mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
⊢ (∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ∞) →
    (∫⁻ (x : α), ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν ∂μ < ∞ ↔
      ∫⁻ (a : α), ENNReal.ofReal (∫⁻ (a_1 : β), ‖f (a, a_1)‖ₑ ∂ν).toReal ∂μ < ∞)

pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
⊢ ∫⁻ (x : α), ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν ∂μ < ∞ → ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ∞

Tactic:
·

Goal state after:
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
⊢ (∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ∞) →
    (∫⁻ (x : α), ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν ∂μ < ∞ ↔
      ∫⁻ (a : α), ENNReal.ofReal (∫⁻ (a_1 : β), ‖f (a, a_1)‖ₑ ∂ν).toReal ∂μ < ∞)
Module: Mathlib.MeasureTheory.Integral.DivergenceTheorem

Goal state before:
pp: case refine_2
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ∀ (i : Fin (n + 1)), ContinuousOn (fun y ↦ f y i) (Box.Icc I)
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Box.Icc I) volume
hE : CompleteSpace E
A :
  HasIntegral I ⊥ (fun x ↦ ∑ i, (f' x) (e i) i) volume.toBoxAdditive.toSMul
    (∫ (x : Fin (n + 1) → ℝ) in ↑I, ∑ i, (f' x) (e i) i ∂volume)
B :
  HasIntegral I IntegrationParams.GP (fun x ↦ ∑ i, (f' x) (e i) i) BoxAdditiveMap.volume
    (∑ i,
      (BoxIntegral.integral (I.face i) IntegrationParams.GP (fun x ↦ f (i.insertNth (I.upper i) x) i)
          BoxAdditiveMap.volume -
        BoxIntegral.integral (I.face i) IntegrationParams.GP (fun x ↦ f (i.insertNth (I.lower i) x) i)
          BoxAdditiveMap.volume))
i : Fin (n + 1)
x✝ : i ∈ Finset.univ
⊢ BoxIntegral.integral (I.face i) IntegrationParams.GP (fun x ↦ f (i.insertNth (I.lower i) x) i) BoxAdditiveMap.volume =
    ∫ (x : Fin n → ℝ) in ↑(I.face i), f (i.insertNth (I.lower i) x) i

Tactic:
·

Goal state after:
case refine_2
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ∀ (i : Fin (n + 1)), ContinuousOn (fun y ↦ f y i) (Box.Icc I)
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Box.Icc I) volume
hE : CompleteSpace E
A :
  HasIntegral I ⊥ (fun x ↦ ∑ i, (f' x) (e i) i) volume.toBoxAdditive.toSMul
    (∫ (x : Fin (n + 1) → ℝ) in ↑I, ∑ i, (f' x) (e i) i ∂volume)
B :
  HasIntegral I IntegrationParams.GP (fun x ↦ ∑ i, (f' x) (e i) i) BoxAdditiveMap.volume
    (∑ i,
      (BoxIntegral.integral (I.face i) IntegrationParams.GP (fun x ↦ f (i.insertNth (I.upper i) x) i)
          BoxAdditiveMap.volume -
        BoxIntegral.integral (I.face i) IntegrationParams.GP (fun x ↦ f (i.insertNth (I.lower i) x) i)
          BoxAdditiveMap.volume))
i : Fin (n + 1)
x✝ : i ∈ Finset.univ
⊢ BoxIntegral.integral (I.face i) IntegrationParams.GP (fun x ↦ f (i.insertNth (I.lower i) x) i) BoxAdditiveMap.volume =
    ∫ (x : Fin n → ℝ) in ↑(I.face i), f (i.insertNth (I.lower i) x) i
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case refine_1
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 < R
hc : ContinuousOn f (sphere c R)
hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
θ₀ : ℝ
hmem : θ₀ ∈ Ioc 0 (2 * π)
hlt : ‖f (circleMap c R θ₀)‖ < C
⊢ ContinuousOn (fun θ ↦ R * ‖f (circleMap c R θ)‖) (Icc 0 (2 * π))

