l3lab/ntp-mathlib huggingface dataset:
/-- For a sheaf `F`, `F(U ⊔ V)` is the pullback of `F(U) ⟶ F(U ⊓ V)` and `F(V) ⟶ F(U ⊓ V)`. -/
def isLimitPullbackCone : IsLimit (interUnionPullbackCone F U V) := by
  let ι : ULift.{w} WalkingPair → Opens X := fun ⟨j⟩ => WalkingPair.casesOn j U V
  have hι : U ⊔ V = iSup ι := by
    ext
    rw [Opens.coe_iSup, Set.mem_iUnion]
    constructor
    · rintro (h | h)
      exacts [⟨⟨WalkingPair.left⟩, h⟩, ⟨⟨WalkingPair.right⟩, h⟩]
    · rintro ⟨⟨_ | _⟩, h⟩
      exacts [Or.inl h, Or.inr h]
  apply PullbackCone.isLimitAux'
  intro s
  use interUnionPullbackConeLift F U V s
  refine' ⟨_, _, _⟩
  · apply interUnionPullbackConeLift_left
  · apply interUnionPullbackConeLift_right
  · intro m h₁ h₂
    rw [← cancel_mono (F.1.map (eqToHom hι.symm).op)]
    apply (F.presheaf.isSheaf_iff_isSheafPairwiseIntersections.mp F.2 ι).some.hom_ext
    rintro ((_ | _) | (_ | _)) <;>
    rw [Category.assoc, Category.assoc]
    · erw [← F.1.map_comp]
      convert h₁
      apply interUnionPullbackConeLift_left
    · erw [← F.1.map_comp]
      convert h₂
      apply interUnionPullbackConeLift_right
    all_goals
      dsimp only [Functor.op, Pairwise.cocone_ι_app, Functor.mapCone_π_app, Cocone.op,
        Pairwise.coconeιApp, unop_op, op_comp, NatTrans.op]
      simp_rw [F.1.map_comp, ← Category.assoc]
      congr 1
      simp_rw [Category.assoc, ← F.1.map_comp]
    · convert h₁
      apply interUnionPullbackConeLift_left
    · 

Proof state:
case property.refine'_3.mk.pair.up.right.e_a
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s✝ : PullbackCone (F.val.map (homOfLE (_ : U ⊓ V ≤ U)).op) (F.val.map (homOfLE (_ : U ⊓ V ≤ V)).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X :=
  fun x =>
    match x with
    | { down := j } => WalkingPair.casesOn j U V
hι : U ⊔ V = iSup ι
s : PullbackCone (F.val.map (homOfLE (_ : U ⊓ V ≤ U)).op) (F.val.map (homOfLE (_ : U ⊓ V ≤ V)).op)
m : s.pt ⟶ (interUnionPullbackCone F U V).pt
h₁ : m ≫ PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s
h₂ : m ≫ PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s
a✝ : ULift.{w, 0} WalkingPair
⊢ m ≫
      F.val.map
        ((eqToHom (_ : ⨆ x, WalkingPair.rec U V x.down = U ⊔ V)).op ≫
          (homOfLE (_ : WalkingPair.rec U V { down := WalkingPair.right }.down ≤ ⨆ x, WalkingPair.rec U V x.down)).op) =
    interUnionPullbackConeLift F U V s ≫
      F.val.map
        ((eqToHom (_ : ⨆ x, WalkingPair.rec U V x.down = U ⊔ V)).op ≫
          (homOfLE (_ : WalkingPair.rec U V { down := WalkingPair.right }.down ≤ ⨆ x, WalkingPair.rec U V x.down)).op)

