Theorem: second_fourier_integrable_aux1a (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
hσ : 1 < σ'
⊢ IntegrableOn (fun x => cexp (-(↑x * (↑σ' - 1)))) (Ici (-x.log)) volume
Tactic:
norm_cast
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
hσ : 1 < σ'
⊢ IntegrableOn (fun x => ↑(rexp (-(x * (σ' - 1))))) (Ici (-x.log)) volume

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
hσ : 1 < σ'
⊢ IntegrableOn (fun x => ↑(rexp (-(x * (σ' - 1))))) (Ici (-x.log)) volume
Tactic:
suffices IntegrableOn (fun (x : ℝ) ↦ (rexp (-(x * (σ' - 1))))) (Ici (-x.log)) _ from this.ofReal
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
hσ : 1 < σ'
⊢ IntegrableOn (fun x => rexp (-(x * (σ' - 1)))) (Ici (-x.log)) volume

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
hσ : 1 < σ'
⊢ IntegrableOn (fun x => rexp (-(x * (σ' - 1)))) (Ici (-x.log)) volume
Tactic:
simp_rw [fun (a x : ℝ) ↦ (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]
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
hσ : 1 < σ'
⊢ IntegrableOn (fun x => rexp (-(σ' - 1) * x)) (Ioi (-x.log)) volume

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
hσ : 1 < σ'
⊢ IntegrableOn (fun x => rexp (-(σ' - 1) * x)) (Ioi (-x.log)) volume
Tactic:
apply exp_neg_integrableOn_Ioi
State after:
case 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
hσ : 1 < σ'
⊢ 0 < σ' - 1

State before:
case 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
hσ : 1 < σ'
⊢ 0 < σ' - 1
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
hσ : 1 < σ'
a x : ℝ
⊢ -(x * a) = -a * x
Tactic:
ring
State after:
no goals