pp: case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 < R
hc : ContinuousOn f (sphere c R)
hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
θ₀ : ℝ
hmem : θ₀ ∈ Ioc 0 (2 * π)
hlt : ‖f (circleMap c R θ₀)‖ < C
θ : ℝ
x✝ : θ ∈ Ioc 0 (2 * π)
⊢ R * ‖f (circleMap c R θ)‖ ≤ R * C

pp: case refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 < R
hc : ContinuousOn f (sphere c R)
hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
θ₀ : ℝ
hmem : θ₀ ∈ Ioc 0 (2 * π)
hlt : ‖f (circleMap c R θ₀)‖ < C
⊢ R * ‖f (circleMap c R θ₀)‖ < R * C

Tactic:
·

Goal state after:
case refine_1
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 < R
hc : ContinuousOn f (sphere c R)
hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
θ₀ : ℝ
hmem : θ₀ ∈ Ioc 0 (2 * π)
hlt : ‖f (circleMap c R θ₀)‖ < C
⊢ ContinuousOn (fun θ ↦ R * ‖f (circleMap c R θ)‖) (Icc 0 (2 * π))
Module: Mathlib.MeasureTheory.Integral.DivergenceTheorem

Goal state before:
pp: case refine_2
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ∀ (i : Fin (n + 1)), ContinuousOn (fun y ↦ f y i) (Box.Icc I)
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Box.Icc I) volume
hE : CompleteSpace E
A :
  HasIntegral I ⊥ (fun x ↦ ∑ i, (f' x) (e i) i) volume.toBoxAdditive.toSMul
    (∫ (x : Fin (n + 1) → ℝ) in ↑I, ∑ i, (f' x) (e i) i ∂volume)
B :
  HasIntegral I IntegrationParams.GP (fun x ↦ ∑ i, (f' x) (e i) i) BoxAdditiveMap.volume
    (∑ i,
      (BoxIntegral.integral (I.face i) IntegrationParams.GP (fun x ↦ f (i.insertNth (I.upper i) x) i)
          BoxAdditiveMap.volume -
        BoxIntegral.integral (I.face i) IntegrationParams.GP (fun x ↦ f (i.insertNth (I.lower i) x) i)
          BoxAdditiveMap.volume))
i : Fin (n + 1)
x✝ : i ∈ Finset.univ
⊢ BoxIntegral.integral (I.face i) IntegrationParams.GP (fun x ↦ f (i.insertNth (I.lower i) x) i) BoxAdditiveMap.volume =
    ∫ (x : Fin n → ℝ) in ↑(I.face i), f (i.insertNth (I.lower i) x) i

Tactic:
· have := Box.continuousOn_face_Icc (Hc i) (Set.left_mem_Icc.2 (I.lower_le_upper i))
    have := (this.integrableOn_compact (μ := volume) (Box.isCompact_Icc _)).mono_set
      Box.coe_subset_Icc
    exact (this.hasBoxIntegral ⊥ rfl).integral_eq

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 < R
hc : ContinuousOn f (sphere c R)
hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
θ₀ : ℝ
hmem : θ₀ ∈ Ioc 0 (2 * π)
hlt : ‖f (circleMap c R θ₀)‖ < C
θ : ℝ
x✝ : θ ∈ Ioc 0 (2 * π)
⊢ R * ‖f (circleMap c R θ)‖ ≤ R * C

pp: case refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 < R
hc : ContinuousOn f (sphere c R)
hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
θ₀ : ℝ
hmem : θ₀ ∈ Ioc 0 (2 * π)
hlt : ‖f (circleMap c R θ₀)‖ < C
⊢ R * ‖f (circleMap c R θ₀)‖ < R * C

Tactic:
·

Goal state after:
case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 < R
hc : ContinuousOn f (sphere c R)
hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
θ₀ : ℝ
hmem : θ₀ ∈ Ioc 0 (2 * π)
hlt : ‖f (circleMap c R θ₀)‖ < C
θ : ℝ
x✝ : θ ∈ Ioc 0 (2 * π)
⊢ R * ‖f (circleMap c R θ)‖ ≤ R * C
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
⊢ ∫⁻ (x : α), ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν ∂μ < ∞ → ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ∞

