Theorem: limiting_fourier_lim3 (PrimeNumberTheoremAnd/Wiener.lean)

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
⊢ 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)))
Tactic:
by_cases hh : tsupport ψ = ∅
State after:
case pos
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 = ∅
⊢ 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)))

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 = ∅
⊢ 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)))

State before:
case pos
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 = ∅
⊢ 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)))

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 = ∅
⊢ 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)))
Tactic:
simp [tsupport_eq_empty_iff.mp hh]
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 = ∅
⊢ 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)))

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 = ∅
⊢ 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)))
Tactic:
obtain ⟨a₀, ha₀⟩ := Set.nonempty_iff_ne_empty.mpr hh
State after:
case neg.intro
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
⊢ 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)))

State before:
case neg.intro
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
⊢ 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)))
Tactic:
let S : Set ℂ := Set.reProdIm (Icc 1 2) (tsupport ψ)
State after:
case neg.intro
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
⊢ 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)))

State before:
case neg.intro
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
⊢ 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)))
Tactic:
have l2 : S ⊆ {s : ℂ | 1 ≤ s.re} := fun z hz => (mem_reProdIm.mp hz).1.1
State after:
case neg.intro
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}
⊢ 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)))

State before:
case neg.intro
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
⊢ 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)))
Tactic:
have l4 : S.Nonempty := ⟨1 + a₀ * I, by simp [S, mem_reProdIm, ha₀]⟩
State after:
case neg.intro
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
⊢ 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)))

State before:
case neg.intro
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
⊢ 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)))
Tactic:
obtain ⟨z, -, hmax⟩ := l1.exists_isMaxOn l4 l3
State after:
case neg.intro.intro.intro
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
⊢ 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)))

State before:
case neg.intro.intro.intro
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
⊢ 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)))
Tactic:
let MG := ‖G z‖
State after:
case neg.intro.intro.intro
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‖
⊢ 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)))

State before:
case neg.intro.intro.intro
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‖
⊢ 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)))
Tactic:
let bound (a : ℝ) : ℝ := MG * ‖ψ a‖
State after:
case neg.intro.intro.intro
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‖
⊢ 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)))

State before:
case neg.intro.intro.intro
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‖
⊢ 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)))
Tactic:
apply tendsto_integral_filter_of_dominated_convergence (bound := bound)
State after:
case neg.intro.intro.intro.hF_meas
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‖
⊢ ∀ᶠ (n : ℝ) in 𝓝[>] 1, AEStronglyMeasurable (fun a => G (↑n + ↑a * I) * ψ.toFun a * ↑x ^ (↑a * I)) volume

case neg.intro.intro.intro.h_bound
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‖
⊢ ∀ᶠ (n : ℝ) in 𝓝[>] 1, ∀ᵐ (a : ℝ), ‖G (↑n + ↑a * I) * ψ.toFun a * ↑x ^ (↑a * I)‖ ≤ bound a

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

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)))

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
⊢ IsCompact S
Tactic:
refine Metric.isCompact_iff_isClosed_bounded.mpr ⟨?_, ?_⟩
State after:
case 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
⊢ IsClosed S

case 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
⊢ Bornology.IsBounded S

State before:
case 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
⊢ IsClosed S
Tactic:
exact isClosed_Icc.reProdIm (isClosed_tsupport ψ)
State after:
no goals

State before:
case 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
⊢ Bornology.IsBounded S
Tactic:
exact (Metric.isBounded_Icc 1 2).reProdIm ψ.h2.isBounded
State after:
no goals

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
⊢ 1 + ↑a₀ * I ∈ S
Tactic:
simp [S, mem_reProdIm, ha₀]
State after:
no goals

State before:
case neg.intro.intro.intro.hF_meas
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‖
⊢ ∀ᶠ (n : ℝ) in 𝓝[>] 1, AEStronglyMeasurable (fun a => G (↑n + ↑a * I) * ψ.toFun a * ↑x ^ (↑a * I)) volume
Tactic:
apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp))
State after:
case neg.intro.intro.intro.hF_meas
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 ∈ Icc 1 2, AEStronglyMeasurable (fun a => G (↑x_1 + ↑a * I) * ψ.toFun a * ↑x ^ (↑a * I)) volume

State before:
case neg.intro.intro.intro.hF_meas
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 ∈ Icc 1 2, AEStronglyMeasurable (fun a => G (↑x_1 + ↑a * I) * ψ.toFun a * ↑x ^ (↑a * I)) volume
Tactic:
intro u hu
State after:
case neg.intro.intro.intro.hF_meas
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
⊢ AEStronglyMeasurable (fun a => G (↑u + ↑a * I) * ψ.toFun a * ↑x ^ (↑a * I)) volume

