Theorem: ResidueTheoremOnRectangleWithSimplePole (PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean)

State before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
⊢ RectangleIntegral' f z w = A
Tactic:
have principalPart' : Set.EqOn f (g + (fun s ↦ A / (s - p))) (Rectangle z w \ {p}) :=
  fun s hs => by rw [Pi.add_apply, ← principalPart hs, Pi.sub_apply, sub_add_cancel]
State after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
⊢ RectangleIntegral' f z w = A

State before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
⊢ RectangleIntegral' f z w = A
Tactic:
have : Set.EqOn f (g + (fun s ↦ A / (s - p))) (RectangleBorder z w) :=
  principalPart'.mono <| Set.subset_diff.mpr ⟨rectangleBorder_subset_rectangle z w,
    disjoint_singleton_right.mpr (not_mem_rectangleBorder_of_rectangle_mem_nhds pInRectInterior)⟩
State after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
⊢ RectangleIntegral' f z w = A

State before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
⊢ RectangleIntegral' f z w = A
Tactic:
rw [RectangleIntegral'_congr this]
State after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
⊢ RectangleIntegral' (g + fun s => A / (s - p)) z w = A

State before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
⊢ RectangleIntegral' (g + fun s => A / (s - p)) z w = A
Tactic:
have t1 : RectangleBorderIntegrable g z w := gHolo.rectangleBorderIntegrable
State after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
⊢ RectangleIntegral' (g + fun s => A / (s - p)) z w = A

State before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
t2 : HolomorphicOn (fun s => A / (s - p)) (Rectangle z w \ {p})
⊢ RectangleIntegral' (g + fun s => A / (s - p)) z w = A
Tactic:
have t3 : RectangleBorderIntegrable (fun s ↦ A / (s - p)) z w :=
  HolomorphicOn.rectangleBorderIntegrable' t2 pInRectInterior
State after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
t2 : HolomorphicOn (fun s => A / (s - p)) (Rectangle z w \ {p})
t3 : RectangleBorderIntegrable (fun s => A / (s - p)) z w
⊢ RectangleIntegral' (g + fun s => A / (s - p)) z w = A

State before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
t2 : HolomorphicOn (fun s => A / (s - p)) (Rectangle z w \ {p})
t3 : RectangleBorderIntegrable (fun s => A / (s - p)) z w
⊢ RectangleIntegral' (g + fun s => A / (s - p)) z w = A
Tactic:
rw [RectangleIntegral', RectangleBorderIntegrable.add t1 t3, smul_add]
State after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
t2 : HolomorphicOn (fun s => A / (s - p)) (Rectangle z w \ {p})
t3 : RectangleBorderIntegrable (fun s => A / (s - p)) z w
⊢ (1 / (2 * ↑π * I)) • RectangleIntegral g z w + (1 / (2 * ↑π * I)) • RectangleIntegral (fun s => A / (s - p)) z w = A

State before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
t2 : HolomorphicOn (fun s => A / (s - p)) (Rectangle z w \ {p})
t3 : RectangleBorderIntegrable (fun s => A / (s - p)) z w
⊢ (1 / (2 * ↑π * I)) • RectangleIntegral g z w + (1 / (2 * ↑π * I)) • RectangleIntegral (fun s => A / (s - p)) z w = A
Tactic:
rw [gHolo.vanishesOnRectangle (by rfl), smul_zero, zero_add]
State after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
t2 : HolomorphicOn (fun s => A / (s - p)) (Rectangle z w \ {p})
t3 : RectangleBorderIntegrable (fun s => A / (s - p)) z w
⊢ (1 / (2 * ↑π * I)) • RectangleIntegral (fun s => A / (s - p)) z w = A

State before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
t2 : HolomorphicOn (fun s => A / (s - p)) (Rectangle z w \ {p})
t3 : RectangleBorderIntegrable (fun s => A / (s - p)) z w
⊢ (1 / (2 * ↑π * I)) • RectangleIntegral (fun s => A / (s - p)) z w = A
Tactic:
exact ResidueTheoremInRectangle zRe_le_wRe zIm_le_wIm pInRectInterior
State after:
no goals