Tactic:
·

Goal state after:
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
⊢ ∫⁻ (x : α), ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν ∂μ < ∞ → ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ∞
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 < R
hc : ContinuousOn f (sphere c R)
hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
θ₀ : ℝ
hmem : θ₀ ∈ Ioc 0 (2 * π)
hlt : ‖f (circleMap c R θ₀)‖ < C
⊢ R * ‖f (circleMap c R θ₀)‖ < R * C

Tactic:
·

Goal state after:
case refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 < R
hc : ContinuousOn f (sphere c R)
hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
θ₀ : ℝ
hmem : θ₀ ∈ Ioc 0 (2 * π)
hlt : ‖f (circleMap c R θ₀)‖ < C
⊢ R * ‖f (circleMap c R θ₀)‖ < R * C
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:
pp: case h₁
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : AEStronglyMeasurable f (μ.prod ν)
⊢ (∀ᵐ (x : α) ∂μ, HasFiniteIntegral (fun y ↦ AEStronglyMeasurable.mk f h1f (x, y)) ν) ↔
    ∀ᵐ (x : α) ∂μ, HasFiniteIntegral (fun y ↦ f (x, y)) ν

pp: case h₂
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : AEStronglyMeasurable f (μ.prod ν)
⊢ HasFiniteIntegral (fun x ↦ ∫ (y : β), ‖AEStronglyMeasurable.mk f h1f (x, y)‖ ∂ν) μ ↔
    HasFiniteIntegral (fun x ↦ ∫ (y : β), ‖f (x, y)‖ ∂ν) μ

Tactic:
·

Goal state after:
case h₁
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : AEStronglyMeasurable f (μ.prod ν)
⊢ (∀ᵐ (x : α) ∂μ, HasFiniteIntegral (fun y ↦ AEStronglyMeasurable.mk f h1f (x, y)) ν) ↔
    ∀ᵐ (x : α) ∂μ, HasFiniteIntegral (fun y ↦ f (x, y)) ν
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:
pp: case h₂
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : AEStronglyMeasurable f (μ.prod ν)
⊢ HasFiniteIntegral (fun x ↦ ∫ (y : β), ‖AEStronglyMeasurable.mk f h1f (x, y)‖ ∂ν) μ ↔
    HasFiniteIntegral (fun x ↦ ∫ (y : β), ‖f (x, y)‖ ∂ν) μ

Tactic:
·

Goal state after:
case h₂
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : AEStronglyMeasurable f (μ.prod ν)
⊢ HasFiniteIntegral (fun x ↦ ∫ (y : β), ‖AEStronglyMeasurable.mk f h1f (x, y)‖ ∂ν) μ ↔
    HasFiniteIntegral (fun x ↦ ∫ (y : β), ‖f (x, y)‖ ∂ν) μ
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case pos
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f f' : ℂ → E
c : ℂ
R : ℝ
h : ∀ z ∈ sphere c |R|, HasDerivWithinAt f (f' z) (sphere c |R|) z
hi : CircleIntegrable f' c R
⊢ ∮ (z : ℂ) in C(c, R), f' z = 0

pp: case neg
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f f' : ℂ → E
c : ℂ
R : ℝ
h : ∀ z ∈ sphere c |R|, HasDerivWithinAt f (f' z) (sphere c |R|) z
hi : ¬CircleIntegrable f' c R
⊢ ∮ (z : ℂ) in C(c, R), f' z = 0

Tactic:
·

Goal state after:
case pos
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f f' : ℂ → E
c : ℂ
R : ℝ
h : ∀ z ∈ sphere c |R|, HasDerivWithinAt f (f' z) (sphere c |R|) z
hi : CircleIntegrable f' c R
⊢ ∮ (z : ℂ) in C(c, R), f' z = 0
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case neg
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f f' : ℂ → E
c : ℂ
R : ℝ
h : ∀ z ∈ sphere c |R|, HasDerivWithinAt f (f' z) (sphere c |R|) z
hi : ¬CircleIntegrable f' c R
⊢ ∮ (z : ℂ) in C(c, R), f' z = 0