Next tactic:
convert h₂
/-- For a sheaf `F`, `F(U ⊔ V)` is the pullback of `F(U) ⟶ F(U ⊓ V)` and `F(V) ⟶ F(U ⊓ V)`. -/
def isLimitPullbackCone : IsLimit (interUnionPullbackCone F U V) := by
  let ι : ULift.{w} WalkingPair → Opens X := fun ⟨j⟩ => WalkingPair.casesOn j U V
  have hι : U ⊔ V = iSup ι := by
    ext
    rw [Opens.coe_iSup, Set.mem_iUnion]
    constructor
    · rintro (h | h)
      exacts [⟨⟨WalkingPair.left⟩, h⟩, ⟨⟨WalkingPair.right⟩, h⟩]
    · rintro ⟨⟨_ | _⟩, h⟩
      exacts [Or.inl h, Or.inr h]
  apply PullbackCone.isLimitAux'
  intro s
  use interUnionPullbackConeLift F U V s
  refine' ⟨_, _, _⟩
  · apply interUnionPullbackConeLift_left
  · apply interUnionPullbackConeLift_right
  · intro m h₁ h₂
    rw [← cancel_mono (F.1.map (eqToHom hι.symm).op)]
    apply (F.presheaf.isSheaf_iff_isSheafPairwiseIntersections.mp F.2 ι).some.hom_ext
    rintro ((_ | _) | (_ | _)) <;>
    rw [Category.assoc, Category.assoc]
    · erw [← F.1.map_comp]
      convert h₁
      apply interUnionPullbackConeLift_left
    · erw [← F.1.map_comp]
      convert h₂
      apply interUnionPullbackConeLift_right
    all_goals
      dsimp only [Functor.op, Pairwise.cocone_ι_app, Functor.mapCone_π_app, Cocone.op,
        Pairwise.coconeιApp, unop_op, op_comp, NatTrans.op]
      simp_rw [F.1.map_comp, ← Category.assoc]
      congr 1
      simp_rw [Category.assoc, ← F.1.map_comp]
    · convert h₁
      apply interUnionPullbackConeLift_left
    · convert h₂
      

Proof state:
case h.e'_3
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s✝ : PullbackCone (F.val.map (homOfLE (_ : U ⊓ V ≤ U)).op) (F.val.map (homOfLE (_ : U ⊓ V ≤ V)).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X :=
  fun x =>
    match x with
    | { down := j } => WalkingPair.casesOn j U V
hι : U ⊔ V = iSup ι
s : PullbackCone (F.val.map (homOfLE (_ : U ⊓ V ≤ U)).op) (F.val.map (homOfLE (_ : U ⊓ V ≤ V)).op)
m : s.pt ⟶ (interUnionPullbackCone F U V).pt
h₁ : m ≫ PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s
h₂ : m ≫ PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s
a✝ : ULift.{w, 0} WalkingPair
⊢ interUnionPullbackConeLift F U V s ≫
      F.val.map
        ((eqToHom (_ : ⨆ x, WalkingPair.rec U V x.down = U ⊔ V)).op ≫
          (homOfLE (_ : WalkingPair.rec U V { down := WalkingPair.right }.down ≤ ⨆ x, WalkingPair.rec U V x.down)).op) =
    PullbackCone.snd s

Next tactic:
apply interUnionPullbackConeLift_right
/-- Suppose that a function `f : ℂ → E` is continuous on a closed rectangle with opposite corners at
`z w : ℂ`, is *real* differentiable at all but countably many points of the corresponding open
rectangle, and $\frac{\partial f}{\partial \bar z}$ is integrable on this rectangle. Then the
integral of `f` over the boundary of the rectangle is equal to the integral of
$2i\frac{\partial f}{\partial \bar z}=i\frac{\partial f}{\partial x}-\frac{\partial f}{\partial y}$
over the rectangle. -/
theorem integral_boundary_rect_of_hasFDerivAt_real_off_countable (f : ℂ → E) (f' : ℂ → ℂ →L[ℝ] E)
    (z w : ℂ) (s : Set ℂ) (hs : s.Countable)
    (Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]]))
    (Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s,
      HasFDerivAt f (f' x) x)
    (Hi : IntegrableOn (fun z => I • f' z 1 - f' z I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) :
    (∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) +
      I • (∫ y : ℝ in z.im..w.im, f (re w + y * I)) -
      I • ∫ y : ℝ in z.im..w.im, f (re z + y * I) =
      ∫ x : ℝ in z.re..w.re, ∫ y : ℝ in z.im..w.im, I • f' (x + y * I) 1 - f' (x + y * I) I := by
  

