Theorem: Perron.residueAtNegOne (PrimeNumberTheoremAnd/PerronFormula.lean)

State before:
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
⊢ ∀ᶠ (c : ℝ) in 𝓝[>] 0, RectangleIntegral' (f x) (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) = -↑x⁻¹
Tactic:
filter_upwards [Ioo_mem_nhdsWithin_Ioi' (by linarith : (0 : ℝ) < 1 / 2), diffBddAtNegOne xpos]
State after:
case h
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
⊢ ∀ a ∈ Ioo 0 (1 / 2),
    BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) a \ {-1})) →
      RectangleIntegral' (f x) (-↑a - ↑a * I - 1) (↑a + ↑a * I - 1) = -↑x⁻¹

State before:
case h
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
⊢ ∀ a ∈ Ioo 0 (1 / 2),
    BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) a \ {-1})) →
      RectangleIntegral' (f x) (-↑a - ↑a * I - 1) (↑a + ↑a * I - 1) = -↑x⁻¹
Tactic:
intro c hc bddAbove
State after:
case h
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
hc : c ∈ Ioo 0 (1 / 2)
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
⊢ RectangleIntegral' (f x) (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) = -↑x⁻¹

State before:
case h
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
hc : c ∈ Ioo 0 (1 / 2)
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
⊢ RectangleIntegral' (f x) (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) = -↑x⁻¹
Tactic:
obtain ⟨cpos, _⟩ := hc
State after:
case h.intro
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
⊢ RectangleIntegral' (f x) (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) = -↑x⁻¹

State before:
case h.intro
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
⊢ RectangleIntegral' (f x) (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) = -↑x⁻¹
Tactic:
have h_mem {s : ℂ} (hs : s ∈ Square (-1) c) :
    -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c := by
  rw [Square, mem_Rect (by simpa using by linarith) (by simp [cpos.le])] at hs
  simpa using hs
State after:
case h.intro
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
⊢ RectangleIntegral' (f x) (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) = -↑x⁻¹

State before:
case h.intro
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
⊢ RectangleIntegral' (f x) (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) = -↑x⁻¹
Tactic:
have RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ := by
  refine fun s ⟨hs, hs1⟩ ↦ not_or.mpr ⟨?_, hs1⟩
  simpa [Complex.ext_iff] using fun _ _ ↦ by linarith [h_mem hs]
State after:
case h.intro
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
⊢ RectangleIntegral' (f x) (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) = -↑x⁻¹

State before:
case h.intro
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
⊢ RectangleIntegral' (f x) (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) = -↑x⁻¹
Tactic:
have fHolo : HolomorphicOn (f x) (Square (-1) c \ {-1}) := (isHolomorphicOn xpos).mono RectSub
State after:
case h.intro
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
fHolo : HolomorphicOn (f x) (Square (-1) c \ {-1})
⊢ RectangleIntegral' (f x) (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) = -↑x⁻¹

State before:
case h.intro
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
fHolo : HolomorphicOn (f x) (Square (-1) c \ {-1})
⊢ RectangleIntegral' (f x) (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) = -↑x⁻¹
Tactic:
have f1Holo : HolomorphicOn ((f x) - (fun (s : ℂ) ↦ -x⁻¹ / (s + 1))) (Square (-1) c \ {-1}) := by
  refine fHolo.sub <| (differentiableOn_const _).neg.div ?_ fun x hx ↦ sPlusOneNeZero hx.2
  exact differentiableOn_id.add (differentiableOn_const 1)
State after:
case h.intro
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
fHolo : HolomorphicOn (f x) (Square (-1) c \ {-1})
f1Holo : HolomorphicOn (f x - fun s => -↑x⁻¹ / (s + 1)) (Square (-1) c \ {-1})
⊢ RectangleIntegral' (f x) (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) = -↑x⁻¹

State before:
case h.intro
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
fHolo : HolomorphicOn (f x) (Square (-1) c \ {-1})
f1Holo : HolomorphicOn (f x - fun s => -↑x⁻¹ / (s + 1)) (Square (-1) c \ {-1})
⊢ RectangleIntegral' (f x) (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) = -↑x⁻¹
Tactic:
have RectMemNhds : Square (-1) c ∈ 𝓝 (-1) := square_mem_nhds (-1) (ne_of_gt cpos)
State after:
case h.intro
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
fHolo : HolomorphicOn (f x) (Square (-1) c \ {-1})
f1Holo : HolomorphicOn (f x - fun s => -↑x⁻¹ / (s + 1)) (Square (-1) c \ {-1})
RectMemNhds : Square (-1) c ∈ 𝓝 (-1)
⊢ RectangleIntegral' (f x) (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) = -↑x⁻¹

