internlm/Lean-Github huggingface dataset:
Theorem: limiting_fourier_lim3
Proof state before:
case neg
n : ℕ
A a b c d u✝ x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
u : ℝ
hu : u ∈ Icc 1 2
v : ℝ
h : v ∉ tsupport ψ.toFun
⊢ ‖G (↑u + ↑v * I) * ψ.toFun v * ↑x ^ (↑v * I)‖ ≤ bound v
Tactic:
have : v ∉ Function.support ψ := fun a ↦ h (subset_tsupport ψ a)
Proof state after:
case neg
n : ℕ
A a b c d u✝ x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
u : ℝ
hu : u ∈ Icc 1 2
v : ℝ
h : v ∉ tsupport ψ.toFun
this : v ∉ Function.support ψ.toFun
⊢ ‖G (↑u + ↑v * I) * ψ.toFun v * ↑x ^ (↑v * I)‖ ≤ bound v
Theorem: limiting_fourier_lim3
Proof state before:
case neg
n : ℕ
A a b c d u✝ x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
u : ℝ
hu : u ∈ Icc 1 2
v : ℝ
h : v ∉ tsupport ψ.toFun
this : v ∉ Function.support ψ.toFun
⊢ ‖G (↑u + ↑v * I) * ψ.toFun v * ↑x ^ (↑v * I)‖ ≤ bound v
Tactic:
simp at this
Proof state after:
case neg
n : ℕ
A a b c d u✝ x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
u : ℝ
hu : u ∈ Icc 1 2
v : ℝ
h : v ∉ tsupport ψ.toFun
this : ψ.toFun v = 0
⊢ ‖G (↑u + ↑v * I) * ψ.toFun v * ↑x ^ (↑v * I)‖ ≤ bound v
Theorem: limiting_fourier_lim3
Proof state before:
case neg
n : ℕ
A a b c d u✝ x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
u : ℝ
hu : u ∈ Icc 1 2
v : ℝ
h : v ∉ tsupport ψ.toFun
this : ψ.toFun v = 0
⊢ ‖G (↑u + ↑v * I) * ψ.toFun v * ↑x ^ (↑v * I)‖ ≤ bound v
Tactic:
simp [this, bound]
Proof state after:
no goals
Theorem: limiting_fourier_lim3
Proof state before:
case neg.intro.intro.intro.bound_integrable
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
⊢ Integrable bound volume
Tactic:
suffices h : Continuous bound by exact h.integrable_of_hasCompactSupport ψ.h2.norm.mul_left
Proof state after:
case neg.intro.intro.intro.bound_integrable
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
⊢ Continuous bound
Theorem: limiting_fourier_lim3
Proof state before:
case neg.intro.intro.intro.bound_integrable
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
⊢ Continuous bound
Tactic:
have := ψ.h1.continuous
Proof state after:
case neg.intro.intro.intro.bound_integrable
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
this : Continuous ψ.toFun
⊢ Continuous bound
Theorem: limiting_fourier_lim3
Proof state before:
case neg.intro.intro.intro.bound_integrable
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
this : Continuous ψ.toFun
⊢ Continuous bound
Tactic:
continuity
Proof state after:
no goals
Theorem: limiting_fourier_lim3
Proof state before:
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
h : Continuous bound
⊢ Integrable bound volume
Tactic:
exact h.integrable_of_hasCompactSupport ψ.h2.norm.mul_left
Proof state after:
no goals
Theorem: limiting_fourier_lim3
Proof state before:
case neg.intro.intro.intro.h_lim
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
⊢ ∀ᵐ (a : ℝ),
