.e_x.e_a
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ Ο βn * (βn ^ β(-x) * βn ^ (-(I * βy))) = βn ^ β(-x) * (Ο βn * βn ^ (-(I * βy))) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.e'_5.h.e'_6.h.e'_6.h.e'_1.h.e'_3.e_x.e_a
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ Ο βn * βn ^ (-(βx + I * βy)) = β(βn ^ (-x)) * (Ο βn * βn ^ (-(I * βy)))
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg_dirichlet | [106, 1] | [135, 37] | ring | case h.e'_4.h.e'_5.h.e'_6.h.e'_6.h.e'_1.h.e'_3.e_x.e_a
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ Ο βn * (βn ^ β(-x) * βn ^ (-(I * βy))) = βn ^ β(-x) * (Ο βn * βn ^ (-(I * βy))) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.e'_5.h.e'_6.h.e'_6.h.e'_1.h.e'_3.e_x.e_a
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ Ο βn * (βn ^ β(-x) * βn ^ (-(I * βy))) = βn ^ β(-x) * (Ο βn * βn ^ (-(I * βy)))
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg_dirichlet | [106, 1] | [135, 37] | norm_cast | N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ βn β 0 | N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ Β¬n = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ βn β 0
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg_dirichlet | [106, 1] | [135, 37] | linarith | N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ Β¬n = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ Β¬n = 0
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg_dirichlet | [106, 1] | [135, 37] | rw [neg_add, cpow_add _ _ <| by norm_cast; linarith, β ofReal_neg,
ofReal_cpow n.cast_nonneg (-x), ofReal_natCast,
show -(2 * I * y) = (2 : β) * (-I * y) by ring, cpow_nat_mul] | case h.e'_4.h.e'_6.h.e'_1.h.e'_3.h.e'_1.h.e'_6
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy)) = β(βn ^ (-x)) * (Ο βn * βn ^ (-(I * βy))) ^ 2 | case h.e'_4.h.e'_6.h.e'_1.h.e'_3.h.e'_1.h.e'_6
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ Ο βn ^ 2 * (βn ^ β(-x) * (βn ^ (-I * βy)) ^ 2) = βn ^ β(-x) * (Ο βn * βn ^ (-(I * βy))) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.e'_6.h.e'_1.h.e'_3.h.e'_1.h.e'_6
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy)) = β(βn ^ (-x)) * (Ο βn * βn ^ (-(I * βy))) ^ 2
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg_dirichlet | [106, 1] | [135, 37] | ring_nf | case h.e'_4.h.e'_6.h.e'_1.h.e'_3.h.e'_1.h.e'_6
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ Ο βn ^ 2 * (βn ^ β(-x) * (βn ^ (-I * βy)) ^ 2) = βn ^ β(-x) * (Ο βn * βn ^ (-(I * βy))) ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.e'_6.h.e'_1.h.e'_3.h.e'_1.h.e'_6
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ Ο βn ^ 2 * (βn ^ β(-x) * (βn ^ (-I * βy)) ^ 2) = βn ^ β(-x) * (Ο βn * βn ^ (-(I * βy))) ^ 2
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg_dirichlet | [106, 1] | [135, 37] | ring | N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ -(2 * I * βy) = β2 * (-I * βy) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : IsUnit βn