State before:
case h.intro
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
fHolo : HolomorphicOn (f x) (Square (-1) c \ {-1})
f1Holo : HolomorphicOn (f x - fun s => -↑x⁻¹ / (s + 1)) (Square (-1) c \ {-1})
RectMemNhds : Square (-1) c ∈ 𝓝 (-1)
⊢ RectangleIntegral' (f x) (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) = -↑x⁻¹
Tactic:
obtain ⟨g, gHolo, g_eq_fDiff⟩ := existsDifferentiableOn_of_bddAbove RectMemNhds f1Holo bddAbove
State after:
case h.intro.intro.intro
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
fHolo : HolomorphicOn (f x) (Square (-1) c \ {-1})
f1Holo : HolomorphicOn (f x - fun s => -↑x⁻¹ / (s + 1)) (Square (-1) c \ {-1})
RectMemNhds : Square (-1) c ∈ 𝓝 (-1)
g : ℂ → ℂ
gHolo : HolomorphicOn g (Square (-1) c)
g_eq_fDiff : EqOn (f x - fun s => -↑x⁻¹ / (s + 1)) g (Square (-1) c \ {-1})
⊢ RectangleIntegral' (f x) (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) = -↑x⁻¹

State before:
case h.intro.intro.intro
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
fHolo : HolomorphicOn (f x) (Square (-1) c \ {-1})
f1Holo : HolomorphicOn (f x - fun s => -↑x⁻¹ / (s + 1)) (Square (-1) c \ {-1})
RectMemNhds : Square (-1) c ∈ 𝓝 (-1)
g : ℂ → ℂ
gHolo : HolomorphicOn g (Square (-1) c)
g_eq_fDiff : EqOn (f x - fun s => -↑x⁻¹ / (s + 1)) g (Square (-1) c \ {-1})
⊢ RectangleIntegral' (f x) (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) = -↑x⁻¹
Tactic:
simp_rw [Square] at fHolo gHolo RectMemNhds
State after:
case h.intro.intro.intro
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
fHolo : HolomorphicOn (f x) (Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1) \ {-1})
f1Holo : HolomorphicOn (f x - fun s => -↑x⁻¹ / (s + 1)) (Square (-1) c \ {-1})
RectMemNhds : Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1) ∈ 𝓝 (-1)
g : ℂ → ℂ
gHolo : HolomorphicOn g (Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1))
g_eq_fDiff : EqOn (f x - fun s => -↑x⁻¹ / (s + 1)) g (Square (-1) c \ {-1})
⊢ RectangleIntegral' (f x) (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) = -↑x⁻¹

State before:
case h.intro.intro.intro
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
fHolo : HolomorphicOn (f x) (Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1) \ {-1})
f1Holo : HolomorphicOn (f x - fun s => -↑x⁻¹ / (s + 1)) (Square (-1) c \ {-1})
RectMemNhds : Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1) ∈ 𝓝 (-1)
g : ℂ → ℂ
gHolo : HolomorphicOn g (Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1))
g_eq_fDiff : EqOn (f x - fun s => -↑x⁻¹ / (s + 1)) g (Square (-1) c \ {-1})
⊢ RectangleIntegral' (f x) (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) = -↑x⁻¹
Tactic:
refine ResidueTheoremOnRectangleWithSimplePole ?_ ?_ RectMemNhds gHolo ?_
State after:
case h.intro.intro.intro.refine_1
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
fHolo : HolomorphicOn (f x) (Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1) \ {-1})
f1Holo : HolomorphicOn (f x - fun s => -↑x⁻¹ / (s + 1)) (Square (-1) c \ {-1})
RectMemNhds : Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1) ∈ 𝓝 (-1)
g : ℂ → ℂ
gHolo : HolomorphicOn g (Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1))
g_eq_fDiff : EqOn (f x - fun s => -↑x⁻¹ / (s + 1)) g (Square (-1) c \ {-1})
⊢ (-↑c - ↑c * I - 1).re ≤ (↑c + ↑c * I - 1).re