State before:
case neg.intro.intro.intro.hF_meas
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
⊢ AEStronglyMeasurable (fun a => G (↑u + ↑a * I) * ψ.toFun a * ↑x ^ (↑a * I)) volume
Tactic:
apply Continuous.aestronglyMeasurable
State after:
case neg.intro.intro.intro.hF_meas.hf
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
⊢ Continuous fun a => G (↑u + ↑a * I) * ψ.toFun a * ↑x ^ (↑a * I)

State before:
case neg.intro.intro.intro.hF_meas.hf
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
⊢ Continuous fun a => G (↑u + ↑a * I) * ψ.toFun a * ↑x ^ (↑a * I)
Tactic:
apply Continuous.mul
State after:
case neg.intro.intro.intro.hF_meas.hf.hf
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
⊢ Continuous fun x => G (↑u + ↑x * I) * ψ.toFun x

case neg.intro.intro.intro.hF_meas.hf.hg
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
⊢ Continuous fun x_1 => ↑x ^ (↑x_1 * I)

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‖
⊢ 1 ∈ Ico 1 2
Tactic:
simp
State after:
no goals

State before:
case neg.intro.intro.intro.hF_meas.hf.hf
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
⊢ Continuous fun x => G (↑u + ↑x * I) * ψ.toFun x
Tactic:
exact (hG.comp_continuous (by continuity) (by simp [hu.1])).mul ψ.h1.continuous
State after:
no goals

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‖
u : ℝ
hu : u ∈ Icc 1 2
⊢ Continuous fun x => ↑u + ↑x * I
Tactic:
continuity
State after:
no goals

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‖
u : ℝ
hu : u ∈ Icc 1 2
⊢ ∀ (x : ℝ), ↑u + ↑x * I ∈ {s | 1 ≤ s.re}
Tactic:
simp [hu.1]
State after:
no goals

State before:
case neg.intro.intro.intro.hF_meas.hf.hg
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
⊢ Continuous fun x_1 => ↑x ^ (↑x_1 * I)
Tactic:
apply Continuous.const_cpow (by continuity)
State after:
case neg.intro.intro.intro.hF_meas.hf.hg
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
⊢ ↑x ≠ 0 ∨ ∀ (a : ℝ), ↑a * I ≠ 0

State before:
case neg.intro.intro.intro.hF_meas.hf.hg
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
⊢ ↑x ≠ 0 ∨ ∀ (a : ℝ), ↑a * I ≠ 0
Tactic:
simp
State after:
case neg.intro.intro.intro.hF_meas.hf.hg
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
⊢ ¬x = 0

State before:
case neg.intro.intro.intro.hF_meas.hf.hg
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
⊢ ¬x = 0
Tactic:
linarith
State after:
no goals

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‖
u : ℝ
hu : u ∈ Icc 1 2
⊢ Continuous fun x => ↑x * I
Tactic:
continuity
State after:
no goals

State before:
case neg.intro.intro.intro.h_bound
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‖
⊢ ∀ᶠ (n : ℝ) in 𝓝[>] 1, ∀ᵐ (a : ℝ), ‖G (↑n + ↑a * I) * ψ.toFun a * ↑x ^ (↑a * I)‖ ≤ bound a
Tactic:
apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp))
State after:
case neg.intro.intro.intro.h_bound
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 ∈ Icc 1 2, ∀ᵐ (a : ℝ), ‖G (↑x_1 + ↑a * I) * ψ.toFun a * ↑x ^ (↑a * I)‖ ≤ bound a

State before:
case neg.intro.intro.intro.h_bound
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 ∈ Icc 1 2, ∀ᵐ (a : ℝ), ‖G (↑x_1 + ↑a * I) * ψ.toFun a * ↑x ^ (↑a * I)‖ ≤ bound a
Tactic:
intro u hu
State after:
case neg.intro.intro.intro.h_bound
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
⊢ ∀ᵐ (a : ℝ), ‖G (↑u + ↑a * I) * ψ.toFun a * ↑x ^ (↑a * I)‖ ≤ bound a

State before:
case neg.intro.intro.intro.h_bound
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
⊢ ∀ᵐ (a : ℝ), ‖G (↑u + ↑a * I) * ψ.toFun a * ↑x ^ (↑a * I)‖ ≤ bound a
Tactic:
apply eventually_of_forall
State after:
case neg.intro.intro.intro.h_bound.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‖
u : ℝ
hu : u ∈ Icc 1 2
⊢ ∀ (x_1 : ℝ), ‖G (↑u + ↑x_1 * I) * ψ.toFun x_1 * ↑x ^ (↑x_1 * I)‖ ≤ bound x_1