    Tendsto (fun n => G (↑n + ↑a * I) * ψ.toFun a * ↑x ^ (↑a * I)) (𝓝[>] 1)
      (𝓝 (G (1 + ↑a * I) * ψ.toFun a * ↑x ^ (↑a * I)))
Tactic:
apply eventually_of_forall
Proof state after:
case neg.intro.intro.intro.h_lim.hp
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
⊢ ∀ (x_1 : ℝ),
    Tendsto (fun n => G (↑n + ↑x_1 * I) * ψ.toFun x_1 * ↑x ^ (↑x_1 * I)) (𝓝[>] 1)
      (𝓝 (G (1 + ↑x_1 * I) * ψ.toFun x_1 * ↑x ^ (↑x_1 * I)))
Theorem: limiting_fourier_lim3
Proof state before:
case neg.intro.intro.intro.h_lim.hp
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
⊢ ∀ (x_1 : ℝ),
    Tendsto (fun n => G (↑n + ↑x_1 * I) * ψ.toFun x_1 * ↑x ^ (↑x_1 * I)) (𝓝[>] 1)
      (𝓝 (G (1 + ↑x_1 * I) * ψ.toFun x_1 * ↑x ^ (↑x_1 * I)))
Tactic:
intro t
Proof state after:
case neg.intro.intro.intro.h_lim.hp
n : ℕ
A a b c d u x y t✝ σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
t : ℝ
⊢ Tendsto (fun n => G (↑n + ↑t * I) * ψ.toFun t * ↑x ^ (↑t * I)) (𝓝[>] 1)
    (𝓝 (G (1 + ↑t * I) * ψ.toFun t * ↑x ^ (↑t * I)))
Theorem: limiting_fourier_lim3
Proof state before:
case neg.intro.intro.intro.h_lim.hp
n : ℕ
A a b c d u x y t✝ σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
t : ℝ
⊢ Tendsto (fun n => G (↑n + ↑t * I) * ψ.toFun t * ↑x ^ (↑t * I)) (𝓝[>] 1)
    (𝓝 (G (1 + ↑t * I) * ψ.toFun t * ↑x ^ (↑t * I)))
Tactic:
apply Tendsto.mul_const
Proof state after:
case neg.intro.intro.intro.h_lim.hp.h
n : ℕ
A a b c d u x y t✝ σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
t : ℝ
⊢ Tendsto (fun k => G (↑k + ↑t * I) * ψ.toFun t) (𝓝[>] 1) (𝓝 (G (1 + ↑t * I) * ψ.toFun t))
Theorem: limiting_fourier_lim3
Proof state before:
case neg.intro.intro.intro.h_lim.hp.h
n : ℕ
A a b c d u x y t✝ σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
t : ℝ
⊢ Tendsto (fun k => G (↑k + ↑t * I) * ψ.toFun t) (𝓝[>] 1) (𝓝 (G (1 + ↑t * I) * ψ.toFun t))
Tactic:
apply Tendsto.mul_const
Proof state after:
case neg.intro.intro.intro.h_lim.hp.h.h
n : ℕ
A a b c d u x y t✝ σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
t : ℝ
⊢ Tendsto (fun k => G (↑k + ↑t * I)) (𝓝[>] 1) (𝓝 (G (1 + ↑t * I)))
Theorem: limiting_fourier_lim3
Proof state before:
case neg.intro.intro.intro.h_lim.hp.h.h
n : ℕ
A a b c d u x y t✝ σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
t : ℝ
⊢ Tendsto (fun k => G (↑k + ↑t * I)) (𝓝[>] 1) (𝓝 (G (1 + ↑t * I)))
Tactic:
refine (hG (1 + t * I) (by simp)).tendsto.comp <| tendsto_nhdsWithin_iff.mpr ⟨?_, ?_⟩
Proof state after:
case neg.intro.intro.intro.h_lim.hp.h.h.refine_1
n : ℕ
A a b c d u x y t✝ σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
t : ℝ
⊢ Tendsto (fun k => ↑k + ↑t * I) (𝓝[>] 1) (𝓝 (1 + ↑t * I))

case neg.intro.intro.intro.h_lim.hp.h.h.refine_2
n : ℕ
A a b c d u x y t✝ σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
t : ℝ
⊢ ∀ᶠ (n : ℝ) in 𝓝[>] 1, ↑n + ↑t * I ∈ {s | 1 ≤ s.re}
Theorem: limiting_fourier_lim3
Proof state before:
n : ℕ