haβ : 0 β€ βn ^ (-x)
haβ : βn ^ (-x) < 1
hz : βΟ βn * βn ^ (-(I * βy))β = 1
β’ -(2 * I * βy) = β2 * (-I * βy)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | re_log_comb_nonneg_dirichlet | [106, 1] | [135, 37] | simp [MulChar.map_nonunit _ hn'] | case neg
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : Β¬IsUnit βn
β’ 0 β€
3 * (-(1 - 1 βn * βn ^ (-βx)).log).re + 4 * (-(1 - Ο βn * βn ^ (-(βx + I * βy))).log).re +
(-(1 - Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy))).log).re | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
N : β
Ο : DirichletCharacter β N
n : β
hn : 2 β€ n
x y : β
hx : 1 < x
hn' : Β¬IsUnit βn
β’ 0 β€
3 * (-(1 - 1 βn * βn ^ (-βx)).log).re + 4 * (-(1 - Ο βn * βn ^ (-(βx + I * βy))).log).re +
(-(1 - Ο βn ^ 2 * βn ^ (-(βx + 2 * I * βy))).log).re
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | one_lt_re_of_pos | [138, 1] | [141, 92] | simp only [add_re, one_re, ofReal_re, lt_add_iff_pos_right, hx, mul_re, I_re, zero_mul, I_im,
ofReal_im, mul_zero, sub_self, add_zero, re_ofNat, im_ofNat, mul_one, mul_im, and_self] | x y : β
hx : 0 < x
β’ 1 < (1 + βx).re β§ 1 < (1 + βx + I * βy).re β§ 1 < (1 + βx + 2 * I * βy).re | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : β
hx : 0 < x
β’ 1 < (1 + βx).re β§ 1 < (1 + βx + I * βy).re β§ 1 < (1 + βx + 2 * I * βy).re
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | norm_dirichlet_product_ge_one | [147, 1] | [174, 33] | let Οβ := (1 : DirichletCharacter β N) | N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
β’ βL (fun n => 1 βn) (1 + βx) ^ 3 * L (fun n => Ο βn) (1 + βx + I * βy) ^ 4 *
L (fun n => (Ο ^ 2) βn) (1 + βx + 2 * I * βy)β β₯
1 | N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
β’ βL (fun n => 1 βn) (1 + βx) ^ 3 * L (fun n => Ο βn) (1 + βx + I * βy) ^ 4 *
L (fun n => (Ο ^ 2) βn) (1 + βx + 2 * I * βy)β β₯
1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
β’ βL (fun n => 1 βn) (1 + βx) ^ 3 * L (fun n => Ο βn) (1 + βx + I * βy) ^ 4 *
L (fun n => (Ο ^ 2) βn) (1 + βx + 2 * I * βy)β β₯
1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | norm_dirichlet_product_ge_one | [147, 1] | [174, 33] | have β¨hβ, hβ, hββ© := one_lt_re_of_pos y hx | N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
β’ βL (fun n => 1 βn) (1 + βx) ^ 3 * L (fun n => Ο βn) (1 + βx + I * βy) ^ 4 *
L (fun n => (Ο ^ 2) βn) (1 + βx + 2 * I * βy)β β₯
1 | N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
hβ : 1 < (1 + βx).re
hβ : 1 < (1 + βx + I * βy).re
hβ : 1 < (1 + βx + 2 * I * βy).re
β’ βL (fun n => 1 βn) (1 + βx) ^ 3 * L (fun n => Ο βn) (1 + βx + I * βy) ^ 4 *
L (fun n => (Ο ^ 2) βn) (1 + βx + 2 * I * βy)β β₯
1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
β’ βL (fun n => 1 βn) (1 + βx) ^ 3 * L (fun n => Ο βn) (1 + βx + I * βy) ^ 4 *
L (fun n => (Ο ^ 2) βn) (1 + βx + 2 * I * βy)β β₯