Proof state:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
f' : ℂ → ℂ →L[ℝ] E
z w : ℂ
s : Set ℂ
hs : Set.Countable s
Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])
Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun z => I • (f' z) 1 - (f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])
⊢ (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * I)) +
        I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)) -
      I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * I) =
    ∫ (x : ℝ) in z.re..w.re, ∫ (y : ℝ) in z.im..w.im, I • (f' (↑x + ↑y * I)) 1 - (f' (↑x + ↑y * I)) I

Next tactic:
set e : (ℝ × ℝ) ≃L[ℝ] ℂ := equivRealProdClm.symm
/-- Suppose that a function `f : ℂ → E` is continuous on a closed rectangle with opposite corners at
`z w : ℂ`, is *real* differentiable at all but countably many points of the corresponding open
rectangle, and $\frac{\partial f}{\partial \bar z}$ is integrable on this rectangle. Then the
integral of `f` over the boundary of the rectangle is equal to the integral of
$2i\frac{\partial f}{\partial \bar z}=i\frac{\partial f}{\partial x}-\frac{\partial f}{\partial y}$
over the rectangle. -/
theorem integral_boundary_rect_of_hasFDerivAt_real_off_countable (f : ℂ → E) (f' : ℂ → ℂ →L[ℝ] E)
    (z w : ℂ) (s : Set ℂ) (hs : s.Countable)
    (Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]]))
    (Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s,
      HasFDerivAt f (f' x) x)
    (Hi : IntegrableOn (fun z => I • f' z 1 - f' z I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) :
    (∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) +
      I • (∫ y : ℝ in z.im..w.im, f (re w + y * I)) -
      I • ∫ y : ℝ in z.im..w.im, f (re z + y * I) =
      ∫ x : ℝ in z.re..w.re, ∫ y : ℝ in z.im..w.im, I • f' (x + y * I) 1 - f' (x + y * I) I := by
  set e : (ℝ × ℝ) ≃L[ℝ] ℂ := equivRealProdClm.symm
  

Proof state:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
f' : ℂ → ℂ →L[ℝ] E
z w : ℂ
s : Set ℂ
hs : Set.Countable s
Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])
Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun z => I • (f' z) 1 - (f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])
e : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm
⊢ (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * I)) +
        I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)) -
      I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * I) =
    ∫ (x : ℝ) in z.re..w.re, ∫ (y : ℝ) in z.im..w.im, I • (f' (↑x + ↑y * I)) 1 - (f' (↑x + ↑y * I)) I

Next tactic:
have he : ∀ x y : ℝ, ↑x + ↑y * I = e (x, y) := fun x y => (mk_eq_add_mul_I x y).symm
/-- Suppose that a function `f : ℂ → E` is continuous on a closed rectangle with opposite corners at
`z w : ℂ`, is *real* differentiable at all but countably many points of the corresponding open
rectangle, and $\frac{\partial f}{\partial \bar z}$ is integrable on this rectangle. Then the
integral of `f` over the boundary of the rectangle is equal to the integral of
$2i\frac{\partial f}{\partial \bar z}=i\frac{\partial f}{\partial x}-\frac{\partial f}{\partial y}$
over the rectangle. -/
theorem integral_boundary_rect_of_hasFDerivAt_real_off_countable (f : ℂ → E) (f' : ℂ → ℂ →L[ℝ] E)
    (z w : ℂ) (s : Set ℂ) (hs : s.Countable)
    (Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]]))
    (Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s,
      HasFDerivAt f (f' x) x)
    (Hi : IntegrableOn (fun z => I • f' z 1 - f' z I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) :
    (∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) +
      I • (∫ y : ℝ in z.im..w.im, f (re w + y * I)) -
      I • ∫ y : ℝ in z.im..w.im, f (re z + y * I) =
      ∫ x : ℝ in z.re..w.re, ∫ y : ℝ in z.im..w.im, I • f' (x + y * I) 1 - f' (x + y * I) I := by
  set e : (ℝ × ℝ) ≃L[ℝ] ℂ := equivRealProdClm.symm
  have he : ∀ x y : ℝ, ↑x + ↑y * I = e (x, y) := fun x y => (mk_eq_add_mul_I x y).symm
  