A a b c d u x y t✝ σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
t : ℝ
⊢ 1 + ↑t * I ∈ {s | 1 ≤ s.re}
Tactic:
simp
Proof state after:
no goals
Theorem: limiting_fourier_lim3
Proof state before:
case neg.intro.intro.intro.h_lim.hp.h.h.refine_1
n : ℕ
A a b c d u x y t✝ σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
t : ℝ
⊢ Tendsto (fun k => ↑k + ↑t * I) (𝓝[>] 1) (𝓝 (1 + ↑t * I))
Tactic:
exact ((continuous_ofReal.tendsto _).add tendsto_const_nhds).mono_left nhdsWithin_le_nhds
Proof state after:
no goals
Theorem: limiting_fourier_lim3
Proof state before:
case neg.intro.intro.intro.h_lim.hp.h.h.refine_2
n : ℕ
A a b c d u x y t✝ σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx : 1 ≤ x
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
t : ℝ
⊢ ∀ᶠ (n : ℝ) in 𝓝[>] 1, ↑n + ↑t * I ∈ {s | 1 ≤ s.re}
Tactic:
exact eventually_nhdsWithin_of_forall (fun x (hx : 1 < x) => by simp [hx.le])
Proof state after:
no goals
Theorem: limiting_fourier_lim3
Proof state before:
n : ℕ
A a b c d u x✝ y t✝ σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hG : ContinuousOn G {s | 1 ≤ s.re}
ψ : CS 2 ℂ
hx✝ : 1 ≤ x✝
hh : ¬tsupport ψ.toFun = ∅
a₀ : ℝ
ha₀ : a₀ ∈ tsupport ψ.toFun
S : Set ℂ := Icc 1 2 ×ℂ tsupport ψ.toFun
l1 : IsCompact S
l2 : S ⊆ {s | 1 ≤ s.re}
l3 : ContinuousOn (fun x => ‖G x‖) S
l4 : S.Nonempty
z : ℂ
hmax : IsMaxOn (fun x => ‖G x‖) S z
MG : ℝ := ‖G z‖
bound : ℝ → ℝ := fun a => MG * ‖ψ.toFun a‖
t x : ℝ
hx : 1 < x
⊢ ↑x + ↑t * I ∈ {s | 1 ≤ s.re}
Tactic:
simp [hx.le]
Proof state after:
no goals
Theorem: limiting_fourier
Proof state before:
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hcheby : cheby f
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries f s - ↑A / (s - 1)) {s | 1 < s.re}
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm f σ')
ψ : CS 2 ℂ
hx : 1 ≤ x
⊢ ∑' (n : ℕ), f n / ↑n * 𝓕 ψ.toFun (1 / (2 * π) * (↑n / x).log) -
      ↑A * ∫ (u : ℝ) in Ici (-x.log), 𝓕 ψ.toFun (u / (2 * π)) =
    ∫ (t : ℝ), G (1 + ↑t * I) * ψ.toFun t * ↑x ^ (↑t * I)
Tactic:
have l1 := limiting_fourier_lim1 hcheby ψ (by linarith)
Proof state after:
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hcheby : cheby f
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries f s - ↑A / (s - 1)) {s | 1 < s.re}
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm f σ')
ψ : CS 2 ℂ
hx : 1 ≤ x
l1 :
  Tendsto (fun σ' => ∑' (n : ℕ), term f (↑σ') n * 𝓕 (W21.ofCS2 ψ).toFun (1 / (2 * π) * (↑n / x).log)) (𝓝[>] 1)
    (𝓝 (∑' (n : ℕ), f n / ↑n * 𝓕 (W21.ofCS2 ψ).toFun (1 / (2 * π) * (↑n / x).log)))
⊢ ∑' (n : ℕ), f n / ↑n * 𝓕 ψ.toFun (1 / (2 * π) * (↑n / x).log) -
      ↑A * ∫ (u : ℝ) in Ici (-x.log), 𝓕 ψ.toFun (u / (2 * π)) =
    ∫ (t : ℝ), G (1 + ↑t * I) * ψ.toFun t * ↑x ^ (↑t * I)
Theorem: limiting_fourier
Proof state before:
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hcheby : cheby f
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries f s - ↑A / (s - 1)) {s | 1 < s.re}