case h.intro.intro.intro.refine_2
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
fHolo : HolomorphicOn (f x) (Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1) \ {-1})
f1Holo : HolomorphicOn (f x - fun s => -↑x⁻¹ / (s + 1)) (Square (-1) c \ {-1})
RectMemNhds : Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1) ∈ 𝓝 (-1)
g : ℂ → ℂ
gHolo : HolomorphicOn g (Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1))
g_eq_fDiff : EqOn (f x - fun s => -↑x⁻¹ / (s + 1)) g (Square (-1) c \ {-1})
⊢ (-↑c - ↑c * I - 1).im ≤ (↑c + ↑c * I - 1).im

case h.intro.intro.intro.refine_3
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
fHolo : HolomorphicOn (f x) (Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1) \ {-1})
f1Holo : HolomorphicOn (f x - fun s => -↑x⁻¹ / (s + 1)) (Square (-1) c \ {-1})
RectMemNhds : Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1) ∈ 𝓝 (-1)
g : ℂ → ℂ
gHolo : HolomorphicOn g (Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1))
g_eq_fDiff : EqOn (f x - fun s => -↑x⁻¹ / (s + 1)) g (Square (-1) c \ {-1})
⊢ EqOn (f x - fun s => -↑x⁻¹ / (s - -1)) g (Rectangle (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) \ {-1})

State before:
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
⊢ 0 < 1 / 2
Tactic:
linarith
State after:
no goals

State before:
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
s : ℂ
hs : s ∈ Square (-1) c
⊢ -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
Tactic:
rw [Square, mem_Rect (by simpa using by linarith) (by simp [cpos.le])] at hs
State after:
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
s : ℂ
hs :
  (-↑c - ↑c * I + -1).re ≤ s.re ∧
    s.re ≤ (↑c + ↑c * I + -1).re ∧ (-↑c - ↑c * I + -1).im ≤ s.im ∧ s.im ≤ (↑c + ↑c * I + -1).im
⊢ -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c

State before:
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
s : ℂ
hs :
  (-↑c - ↑c * I + -1).re ≤ s.re ∧
    s.re ≤ (↑c + ↑c * I + -1).re ∧ (-↑c - ↑c * I + -1).im ≤ s.im ∧ s.im ≤ (↑c + ↑c * I + -1).im
⊢ -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
Tactic:
simpa using hs
State after:
no goals

State before:
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
s : ℂ
hs : s ∈ Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1)
⊢ (-↑c - ↑c * I + -1).re ≤ (↑c + ↑c * I + -1).re
Tactic:
simpa using by linarith
State after:
no goals

State before:
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
s : ℂ
hs : s ∈ Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1)
⊢ 0 ≤ c
Tactic:
linarith
State after:
no goals

State before:
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
s : ℂ
hs : s ∈ Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1)
⊢ (-↑c - ↑c * I + -1).im ≤ (↑c + ↑c * I + -1).im
Tactic:
simp [cpos.le]
State after:
no goals

State before:
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
⊢ Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
Tactic:
refine fun s ⟨hs, hs1⟩ ↦ not_or.mpr ⟨?_, hs1⟩
State after:
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
s : ℂ
x✝ : s ∈ Square (-1) c \ {-1}
hs : s ∈ Square (-1) c
hs1 : s ∉ {-1}
⊢ ¬s = 0

State before:
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
s : ℂ
x✝ : s ∈ Square (-1) c \ {-1}
hs : s ∈ Square (-1) c
hs1 : s ∉ {-1}
⊢ ¬s = 0
Tactic:
simpa [Complex.ext_iff] using fun _ _ ↦ by linarith [h_mem hs]
State after:
no goals

State before:
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
s : ℂ
x✝² : s ∈ Square (-1) c \ {-1}
hs : s ∈ Square (-1) c
hs1 : s ∉ {-1}
x✝¹ : s.re = 0
x✝ : s.im = 0
⊢ False
Tactic:
linarith [h_mem hs]
State after:
no goals

State before:
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
fHolo : HolomorphicOn (f x) (Square (-1) c \ {-1})
⊢ HolomorphicOn (f x - fun s => -↑x⁻¹ / (s + 1)) (Square (-1) c \ {-1})
Tactic:
refine fHolo.sub <| (differentiableOn_const _).neg.div ?_ fun x hx ↦ sPlusOneNeZero hx.2
State after:
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
fHolo : HolomorphicOn (f x) (Square (-1) c \ {-1})
⊢ DifferentiableOn ℂ (fun y => y + 1) (Square (-1) c \ {-1})

State before:
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
fHolo : HolomorphicOn (f x) (Square (-1) c \ {-1})
⊢ DifferentiableOn ℂ (fun y => y + 1) (Square (-1) c \ {-1})
Tactic:
exact differentiableOn_id.add (differentiableOn_const 1)
State after:
no goals