1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | norm_dirichlet_product_ge_one | [147, 1] | [174, 33] | have hxβ : 1 + (x : β) = (1 + x : β).re := by simp only [add_re, one_re, ofReal_re, ofReal_add, ofReal_one] | N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
hβ : 1 < (1 + βx).re
hβ : 1 < (1 + βx + I * βy).re
hβ : 1 < (1 + βx + 2 * I * βy).re
β’ βL (fun n => 1 βn) (1 + βx) ^ 3 * L (fun n => Ο βn) (1 + βx + I * βy) ^ 4 *
L (fun n => (Ο ^ 2) βn) (1 + βx + 2 * I * βy)β β₯
1 | N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
hβ : 1 < (1 + βx).re
hβ : 1 < (1 + βx + I * βy).re
hβ : 1 < (1 + βx + 2 * I * βy).re
hxβ : 1 + βx = β(1 + βx).re
β’ βL (fun n => 1 βn) (1 + βx) ^ 3 * L (fun n => Ο βn) (1 + βx + I * βy) ^ 4 *
L (fun n => (Ο ^ 2) βn) (1 + βx + 2 * I * βy)β β₯
1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
hβ : 1 < (1 + βx).re
hβ : 1 < (1 + βx + I * βy).re
hβ : 1 < (1 + βx + 2 * I * βy).re
β’ βL (fun n => 1 βn) (1 + βx) ^ 3 * L (fun n => Ο βn) (1 + βx + I * βy) ^ 4 *
L (fun n => (Ο ^ 2) βn) (1 + βx + 2 * I * βy)β β₯
1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | norm_dirichlet_product_ge_one | [147, 1] | [174, 33] | have hsumβ :=
(hasSum_re (summable_neg_log_one_sub_char_mul_prime_cpow Οβ hβ).hasSum).summable.mul_left 3 | N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
hβ : 1 < (1 + βx).re
hβ : 1 < (1 + βx + I * βy).re
hβ : 1 < (1 + βx + 2 * I * βy).re
hxβ : 1 + βx = β(1 + βx).re
β’ βL (fun n => 1 βn) (1 + βx) ^ 3 * L (fun n => Ο βn) (1 + βx + I * βy) ^ 4 *
L (fun n => (Ο ^ 2) βn) (1 + βx + 2 * I * βy)β β₯
1 | N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
hβ : 1 < (1 + βx).re
hβ : 1 < (1 + βx + I * βy).re
hβ : 1 < (1 + βx + 2 * I * βy).re
hxβ : 1 + βx = β(1 + βx).re
hsumβ : Summable fun i => 3 * (-(1 - Οβ ββi * ββi ^ (-(1 + βx))).log).re
β’ βL (fun n => 1 βn) (1 + βx) ^ 3 * L (fun n => Ο βn) (1 + βx + I * βy) ^ 4 *
L (fun n => (Ο ^ 2) βn) (1 + βx + 2 * I * βy)β β₯
1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
hβ : 1 < (1 + βx).re
hβ : 1 < (1 + βx + I * βy).re
hβ : 1 < (1 + βx + 2 * I * βy).re
hxβ : 1 + βx = β(1 + βx).re
β’ βL (fun n => 1 βn) (1 + βx) ^ 3 * L (fun n => Ο βn) (1 + βx + I * βy) ^ 4 *
L (fun n => (Ο ^ 2) βn) (1 + βx + 2 * I * βy)β β₯
1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | norm_dirichlet_product_ge_one | [147, 1] | [174, 33] | have hsumβ :=
(hasSum_re (summable_neg_log_one_sub_char_mul_prime_cpow Ο hβ).hasSum).summable.mul_left 4 | N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
hβ : 1 < (1 + βx).re
hβ : 1 < (1 + βx + I * βy).re
hβ : 1 < (1 + βx + 2 * I * βy).re
hxβ : 1 + βx = β(1 + βx).re
hsumβ : Summable fun i => 3 * (-(1 - Οβ ββi * ββi ^ (-(1 + βx))).log).re
β’ βL (fun n => 1 βn) (1 + βx) ^ 3 * L (fun n => Ο βn) (1 + βx + I * βy) ^ 4 *
L (fun n => (Ο ^ 2) βn) (1 + βx + 2 * I * βy)β β₯
1 | N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
hβ : 1 < (1 + βx).re
hβ : 1 < (1 + βx + I * βy).re
hβ : 1 < (1 + βx + 2 * I * βy).re
hxβ : 1 + βx = β(1 + βx).re
hsumβ : Summable fun i => 3 * (-(1 - Οβ ββi * ββi ^ (-(1 + βx))).log).re