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm f σ')
ψ : CS 2 ℂ
hx : 1 ≤ x
l1 :
  Tendsto (fun σ' => ∑' (n : ℕ), term f (↑σ') n * 𝓕 (W21.ofCS2 ψ).toFun (1 / (2 * π) * (↑n / x).log)) (𝓝[>] 1)
    (𝓝 (∑' (n : ℕ), f n / ↑n * 𝓕 (W21.ofCS2 ψ).toFun (1 / (2 * π) * (↑n / x).log)))
⊢ ∑' (n : ℕ), f n / ↑n * 𝓕 ψ.toFun (1 / (2 * π) * (↑n / x).log) -
      ↑A * ∫ (u : ℝ) in Ici (-x.log), 𝓕 ψ.toFun (u / (2 * π)) =
    ∫ (t : ℝ), G (1 + ↑t * I) * ψ.toFun t * ↑x ^ (↑t * I)
Tactic:
have l2 := limiting_fourier_lim2 A ψ hx
Proof state after:
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hcheby : cheby f
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries f s - ↑A / (s - 1)) {s | 1 < s.re}
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm f σ')
ψ : CS 2 ℂ
hx : 1 ≤ x
l1 :
  Tendsto (fun σ' => ∑' (n : ℕ), term f (↑σ') n * 𝓕 (W21.ofCS2 ψ).toFun (1 / (2 * π) * (↑n / x).log)) (𝓝[>] 1)
    (𝓝 (∑' (n : ℕ), f n / ↑n * 𝓕 (W21.ofCS2 ψ).toFun (1 / (2 * π) * (↑n / x).log)))
l2 :
  Tendsto
    (fun σ' =>
      ↑A * ↑(x ^ (1 - σ')) * ∫ (u : ℝ) in Ici (-x.log), ↑(rexp (-u * (σ' - 1))) * 𝓕 (W21.ofCS2 ψ).toFun (u / (2 * π)))
    (𝓝[>] 1) (𝓝 (↑A * ∫ (u : ℝ) in Ici (-x.log), 𝓕 (W21.ofCS2 ψ).toFun (u / (2 * π))))
⊢ ∑' (n : ℕ), f n / ↑n * 𝓕 ψ.toFun (1 / (2 * π) * (↑n / x).log) -
      ↑A * ∫ (u : ℝ) in Ici (-x.log), 𝓕 ψ.toFun (u / (2 * π)) =
    ∫ (t : ℝ), G (1 + ↑t * I) * ψ.toFun t * ↑x ^ (↑t * I)
Theorem: limiting_fourier
Proof state before:
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hcheby : cheby f
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries f s - ↑A / (s - 1)) {s | 1 < s.re}
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm f σ')
ψ : CS 2 ℂ
hx : 1 ≤ x
l1 :
  Tendsto (fun σ' => ∑' (n : ℕ), term f (↑σ') n * 𝓕 (W21.ofCS2 ψ).toFun (1 / (2 * π) * (↑n / x).log)) (𝓝[>] 1)
    (𝓝 (∑' (n : ℕ), f n / ↑n * 𝓕 (W21.ofCS2 ψ).toFun (1 / (2 * π) * (↑n / x).log)))
l2 :
  Tendsto
    (fun σ' =>
      ↑A * ↑(x ^ (1 - σ')) * ∫ (u : ℝ) in Ici (-x.log), ↑(rexp (-u * (σ' - 1))) * 𝓕 (W21.ofCS2 ψ).toFun (u / (2 * π)))
    (𝓝[>] 1) (𝓝 (↑A * ∫ (u : ℝ) in Ici (-x.log), 𝓕 (W21.ofCS2 ψ).toFun (u / (2 * π))))
⊢ ∑' (n : ℕ), f n / ↑n * 𝓕 ψ.toFun (1 / (2 * π) * (↑n / x).log) -
      ↑A * ∫ (u : ℝ) in Ici (-x.log), 𝓕 ψ.toFun (u / (2 * π)) =
    ∫ (t : ℝ), G (1 + ↑t * I) * ψ.toFun t * ↑x ^ (↑t * I)
Tactic:
have l3 := limiting_fourier_lim3 hG ψ hx
Proof state after:
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hcheby : cheby f
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries f s - ↑A / (s - 1)) {s | 1 < s.re}
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm f σ')
ψ : CS 2 ℂ
hx : 1 ≤ x
l1 :
  Tendsto (fun σ' => ∑' (n : ℕ), term f (↑σ') n * 𝓕 (W21.ofCS2 ψ).toFun (1 / (2 * π) * (↑n / x).log)) (𝓝[>] 1)
    (𝓝 (∑' (n : ℕ), f n / ↑n * 𝓕 (W21.ofCS2 ψ).toFun (1 / (2 * π) * (↑n / x).log)))
l2 :