Proof state:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
f' : ℂ → ℂ →L[ℝ] E
z w : ℂ
s : Set ℂ
hs : Set.Countable s
Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])
Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun z => I • (f' z) 1 - (f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])
e : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm
he : ∀ (x y : ℝ), ↑x + ↑y * I = e (x, y)
⊢ (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * I)) +
        I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)) -
      I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * I) =
    ∫ (x : ℝ) in z.re..w.re, ∫ (y : ℝ) in z.im..w.im, I • (f' (↑x + ↑y * I)) 1 - (f' (↑x + ↑y * I)) I

Next tactic:
have he₁ : e (1, 0) = 1 := rfl
/-- Suppose that a function `f : ℂ → E` is continuous on a closed rectangle with opposite corners at
`z w : ℂ`, is *real* differentiable at all but countably many points of the corresponding open
rectangle, and $\frac{\partial f}{\partial \bar z}$ is integrable on this rectangle. Then the
integral of `f` over the boundary of the rectangle is equal to the integral of
$2i\frac{\partial f}{\partial \bar z}=i\frac{\partial f}{\partial x}-\frac{\partial f}{\partial y}$
over the rectangle. -/
theorem integral_boundary_rect_of_hasFDerivAt_real_off_countable (f : ℂ → E) (f' : ℂ → ℂ →L[ℝ] E)
    (z w : ℂ) (s : Set ℂ) (hs : s.Countable)
    (Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]]))
    (Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s,
      HasFDerivAt f (f' x) x)
    (Hi : IntegrableOn (fun z => I • f' z 1 - f' z I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) :
    (∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) +
      I • (∫ y : ℝ in z.im..w.im, f (re w + y * I)) -
      I • ∫ y : ℝ in z.im..w.im, f (re z + y * I) =
      ∫ x : ℝ in z.re..w.re, ∫ y : ℝ in z.im..w.im, I • f' (x + y * I) 1 - f' (x + y * I) I := by
  set e : (ℝ × ℝ) ≃L[ℝ] ℂ := equivRealProdClm.symm
  have he : ∀ x y : ℝ, ↑x + ↑y * I = e (x, y) := fun x y => (mk_eq_add_mul_I x y).symm
  have he₁ : e (1, 0) = 1 := rfl; 

Proof state:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
f' : ℂ → ℂ →L[ℝ] E
z w : ℂ
s : Set ℂ
hs : Set.Countable s
Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])
Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun z => I • (f' z) 1 - (f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])
e : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm
he : ∀ (x y : ℝ), ↑x + ↑y * I = e (x, y)
he₁ : e (1, 0) = 1
⊢ (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * I)) +
        I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)) -
      I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * I) =
    ∫ (x : ℝ) in z.re..w.re, ∫ (y : ℝ) in z.im..w.im, I • (f' (↑x + ↑y * I)) 1 - (f' (↑x + ↑y * I)) I