hsumβ : Summable fun i => 4 * (-(1 - Ο ββi * ββi ^ (-(1 + βx + I * βy))).log).re
β’ βL (fun n => 1 βn) (1 + βx) ^ 3 * L (fun n => Ο βn) (1 + βx + I * βy) ^ 4 *
L (fun n => (Ο ^ 2) βn) (1 + βx + 2 * I * βy)β β₯
1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
hβ : 1 < (1 + βx).re
hβ : 1 < (1 + βx + I * βy).re
hβ : 1 < (1 + βx + 2 * I * βy).re
hxβ : 1 + βx = β(1 + βx).re
hsumβ : Summable fun i => 3 * (-(1 - Οβ ββi * ββi ^ (-(1 + βx))).log).re
β’ βL (fun n => 1 βn) (1 + βx) ^ 3 * L (fun n => Ο βn) (1 + βx + I * βy) ^ 4 *
L (fun n => (Ο ^ 2) βn) (1 + βx + 2 * I * βy)β β₯
1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | norm_dirichlet_product_ge_one | [147, 1] | [174, 33] | have hsumβ :=
(hasSum_re (summable_neg_log_one_sub_char_mul_prime_cpow (Ο ^ 2) hβ).hasSum).summable | N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
hβ : 1 < (1 + βx).re
hβ : 1 < (1 + βx + I * βy).re
hβ : 1 < (1 + βx + 2 * I * βy).re
hxβ : 1 + βx = β(1 + βx).re
hsumβ : Summable fun i => 3 * (-(1 - Οβ ββi * ββi ^ (-(1 + βx))).log).re
hsumβ : Summable fun i => 4 * (-(1 - Ο ββi * ββi ^ (-(1 + βx + I * βy))).log).re
β’ βL (fun n => 1 βn) (1 + βx) ^ 3 * L (fun n => Ο βn) (1 + βx + I * βy) ^ 4 *
L (fun n => (Ο ^ 2) βn) (1 + βx + 2 * I * βy)β β₯
1 | N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
hβ : 1 < (1 + βx).re
hβ : 1 < (1 + βx + I * βy).re
hβ : 1 < (1 + βx + 2 * I * βy).re
hxβ : 1 + βx = β(1 + βx).re
hsumβ : Summable fun i => 3 * (-(1 - Οβ ββi * ββi ^ (-(1 + βx))).log).re
hsumβ : Summable fun i => 4 * (-(1 - Ο ββi * ββi ^ (-(1 + βx + I * βy))).log).re
hsumβ : Summable fun x_1 => (-(1 - (Ο ^ 2) ββx_1 * ββx_1 ^ (-(1 + βx + 2 * I * βy))).log).re
β’ βL (fun n => 1 βn) (1 + βx) ^ 3 * L (fun n => Ο βn) (1 + βx + I * βy) ^ 4 *
L (fun n => (Ο ^ 2) βn) (1 + βx + 2 * I * βy)β β₯
1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
hβ : 1 < (1 + βx).re
hβ : 1 < (1 + βx + I * βy).re
hβ : 1 < (1 + βx + 2 * I * βy).re
hxβ : 1 + βx = β(1 + βx).re
hsumβ : Summable fun i => 3 * (-(1 - Οβ ββi * ββi ^ (-(1 + βx))).log).re
hsumβ : Summable fun i => 4 * (-(1 - Ο ββi * ββi ^ (-(1 + βx + I * βy))).log).re
β’ βL (fun n => 1 βn) (1 + βx) ^ 3 * L (fun n => Ο βn) (1 + βx + I * βy) ^ 4 *
L (fun n => (Ο ^ 2) βn) (1 + βx + 2 * I * βy)β β₯
1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | norm_dirichlet_product_ge_one | [147, 1] | [174, 33] | rw [β DirichletCharacter.LSeries_eulerProduct' _ hβ,
β DirichletCharacter.LSeries_eulerProduct' Ο hβ,
β DirichletCharacter.LSeries_eulerProduct' (Ο ^ 2) hβ, β exp_nat_mul, β exp_nat_mul, β exp_add,
β exp_add, norm_eq_abs, abs_exp] | N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
hβ : 1 < (1 + βx).re
hβ : 1 < (1 + βx + I * βy).re
hβ : 1 < (1 + βx + 2 * I * βy).re
hxβ : 1 + βx = β(1 + βx).re
hsumβ : Summable fun i => 3 * (-(1 - Οβ ββi * ββi ^ (-(1 + βx))).log).re
hsumβ : Summable fun i => 4 * (-(1 - Ο ββi * ββi ^ (-(1 + βx + I * βy))).log).re
hsumβ : Summable fun x_1 => (-(1 - (Ο ^ 2) ββx_1 * ββx_1 ^ (-(1 + βx + 2 * I * βy))).log).re