State before:
case neg.intro.intro.intro.h_bound.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‖
u : ℝ
hu : u ∈ Icc 1 2
⊢ ∀ (x_1 : ℝ), ‖G (↑u + ↑x_1 * I) * ψ.toFun x_1 * ↑x ^ (↑x_1 * I)‖ ≤ bound x_1
Tactic:
intro v
State after:
case neg.intro.intro.intro.h_bound.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‖
u : ℝ
hu : u ∈ Icc 1 2
v : ℝ
⊢ ‖G (↑u + ↑v * I) * ψ.toFun v * ↑x ^ (↑v * I)‖ ≤ bound v

State before:
case neg.intro.intro.intro.h_bound.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‖
u : ℝ
hu : u ∈ Icc 1 2
v : ℝ
⊢ ‖G (↑u + ↑v * I) * ψ.toFun v * ↑x ^ (↑v * I)‖ ≤ bound v
Tactic:
by_cases h : v ∈ tsupport ψ
State after:
case pos
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

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

State before:
case pos
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 r1 : u + v * I ∈ S := by simp [S, mem_reProdIm, hu.1, hu.2, h]
State after:
case pos
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
r1 : ↑u + ↑v * I ∈ S
⊢ ‖G (↑u + ↑v * I) * ψ.toFun v * ↑x ^ (↑v * I)‖ ≤ bound v

State before:
case pos
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
r1 : ↑u + ↑v * I ∈ S
⊢ ‖G (↑u + ↑v * I) * ψ.toFun v * ↑x ^ (↑v * I)‖ ≤ bound v
Tactic:
have r2 := isMaxOn_iff.mp hmax _ r1
State after:
case pos
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
r1 : ↑u + ↑v * I ∈ S
r2 : ‖G (↑u + ↑v * I)‖ ≤ ‖G z‖
⊢ ‖G (↑u + ↑v * I) * ψ.toFun v * ↑x ^ (↑v * I)‖ ≤ bound v

State before:
case pos
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
r1 : ↑u + ↑v * I ∈ S
r2 : ‖G (↑u + ↑v * I)‖ ≤ ‖G z‖
⊢ ‖G (↑u + ↑v * I) * ψ.toFun v * ↑x ^ (↑v * I)‖ ≤ bound v
Tactic:
have r4 : (x : ℂ) ≠ 0 := by simp ; linarith
State after:
case pos
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
r1 : ↑u + ↑v * I ∈ S
r2 : ‖G (↑u + ↑v * I)‖ ≤ ‖G z‖
r4 : ↑x ≠ 0
⊢ ‖G (↑u + ↑v * I) * ψ.toFun v * ↑x ^ (↑v * I)‖ ≤ bound v

State before:
case pos
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
r1 : ↑u + ↑v * I ∈ S
r2 : ‖G (↑u + ↑v * I)‖ ≤ ‖G z‖
r4 : ↑x ≠ 0
⊢ ‖G (↑u + ↑v * I) * ψ.toFun v * ↑x ^ (↑v * I)‖ ≤ bound v
Tactic:
have r5 : arg x = 0 := by simp [arg_eq_zero_iff] ; linarith
State after:
case pos
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
r1 : ↑u + ↑v * I ∈ S
r2 : ‖G (↑u + ↑v * I)‖ ≤ ‖G z‖
r4 : ↑x ≠ 0
r5 : (↑x).arg = 0
⊢ ‖G (↑u + ↑v * I) * ψ.toFun v * ↑x ^ (↑v * I)‖ ≤ bound v

State before:
case pos
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
r1 : ↑u + ↑v * I ∈ S
r2 : ‖G (↑u + ↑v * I)‖ ≤ ‖G z‖
r4 : ↑x ≠ 0
r5 : (↑x).arg = 0
⊢ ‖G (↑u + ↑v * I) * ψ.toFun v * ↑x ^ (↑v * I)‖ ≤ bound v
Tactic:
have r3 : ‖(x : ℂ) ^ (v * I)‖ = 1 := by simp [abs_cpow_of_ne_zero r4, r5]
State after:
case pos
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
r1 : ↑u + ↑v * I ∈ S
r2 : ‖G (↑u + ↑v * I)‖ ≤ ‖G z‖
r4 : ↑x ≠ 0
r5 : (↑x).arg = 0
r3 : ‖↑x ^ (↑v * I)‖ = 1
⊢ ‖G (↑u + ↑v * I) * ψ.toFun v * ↑x ^ (↑v * I)‖ ≤ bound v