State before:
case h.intro.intro.intro.refine_1
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
fHolo : HolomorphicOn (f x) (Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1) \ {-1})
f1Holo : HolomorphicOn (f x - fun s => -↑x⁻¹ / (s + 1)) (Square (-1) c \ {-1})
RectMemNhds : Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1) ∈ 𝓝 (-1)
g : ℂ → ℂ
gHolo : HolomorphicOn g (Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1))
g_eq_fDiff : EqOn (f x - fun s => -↑x⁻¹ / (s + 1)) g (Square (-1) c \ {-1})
⊢ (-↑c - ↑c * I - 1).re ≤ (↑c + ↑c * I - 1).re
Tactic:
simpa using cpos.le
State after:
no goals

State before:
case h.intro.intro.intro.refine_2
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
fHolo : HolomorphicOn (f x) (Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1) \ {-1})
f1Holo : HolomorphicOn (f x - fun s => -↑x⁻¹ / (s + 1)) (Square (-1) c \ {-1})
RectMemNhds : Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1) ∈ 𝓝 (-1)
g : ℂ → ℂ
gHolo : HolomorphicOn g (Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1))
g_eq_fDiff : EqOn (f x - fun s => -↑x⁻¹ / (s + 1)) g (Square (-1) c \ {-1})
⊢ (-↑c - ↑c * I - 1).im ≤ (↑c + ↑c * I - 1).im
Tactic:
simpa using cpos.le
State after:
no goals

State before:
case h.intro.intro.intro.refine_3
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
fHolo : HolomorphicOn (f x) (Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1) \ {-1})
f1Holo : HolomorphicOn (f x - fun s => -↑x⁻¹ / (s + 1)) (Square (-1) c \ {-1})
RectMemNhds : Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1) ∈ 𝓝 (-1)
g : ℂ → ℂ
gHolo : HolomorphicOn g (Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1))
g_eq_fDiff : EqOn (f x - fun s => -↑x⁻¹ / (s + 1)) g (Square (-1) c \ {-1})
⊢ EqOn (f x - fun s => -↑x⁻¹ / (s - -1)) g (Rectangle (-↑c - ↑c * I - 1) (↑c + ↑c * I - 1) \ {-1})
Tactic:
convert g_eq_fDiff using 3
State after:
case h.e'_3.h.e'_6.h
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
fHolo : HolomorphicOn (f x) (Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1) \ {-1})
f1Holo : HolomorphicOn (f x - fun s => -↑x⁻¹ / (s + 1)) (Square (-1) c \ {-1})
RectMemNhds : Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1) ∈ 𝓝 (-1)
g : ℂ → ℂ
gHolo : HolomorphicOn g (Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1))
g_eq_fDiff : EqOn (f x - fun s => -↑x⁻¹ / (s + 1)) g (Square (-1) c \ {-1})
x✝ : ℂ
⊢ -↑x⁻¹ / (x✝ - -1) = -↑x⁻¹ / (x✝ + 1)

State before:
case h.e'_3.h.e'_6.h
x σ σ' σ'' T : ℝ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : NoMaxOrder β
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology β
y : β
l : Filter α
xpos : 0 < x
c : ℝ
bddAbove : BddAbove ((Norm.norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \ {-1}))
cpos : 0 < c
right✝ : c < 1 / 2
h_mem : ∀ {s : ℂ}, s ∈ Square (-1) c → -c ≤ s.re + 1 ∧ s.re + 1 ≤ c ∧ -c ≤ s.im ∧ s.im ≤ c
RectSub : Square (-1) c \ {-1} ⊆ {0, -1}ᶜ
fHolo : HolomorphicOn (f x) (Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1) \ {-1})
f1Holo : HolomorphicOn (f x - fun s => -↑x⁻¹ / (s + 1)) (Square (-1) c \ {-1})
RectMemNhds : Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1) ∈ 𝓝 (-1)
g : ℂ → ℂ
gHolo : HolomorphicOn g (Rectangle (-↑c - ↑c * I + -1) (↑c + ↑c * I + -1))
g_eq_fDiff : EqOn (f x - fun s => -↑x⁻¹ / (s + 1)) g (Square (-1) c \ {-1})
x✝ : ℂ
⊢ -↑x⁻¹ / (x✝ - -1) = -↑x⁻¹ / (x✝ + 1)
Tactic:
simp
State after:
no goals