β’ βL (fun n => 1 βn) (1 + βx) ^ 3 * L (fun n => Ο βn) (1 + βx + I * βy) ^ 4 *
L (fun n => (Ο ^ 2) βn) (1 + βx + 2 * I * βy)β β₯
1 | N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
hβ : 1 < (1 + βx).re
hβ : 1 < (1 + βx + I * βy).re
hβ : 1 < (1 + βx + 2 * I * βy).re
hxβ : 1 + βx = β(1 + βx).re
hsumβ : Summable fun i => 3 * (-(1 - Οβ ββi * ββi ^ (-(1 + βx))).log).re
hsumβ : Summable fun i => 4 * (-(1 - Ο ββi * ββi ^ (-(1 + βx + I * βy))).log).re
hsumβ : Summable fun x_1 => (-(1 - (Ο ^ 2) ββx_1 * ββx_1 ^ (-(1 + βx + 2 * I * βy))).log).re
β’ (β3 * β' (p : Primes), -(1 - 1 ββp * ββp ^ (-(1 + βx))).log +
β4 * β' (p : Primes), -(1 - Ο ββp * ββp ^ (-(1 + βx + I * βy))).log +
β' (p : Primes), -(1 - (Ο ^ 2) ββp * ββp ^ (-(1 + βx + 2 * I * βy))).log).re.exp β₯
1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
hβ : 1 < (1 + βx).re
hβ : 1 < (1 + βx + I * βy).re
hβ : 1 < (1 + βx + 2 * I * βy).re
hxβ : 1 + βx = β(1 + βx).re
hsumβ : Summable fun i => 3 * (-(1 - Οβ ββi * ββi ^ (-(1 + βx))).log).re
hsumβ : Summable fun i => 4 * (-(1 - Ο ββi * ββi ^ (-(1 + βx + I * βy))).log).re
hsumβ : Summable fun x_1 => (-(1 - (Ο ^ 2) ββx_1 * ββx_1 ^ (-(1 + βx + 2 * I * βy))).log).re
β’ βL (fun n => 1 βn) (1 + βx) ^ 3 * L (fun n => Ο βn) (1 + βx + I * βy) ^ 4 *
L (fun n => (Ο ^ 2) βn) (1 + βx + 2 * I * βy)β β₯
1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | norm_dirichlet_product_ge_one | [147, 1] | [174, 33] | simp only [Nat.cast_ofNat, add_re, mul_re, re_ofNat, im_ofNat, zero_mul, sub_zero,
Real.one_le_exp_iff] | N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
hβ : 1 < (1 + βx).re
hβ : 1 < (1 + βx + I * βy).re
hβ : 1 < (1 + βx + 2 * I * βy).re
hxβ : 1 + βx = β(1 + βx).re
hsumβ : Summable fun i => 3 * (-(1 - Οβ ββi * ββi ^ (-(1 + βx))).log).re
hsumβ : Summable fun i => 4 * (-(1 - Ο ββi * ββi ^ (-(1 + βx + I * βy))).log).re
hsumβ : Summable fun x_1 => (-(1 - (Ο ^ 2) ββx_1 * ββx_1 ^ (-(1 + βx + 2 * I * βy))).log).re
β’ (β3 * β' (p : Primes), -(1 - 1 ββp * ββp ^ (-(1 + βx))).log +
β4 * β' (p : Primes), -(1 - Ο ββp * ββp ^ (-(1 + βx + I * βy))).log +
β' (p : Primes), -(1 - (Ο ^ 2) ββp * ββp ^ (-(1 + βx + 2 * I * βy))).log).re.exp β₯
1 | N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
hβ : 1 < (1 + βx).re
hβ : 1 < (1 + βx + I * βy).re
hβ : 1 < (1 + βx + 2 * I * βy).re
hxβ : 1 + βx = β(1 + βx).re
hsumβ : Summable fun i => 3 * (-(1 - Οβ ββi * ββi ^ (-(1 + βx))).log).re
hsumβ : Summable fun i => 4 * (-(1 - Ο ββi * ββi ^ (-(1 + βx + I * βy))).log).re
hsumβ : Summable fun x_1 => (-(1 - (Ο ^ 2) ββx_1 * ββx_1 ^ (-(1 + βx + 2 * I * βy))).log).re
β’ 0 β€
3 * (β' (p : Primes), -(1 - 1 ββp * ββp ^ (-(1 + βx))).log).re +
4 * (β' (p : Primes), -(1 - Ο ββp * ββp ^ (-(1 + βx + I * βy))).log).re +
(β' (p : Primes), -(1 - (Ο ^ 2) ββp * ββp ^ (-(1 + βx + 2 * I * βy))).log).re | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
hβ : 1 < (1 + βx).re
hβ : 1 < (1 + βx + I * βy).re