State before:
case pos
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
r1 : ↑u + ↑v * I ∈ S
r2 : ‖G (↑u + ↑v * I)‖ ≤ ‖G z‖
r4 : ↑x ≠ 0
r5 : (↑x).arg = 0
r3 : ‖↑x ^ (↑v * I)‖ = 1
⊢ ‖G (↑u + ↑v * I) * ψ.toFun v * ↑x ^ (↑v * I)‖ ≤ bound v
Tactic:
simp_rw [norm_mul, r3, mul_one]
State after:
case pos
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
r1 : ↑u + ↑v * I ∈ S
r2 : ‖G (↑u + ↑v * I)‖ ≤ ‖G z‖
r4 : ↑x ≠ 0
r5 : (↑x).arg = 0
r3 : ‖↑x ^ (↑v * I)‖ = 1
⊢ ‖G (↑u + ↑v * I)‖ * ‖ψ.toFun v‖ ≤ bound v

State before:
case pos
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
r1 : ↑u + ↑v * I ∈ S
r2 : ‖G (↑u + ↑v * I)‖ ≤ ‖G z‖
r4 : ↑x ≠ 0
r5 : (↑x).arg = 0
r3 : ‖↑x ^ (↑v * I)‖ = 1
⊢ ‖G (↑u + ↑v * I)‖ * ‖ψ.toFun v‖ ≤ bound v
Tactic:
exact mul_le_mul_of_nonneg_right r2 (norm_nonneg _)
State after:
no goals

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‖
u : ℝ
hu : u ∈ Icc 1 2
v : ℝ
h : v ∈ tsupport ψ.toFun
⊢ ↑u + ↑v * I ∈ S
Tactic:
simp [S, mem_reProdIm, hu.1, hu.2, h]
State after:
no goals

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‖
u : ℝ
hu : u ∈ Icc 1 2
v : ℝ
h : v ∈ tsupport ψ.toFun
r1 : ↑u + ↑v * I ∈ S
r2 : ‖G (↑u + ↑v * I)‖ ≤ ‖G z‖
⊢ ↑x ≠ 0
Tactic:
simp
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
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
r1 : ↑u + ↑v * I ∈ S
r2 : ‖G (↑u + ↑v * I)‖ ≤ ‖G z‖
⊢ ¬x = 0

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‖
u : ℝ
hu : u ∈ Icc 1 2
v : ℝ
h : v ∈ tsupport ψ.toFun
r1 : ↑u + ↑v * I ∈ S
r2 : ‖G (↑u + ↑v * I)‖ ≤ ‖G z‖
⊢ ¬x = 0
Tactic:
linarith
State after:
no goals

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‖
u : ℝ
hu : u ∈ Icc 1 2
v : ℝ
h : v ∈ tsupport ψ.toFun
r1 : ↑u + ↑v * I ∈ S
r2 : ‖G (↑u + ↑v * I)‖ ≤ ‖G z‖
r4 : ↑x ≠ 0
⊢ (↑x).arg = 0
Tactic:
simp [arg_eq_zero_iff]
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
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
r1 : ↑u + ↑v * I ∈ S
r2 : ‖G (↑u + ↑v * I)‖ ≤ ‖G z‖
r4 : ↑x ≠ 0
⊢ 0 ≤ x

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‖
u : ℝ
hu : u ∈ Icc 1 2
v : ℝ
h : v ∈ tsupport ψ.toFun
r1 : ↑u + ↑v * I ∈ S
r2 : ‖G (↑u + ↑v * I)‖ ≤ ‖G z‖
r4 : ↑x ≠ 0
⊢ 0 ≤ x
Tactic:
linarith
State after:
no goals

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‖
u : ℝ
hu : u ∈ Icc 1 2
v : ℝ
h : v ∈ tsupport ψ.toFun
r1 : ↑u + ↑v * I ∈ S
r2 : ‖G (↑u + ↑v * I)‖ ≤ ‖G z‖
r4 : ↑x ≠ 0
r5 : (↑x).arg = 0
⊢ ‖↑x ^ (↑v * I)‖ = 1
Tactic:
simp [abs_cpow_of_ne_zero r4, r5]
State after:
no goals

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)
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

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
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

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]
State after:
no goals

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
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

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
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

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
State after:
no goals

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
State after:
no goals

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
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)))

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
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)))

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
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))

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
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)))

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 ⟨?_, ?_⟩
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}

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
State after:
no goals

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
State after:
no goals

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])
State after:
no goals

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]
State after:
no goals