State before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
s : ℂ
hs : s ∈ Rectangle z w \ {p}
⊢ f s = (g + fun s => A / (s - p)) s
Tactic:
rw [Pi.add_apply, ← principalPart hs, Pi.sub_apply, sub_add_cancel]
State after:
no goals

State before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
⊢ HolomorphicOn (fun s => A / (s - p)) (Rectangle z w \ {p})
Tactic:
apply DifferentiableOn.mono (t := {p}ᶜ)
State after:
case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
⊢ DifferentiableOn ℂ (fun s => A / (s - p)) {p}ᶜ

case st
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
⊢ Rectangle z w \ {p} ⊆ {p}ᶜ

State before:
case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
⊢ DifferentiableOn ℂ (fun s => A / (s - p)) {p}ᶜ
Tactic:
apply DifferentiableOn.div
State after:
case h.hc
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
⊢ DifferentiableOn ℂ (fun x => A) {p}ᶜ

case h.hd
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
⊢ DifferentiableOn ℂ (fun x => x - p) {p}ᶜ

case h.hx
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
⊢ ∀ x ∈ {p}ᶜ, x - p ≠ 0

State before:
case h.hc
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
⊢ DifferentiableOn ℂ (fun x => A) {p}ᶜ
Tactic:
exact differentiableOn_const _
State after:
no goals

State before:
case h.hd
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
⊢ DifferentiableOn ℂ (fun x => x - p) {p}ᶜ
Tactic:
exact DifferentiableOn.sub differentiableOn_id (differentiableOn_const _)
State after:
no goals

State before:
case h.hx
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
⊢ ∀ x ∈ {p}ᶜ, x - p ≠ 0
Tactic:
exact fun x hx => by rw [sub_ne_zero]; exact hx
State after:
no goals

State before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x✝ x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
x : ℂ
hx : x ∈ {p}ᶜ
⊢ x - p ≠ 0
Tactic:
rw [sub_ne_zero]
State after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x✝ x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
x : ℂ
hx : x ∈ {p}ᶜ
⊢ x ≠ p

State before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x✝ x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
x : ℂ
hx : x ∈ {p}ᶜ
⊢ x ≠ p
Tactic:
exact hx
State after:
no goals

State before:
case st
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
⊢ Rectangle z w \ {p} ⊆ {p}ᶜ
Tactic:
rintro s ⟨_, hs⟩
State after:
case st.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
s : ℂ
left✝ : s ∈ Rectangle z w
hs : s ∉ {p}
⊢ s ∈ {p}ᶜ

State before:
case st.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
s : ℂ
left✝ : s ∈ Rectangle z w
hs : s ∉ {p}
⊢ s ∈ {p}ᶜ
Tactic:
exact hs
State after:
no goals

State before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f✝ g✝ : ℂ → E
z✝ w✝ p✝ c A✝ : ℂ
x x₁ x₂ y y₁ y₂ σ : ℝ
f g : ℂ → ℂ
z w p A : ℂ
zRe_le_wRe : z.re ≤ w.re
zIm_le_wIm : z.im ≤ w.im
pInRectInterior : Rectangle z w ∈ 𝓝 p
gHolo : HolomorphicOn g (Rectangle z w)
principalPart : EqOn (f - fun s => A / (s - p)) g (Rectangle z w \ {p})
principalPart' : EqOn f (g + fun s => A / (s - p)) (Rectangle z w \ {p})
this : EqOn f (g + fun s => A / (s - p)) (RectangleBorder z w)
t1 : RectangleBorderIntegrable g z w
t2 : HolomorphicOn (fun s => A / (s - p)) (Rectangle z w \ {p})
t3 : RectangleBorderIntegrable (fun s => A / (s - p)) z w
⊢ Rectangle z w ⊆ Rectangle z w
Tactic:
rfl
State after:
no goals