hβ : 1 < (1 + βx + 2 * I * βy).re
hxβ : 1 + βx = β(1 + βx).re
hsumβ : Summable fun i => 3 * (-(1 - Οβ ββi * ββi ^ (-(1 + βx))).log).re
hsumβ : Summable fun i => 4 * (-(1 - Ο ββi * ββi ^ (-(1 + βx + I * βy))).log).re
hsumβ : Summable fun x_1 => (-(1 - (Ο ^ 2) ββx_1 * ββx_1 ^ (-(1 + βx + 2 * I * βy))).log).re
β’ (β3 * β' (p : Primes), -(1 - 1 ββp * ββp ^ (-(1 + βx))).log +
β4 * β' (p : Primes), -(1 - Ο ββp * ββp ^ (-(1 + βx + I * βy))).log +
β' (p : Primes), -(1 - (Ο ^ 2) ββp * ββp ^ (-(1 + βx + 2 * I * βy))).log).re.exp β₯
1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git | 21e07835d1a467b99b5c3c9390d61ae69404445d | EulerProducts/PNT.lean | norm_dirichlet_product_ge_one | [147, 1] | [174, 33] | rw [re_tsum <| summable_neg_log_one_sub_char_mul_prime_cpow _ hβ,
re_tsum <| summable_neg_log_one_sub_char_mul_prime_cpow _ hβ,
re_tsum <| summable_neg_log_one_sub_char_mul_prime_cpow _ hβ, β tsum_mul_left, β tsum_mul_left,
β tsum_add hsumβ hsumβ, β tsum_add (hsumβ.add hsumβ) hsumβ] | N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
hβ : 1 < (1 + βx).re
hβ : 1 < (1 + βx + I * βy).re
hβ : 1 < (1 + βx + 2 * I * βy).re
hxβ : 1 + βx = β(1 + βx).re
hsumβ : Summable fun i => 3 * (-(1 - Οβ ββi * ββi ^ (-(1 + βx))).log).re
hsumβ : Summable fun i => 4 * (-(1 - Ο ββi * ββi ^ (-(1 + βx + I * βy))).log).re
hsumβ : Summable fun x_1 => (-(1 - (Ο ^ 2) ββx_1 * ββx_1 ^ (-(1 + βx + 2 * I * βy))).log).re
β’ 0 β€
3 * (β' (p : Primes), -(1 - 1 ββp * ββp ^ (-(1 + βx))).log).re +
4 * (β' (p : Primes), -(1 - Ο ββp * ββp ^ (-(1 + βx + I * βy))).log).re +
(β' (p : Primes), -(1 - (Ο ^ 2) ββp * ββp ^ (-(1 + βx + 2 * I * βy))).log).re | N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
hβ : 1 < (1 + βx).re
hβ : 1 < (1 + βx + I * βy).re
hβ : 1 < (1 + βx + 2 * I * βy).re
hxβ : 1 + βx = β(1 + βx).re
hsumβ : Summable fun i => 3 * (-(1 - Οβ ββi * ββi ^ (-(1 + βx))).log).re
hsumβ : Summable fun i => 4 * (-(1 - Ο ββi * ββi ^ (-(1 + βx + I * βy))).log).re
hsumβ : Summable fun x_1 => (-(1 - (Ο ^ 2) ββx_1 * ββx_1 ^ (-(1 + βx + 2 * I * βy))).log).re
β’ 0 β€
β' (b : Primes),
(3 * (-(1 - Οβ ββb * ββb ^ (-(1 + βx))).log).re + 4 * (-(1 - Ο ββb * ββb ^ (-(1 + βx + I * βy))).log).re +
(-(1 - (Ο ^ 2) ββb * ββb ^ (-(1 + βx + 2 * I * βy))).log).re) | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
x : β
hx : 0 < x
y : β
Οβ : DirichletCharacter β N := 1
hβ : 1 < (1 + βx).re
hβ : 1 < (1 + βx + I * βy).re
hβ : 1 < (1 + βx + 2 * I * βy).re
hxβ : 1 + βx = β(1 + βx).re
hsumβ : Summable fun i => 3 * (-(1 - Οβ ββi * ββi ^ (-(1 + βx))).log).re
hsumβ : Summable fun i => 4 * (-(1 - Ο ββi * ββi ^ (-(1 + βx + I * βy))).log).re
hsumβ : Summable fun x_1 => (-(1 - (Ο ^ 2) ββx_1 * ββx_1 ^ (-(1 + βx + 2 * I * βy))).log).re
β’ 0 β€
3 * (β' (p : Primes), -(1 - 1 ββp * ββp ^ (-(1 + βx))).log).re +
4 * (β' (p : Primes), -(1 - Ο ββp * ββp ^ (-(1 + βx + I * βy))).log).re +
(β' (p : Primes), -(1 - (Ο ^ 2) ββp * ββp ^ (-(1 + βx + 2 * I * βy))).log).re
TACTIC:
|
Subsets and Splits