  Tendsto
    (fun σ' =>
      ↑A * ↑(x ^ (1 - σ')) * ∫ (u : ℝ) in Ici (-x.log), ↑(rexp (-u * (σ' - 1))) * 𝓕 (W21.ofCS2 ψ).toFun (u / (2 * π)))
    (𝓝[>] 1) (𝓝 (↑A * ∫ (u : ℝ) in Ici (-x.log), 𝓕 (W21.ofCS2 ψ).toFun (u / (2 * π))))
l3 :
  Tendsto (fun σ' => ∫ (t : ℝ), G (↑σ' + ↑t * I) * ψ.toFun t * ↑x ^ (↑t * I)) (𝓝[>] 1)
    (𝓝 (∫ (t : ℝ), G (1 + ↑t * I) * ψ.toFun t * ↑x ^ (↑t * I)))
⊢ ∑' (n : ℕ), f n / ↑n * 𝓕 ψ.toFun (1 / (2 * π) * (↑n / x).log) -
      ↑A * ∫ (u : ℝ) in Ici (-x.log), 𝓕 ψ.toFun (u / (2 * π)) =
    ∫ (t : ℝ), G (1 + ↑t * I) * ψ.toFun t * ↑x ^ (↑t * I)
Theorem: limiting_fourier
Proof state before:
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hcheby : cheby f
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries f s - ↑A / (s - 1)) {s | 1 < s.re}
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm f σ')
ψ : CS 2 ℂ
hx : 1 ≤ x
l1 :
  Tendsto (fun σ' => ∑' (n : ℕ), term f (↑σ') n * 𝓕 (W21.ofCS2 ψ).toFun (1 / (2 * π) * (↑n / x).log)) (𝓝[>] 1)
    (𝓝 (∑' (n : ℕ), f n / ↑n * 𝓕 (W21.ofCS2 ψ).toFun (1 / (2 * π) * (↑n / x).log)))
l2 :
  Tendsto
    (fun σ' =>
      ↑A * ↑(x ^ (1 - σ')) * ∫ (u : ℝ) in Ici (-x.log), ↑(rexp (-u * (σ' - 1))) * 𝓕 (W21.ofCS2 ψ).toFun (u / (2 * π)))
    (𝓝[>] 1) (𝓝 (↑A * ∫ (u : ℝ) in Ici (-x.log), 𝓕 (W21.ofCS2 ψ).toFun (u / (2 * π))))
l3 :
  Tendsto (fun σ' => ∫ (t : ℝ), G (↑σ' + ↑t * I) * ψ.toFun t * ↑x ^ (↑t * I)) (𝓝[>] 1)
    (𝓝 (∫ (t : ℝ), G (1 + ↑t * I) * ψ.toFun t * ↑x ^ (↑t * I)))
⊢ ∑' (n : ℕ), f n / ↑n * 𝓕 ψ.toFun (1 / (2 * π) * (↑n / x).log) -
      ↑A * ∫ (u : ℝ) in Ici (-x.log), 𝓕 ψ.toFun (u / (2 * π)) =
    ∫ (t : ℝ), G (1 + ↑t * I) * ψ.toFun t * ↑x ^ (↑t * I)
Tactic:
apply tendsto_nhds_unique_of_eventuallyEq (l1.sub l2) l3
Proof state after:
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
hcheby : cheby f
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries f s - ↑A / (s - 1)) {s | 1 < s.re}
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm f σ')
ψ : CS 2 ℂ
hx : 1 ≤ x
l1 :
  Tendsto (fun σ' => ∑' (n : ℕ), term f (↑σ') n * 𝓕 (W21.ofCS2 ψ).toFun (1 / (2 * π) * (↑n / x).log)) (𝓝[>] 1)
    (𝓝 (∑' (n : ℕ), f n / ↑n * 𝓕 (W21.ofCS2 ψ).toFun (1 / (2 * π) * (↑n / x).log)))
l2 :
  Tendsto
    (fun σ' =>
      ↑A * ↑(x ^ (1 - σ')) * ∫ (u : ℝ) in Ici (-x.log), ↑(rexp (-u * (σ' - 1))) * 𝓕 (W21.ofCS2 ψ).toFun (u / (2 * π)))
    (𝓝[>] 1) (𝓝 (↑A * ∫ (u : ℝ) in Ici (-x.log), 𝓕 (W21.ofCS2 ψ).toFun (u / (2 * π))))
l3 :
  Tendsto (fun σ' => ∫ (t : ℝ), G (↑σ' + ↑t * I) * ψ.toFun t * ↑x ^ (↑t * I)) (𝓝[>] 1)
    (𝓝 (∫ (t : ℝ), G (1 + ↑t * I) * ψ.toFun t * ↑x ^ (↑t * I)))
⊢ (fun x_1 =>
      ∑' (n : ℕ), term f (↑x_1) n * 𝓕 (W21.ofCS2 ψ).toFun (1 / (2 * π) * (↑n / x).log) -
        ↑A * ↑(x ^ (1 - x_1)) *
          ∫ (u : ℝ) in Ici (-x.log), ↑(rexp (-u * (x_1 - 1))) * 𝓕 (W21.ofCS2 ψ).toFun (u / (2 * π))) =ᶠ[𝓝[>] 1]
    fun σ' => ∫ (t : ℝ), G (↑σ' + ↑t * I) * ψ.toFun t * ↑x ^ (↑t * I)