Tactic:
·

Goal state after:
case neg
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f f' : ℂ → E
c : ℂ
R : ℝ
h : ∀ z ∈ sphere c |R|, HasDerivWithinAt f (f' z) (sphere c |R|) z
hi : ¬CircleIntegrable f' c R
⊢ ∮ (z : ℂ) in C(c, R), f' z = 0
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case inl
n : ℤ
c w : ℂ
hn : n < 0
hw : w ∈ sphere c |0|
⊢ ∮ (z : ℂ) in C(c, 0), (z - w) ^ n = 0

pp: case inr
n : ℤ
c w : ℂ
R : ℝ
hn : n < 0
hw : w ∈ sphere c |R|
h0 : R ≠ 0
⊢ ∮ (z : ℂ) in C(c, R), (z - w) ^ n = 0

Tactic:
·

Goal state after:
case inl
n : ℤ
c w : ℂ
hn : n < 0
hw : w ∈ sphere c |0|
⊢ ∮ (z : ℂ) in C(c, 0), (z - w) ^ n = 0
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case inr
n : ℤ
c w : ℂ
R : ℝ
hn : n < 0
hw : w ∈ sphere c |R|
h0 : R ≠ 0
⊢ ∮ (z : ℂ) in C(c, R), (z - w) ^ n = 0

Tactic:
·

Goal state after:
case inr
n : ℤ
c w : ℂ
R : ℝ
hn : n < 0
hw : w ∈ sphere c |R|
h0 : R ≠ 0
⊢ ∮ (z : ℂ) in C(c, R), (z - w) ^ n = 0
Module: Mathlib.Analysis.BoxIntegral.DivergenceTheorem

Goal state before:
pp: case refine_3
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
i : Fin (n + 1)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J ↦ integral J GP (fun x ↦ f (i.insertNth y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (I.lower i) (I.upper i)) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x ↦ ⋯.toBoxAdditive
F : Fin (n + 1) →ᵇᵃ[↑I] E := BoxAdditiveMap.upperSubLower I i fI fb ⋯
⊢ ∀ (c : ℝ≥0),
    ∀ x ∈ Box.Icc I ∩ s,
      ∀ ε > 0,
        ∃ δ > 0,
          ∀ J ≤ I,
            Box.Icc J ⊆ closedBall x δ →
              x ∈ Box.Icc J →
                (GP.bDistortion = true → J.distortion ≤ c) →
                  dist ((BoxAdditiveMap.volume J) ((f' x) (Pi.single i 1))) (F J) ≤ ε

pp: case refine_4
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
i : Fin (n + 1)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J ↦ integral J GP (fun x ↦ f (i.insertNth y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (I.lower i) (I.upper i)) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x ↦ ⋯.toBoxAdditive
F : Fin (n + 1) →ᵇᵃ[↑I] E := BoxAdditiveMap.upperSubLower I i fI fb ⋯
⊢ ∀ (c : ℝ≥0),
    ∀ x ∈ Box.Icc I \ s,
      ∀ ε > 0,
        ∃ δ > 0,
          ∀ J ≤ I,
            Box.Icc J ⊆ closedBall x δ →
              x ∈ Box.Icc J →
                (GP.bDistortion = true → J.distortion ≤ c) →
                  dist ((BoxAdditiveMap.volume J) ((f' x) (Pi.single i 1))) (F J) ≤
                    ε * (volume.toBoxAdditive.restrict ↑I ⋯) J

