Theorem: hh_antitone (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
a : ℝ
ha : a ∈ Ioo (-1) 1
⊢ AntitoneOn (hh a) (Ioi 0)
Tactic:
have l1 x (hx : x ∈ interior (Ioi 0)) : HasDerivWithinAt (hh a) (hh' a x) (interior (Ioi 0)) x := by
  have : x ≠ 0 := by contrapose! hx ; simp [hx]
  exact (hh_deriv a this).hasDerivWithinAt
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
a : ℝ
ha : a ∈ Ioo (-1) 1
l1 : ∀ x ∈ interior (Ioi 0), HasDerivWithinAt (hh a) (hh' a x) (interior (Ioi 0)) x
⊢ AntitoneOn (hh a) (Ioi 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
a : ℝ
ha : a ∈ Ioo (-1) 1
l1 : ∀ x ∈ interior (Ioi 0), HasDerivWithinAt (hh a) (hh' a x) (interior (Ioi 0)) x
⊢ AntitoneOn (hh a) (Ioi 0)
Tactic:
apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Ioi _) (hh_continuous _) l1 (fun x _ => hh'_nonpos ha)
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
a : ℝ
ha : a ∈ Ioo (-1) 1
x : ℝ
hx : x ∈ interior (Ioi 0)
⊢ HasDerivWithinAt (hh a) (hh' a x) (interior (Ioi 0)) x
Tactic:
have : x ≠ 0 := by contrapose! hx ; simp [hx]
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
a : ℝ
ha : a ∈ Ioo (-1) 1
x : ℝ
hx : x ∈ interior (Ioi 0)
this : x ≠ 0
⊢ HasDerivWithinAt (hh a) (hh' a x) (interior (Ioi 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
a : ℝ
ha : a ∈ Ioo (-1) 1
x : ℝ
hx : x ∈ interior (Ioi 0)
this : x ≠ 0
⊢ HasDerivWithinAt (hh a) (hh' a x) (interior (Ioi 0)) x
Tactic:
exact (hh_deriv a this).hasDerivWithinAt
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
a : ℝ
ha : a ∈ Ioo (-1) 1
x : ℝ
hx : x ∈ interior (Ioi 0)
⊢ x ≠ 0
Tactic:
contrapose! hx
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
a : ℝ
ha : a ∈ Ioo (-1) 1
x : ℝ
hx : x = 0
⊢ x ∉ interior (Ioi 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
a : ℝ
ha : a ∈ Ioo (-1) 1
x : ℝ
hx : x = 0
⊢ x ∉ interior (Ioi 0)
Tactic:
simp [hx]
State after:
no goals