Tactic:
·

Goal state after:
case refine_3
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
i : Fin (n + 1)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J ↦ integral J GP (fun x ↦ f (i.insertNth y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (I.lower i) (I.upper i)) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x ↦ ⋯.toBoxAdditive
F : Fin (n + 1) →ᵇᵃ[↑I] E := BoxAdditiveMap.upperSubLower I i fI fb ⋯
⊢ ∀ (c : ℝ≥0),
    ∀ x ∈ Box.Icc I ∩ s,
      ∀ ε > 0,
        ∃ δ > 0,
          ∀ J ≤ I,
            Box.Icc J ⊆ closedBall x δ →
              x ∈ Box.Icc J →
                (GP.bDistortion = true → J.distortion ≤ c) →
                  dist ((BoxAdditiveMap.volume J) ((f' x) (Pi.single i 1))) (F J) ≤ ε
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case pos
n : ℤ
hn : n ≠ -1
c w : ℂ
R : ℝ
H : w ∈ sphere c |R| ∧ n < -1
⊢ ∮ (z : ℂ) in C(c, R), (z - w) ^ n = 0

pp: case neg
n : ℤ
hn : n ≠ -1
c w : ℂ
R : ℝ
H : w ∈ sphere c |R| → -1 ≤ n
⊢ ∮ (z : ℂ) in C(c, R), (z - w) ^ n = 0

Tactic:
·

Goal state after:
case pos
n : ℤ
hn : n ≠ -1
c w : ℂ
R : ℝ
H : w ∈ sphere c |R| ∧ n < -1
⊢ ∮ (z : ℂ) in C(c, R), (z - w) ^ n = 0
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case e'_9
n : ℤ
hn : n ≠ -1
c w : ℂ
R : ℝ
H : w ∈ sphere c |R| → -1 ≤ n
z : ℂ
hne : z ≠ w ∨ -1 ≤ n
⊢ (z - w) ^ n = ↑(n + 1) * (id z - w) ^ (n + 1 - 1) * 1 / (↑n + 1)

pp: case convert_1
n : ℤ
hn : n ≠ -1
c w : ℂ
R : ℝ
H : w ∈ sphere c |R| → -1 ≤ n
z : ℂ
hne : z ≠ w ∨ -1 ≤ n
⊢ z ≠ w → id z - w ≠ 0

pp: case convert_2
n : ℤ
hn : n ≠ -1
c w : ℂ
R : ℝ
H : w ∈ sphere c |R| → -1 ≤ n
z : ℂ
hne : z ≠ w ∨ -1 ≤ n
⊢ -1 ≤ n → 0 ≤ n + 1

Tactic:
·

Goal state after:
case e'_9
n : ℤ
hn : n ≠ -1
c w : ℂ
R : ℝ
H : w ∈ sphere c |R| → -1 ≤ n
z : ℂ
hne : z ≠ w ∨ -1 ≤ n
⊢ (z - w) ^ n = ↑(n + 1) * (id z - w) ^ (n + 1 - 1) * 1 / (↑n + 1)
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
n : ℕ
⊢ (2 * π)⁻¹ * (|0|⁻¹ ^ n * (|0| * (|0|⁻¹ * ∫ (x : ℝ) in 0..2 * π, ‖f (circleMap c 0 x)‖))) ≤
    ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c 0 θ)‖) * |0|⁻¹ ^ n

pp: case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
n : ℕ
hR : R ≠ 0
⊢ (2 * π)⁻¹ * (|R|⁻¹ ^ n * (|R| * (|R|⁻¹ * ∫ (x : ℝ) in 0..2 * π, ‖f (circleMap c R x)‖))) ≤
    ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c R θ)‖) * |R|⁻¹ ^ n

Tactic:
·

Goal state after:
case inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
n : ℕ
⊢ (2 * π)⁻¹ * (|0|⁻¹ ^ n * (|0| * (|0|⁻¹ * ∫ (x : ℝ) in 0..2 * π, ‖f (circleMap c 0 x)‖))) ≤
    ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c 0 θ)‖) * |0|⁻¹ ^ n
Module: Mathlib.Analysis.BoxIntegral.DivergenceTheorem

Goal state before:
pp: case refine_1
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
i : Fin (n + 1)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J ↦ integral J GP (fun x ↦ f (i.insertNth y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (I.lower i) (I.upper i)) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x ↦ ⋯.toBoxAdditive
F : Fin (n + 1) →ᵇᵃ[↑I] E := BoxAdditiveMap.upperSubLower I i fI fb ⋯
c : ℝ≥0
x : Fin (n + 1) → ℝ
hx : x ∈ Box.Icc I ∩ s
ε : ℝ
ε0 : ε > 0
⊢ ∀ᶠ (δ : ℝ) in 𝓝[>] 0, ∀ y₁ ∈ closedBall x δ ∩ Box.Icc I, ∀ y₂ ∈ closedBall x δ ∩ Box.Icc I, ‖f y₁ - f y₂‖ ≤ ε / 2

pp: case refine_2
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
i : Fin (n + 1)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J ↦ integral J GP (fun x ↦ f (i.insertNth y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (I.lower i) (I.upper i)) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x ↦ ⋯.toBoxAdditive
F : Fin (n + 1) →ᵇᵃ[↑I] E := BoxAdditiveMap.upperSubLower I i fI fb ⋯
c : ℝ≥0
x : Fin (n + 1) → ℝ
hx : x ∈ Box.Icc I ∩ s
ε : ℝ
ε0 : ε > 0
⊢ ∀ᶠ (δ : ℝ) in 𝓝[>] 0, (2 * δ) ^ (n + 1) * ‖(f' x) (Pi.single i 1)‖ ≤ ε / 2

Tactic:
·

Goal state after:
case refine_1
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
i : Fin (n + 1)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J ↦ integral J GP (fun x ↦ f (i.insertNth y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (I.lower i) (I.upper i)) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x ↦ ⋯.toBoxAdditive
F : Fin (n + 1) →ᵇᵃ[↑I] E := BoxAdditiveMap.upperSubLower I i fI fb ⋯
c : ℝ≥0
x : Fin (n + 1) → ℝ
hx : x ∈ Box.Icc I ∩ s
ε : ℝ
ε0 : ε > 0
⊢ ∀ᶠ (δ : ℝ) in 𝓝[>] 0, ∀ y₁ ∈ closedBall x δ ∩ Box.Icc I, ∀ y₂ ∈ closedBall x δ ∩ Box.Icc I, ‖f y₁ - f y₂‖ ≤ ε / 2
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
n : ℕ
hR : R ≠ 0
⊢ (2 * π)⁻¹ * (|R|⁻¹ ^ n * (|R| * (|R|⁻¹ * ∫ (x : ℝ) in 0..2 * π, ‖f (circleMap c R x)‖))) ≤
    ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c R θ)‖) * |R|⁻¹ ^ n

Tactic:
·

Goal state after:
case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
n : ℕ
hR : R ≠ 0
⊢ (2 * π)⁻¹ * (|R|⁻¹ ^ n * (|R| * (|R|⁻¹ * ∫ (x : ℝ) in 0..2 * π, ‖f (circleMap c R x)‖))) ≤
    ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c R θ)‖) * |R|⁻¹ ^ n
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ≥0
n : ℕ
hR : ↑R ^ n = 0
⊢ ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖) * (↑R)⁻¹ ^ n * ↑R ^ n ≤
    (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖

pp: case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ≥0
n : ℕ
hR : ↑R ^ n ≠ 0
⊢ ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖) * (↑R)⁻¹ ^ n * ↑R ^ n ≤
    (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖

Tactic:
·

Goal state after:
case inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ≥0
n : ℕ
hR : ↑R ^ n = 0
⊢ ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖) * (↑R)⁻¹ ^ n * ↑R ^ n ≤
    (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ≥0
n : ℕ
hR : ↑R ^ n ≠ 0
⊢ ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖) * (↑R)⁻¹ ^ n * ↑R ^ n ≤
    (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖

Tactic:
·

Goal state after:
case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ≥0
n : ℕ
hR : ↑R ^ n ≠ 0
⊢ ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖) * (↑R)⁻¹ ^ n * ↑R ^ n ≤
    (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