Next tactic:
have he₂ : e (0, 1) = I := rfl
/-- Suppose that a function `f : ℂ → E` is continuous on a closed rectangle with opposite corners at
`z w : ℂ`, is *real* differentiable at all but countably many points of the corresponding open
rectangle, and $\frac{\partial f}{\partial \bar z}$ is integrable on this rectangle. Then the
integral of `f` over the boundary of the rectangle is equal to the integral of
$2i\frac{\partial f}{\partial \bar z}=i\frac{\partial f}{\partial x}-\frac{\partial f}{\partial y}$
over the rectangle. -/
theorem integral_boundary_rect_of_hasFDerivAt_real_off_countable (f : ℂ → E) (f' : ℂ → ℂ →L[ℝ] E)
    (z w : ℂ) (s : Set ℂ) (hs : s.Countable)
    (Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]]))
    (Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s,
      HasFDerivAt f (f' x) x)
    (Hi : IntegrableOn (fun z => I • f' z 1 - f' z I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) :
    (∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) +
      I • (∫ y : ℝ in z.im..w.im, f (re w + y * I)) -
      I • ∫ y : ℝ in z.im..w.im, f (re z + y * I) =
      ∫ x : ℝ in z.re..w.re, ∫ y : ℝ in z.im..w.im, I • f' (x + y * I) 1 - f' (x + y * I) I := by
  set e : (ℝ × ℝ) ≃L[ℝ] ℂ := equivRealProdClm.symm
  have he : ∀ x y : ℝ, ↑x + ↑y * I = e (x, y) := fun x y => (mk_eq_add_mul_I x y).symm
  have he₁ : e (1, 0) = 1 := rfl; have he₂ : e (0, 1) = I := rfl
  

Proof state:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
f' : ℂ → ℂ →L[ℝ] E
z w : ℂ
s : Set ℂ
hs : Set.Countable s
Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])
Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun z => I • (f' z) 1 - (f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])
e : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm
he : ∀ (x y : ℝ), ↑x + ↑y * I = e (x, y)
he₁ : e (1, 0) = 1
he₂ : e (0, 1) = I
⊢ (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * I)) +
        I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)) -
      I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * I) =
    ∫ (x : ℝ) in z.re..w.re, ∫ (y : ℝ) in z.im..w.im, I • (f' (↑x + ↑y * I)) 1 - (f' (↑x + ↑y * I)) I

Next tactic:
simp only [he] at *
/-- Suppose that a function `f : ℂ → E` is continuous on a closed rectangle with opposite corners at
`z w : ℂ`, is *real* differentiable at all but countably many points of the corresponding open
rectangle, and $\frac{\partial f}{\partial \bar z}$ is integrable on this rectangle. Then the
integral of `f` over the boundary of the rectangle is equal to the integral of
$2i\frac{\partial f}{\partial \bar z}=i\frac{\partial f}{\partial x}-\frac{\partial f}{\partial y}$
over the rectangle. -/
theorem integral_boundary_rect_of_hasFDerivAt_real_off_countable (f : ℂ → E) (f' : ℂ → ℂ →L[ℝ] E)
    (z w : ℂ) (s : Set ℂ) (hs : s.Countable)
    (Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]]))
    (Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s,
      HasFDerivAt f (f' x) x)
    (Hi : IntegrableOn (fun z => I • f' z 1 - f' z I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) :
    (∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) +
      I • (∫ y : ℝ in z.im..w.im, f (re w + y * I)) -
      I • ∫ y : ℝ in z.im..w.im, f (re z + y * I) =
      ∫ x : ℝ in z.re..w.re, ∫ y : ℝ in z.im..w.im, I • f' (x + y * I) 1 - f' (x + y * I) I := by
  set e : (ℝ × ℝ) ≃L[ℝ] ℂ := equivRealProdClm.symm
  have he : ∀ x y : ℝ, ↑x + ↑y * I = e (x, y) := fun x y => (mk_eq_add_mul_I x y).symm
  have he₁ : e (1, 0) = 1 := rfl; have he₂ : e (0, 1) = I := rfl
  simp only [he] at *
  

Proof state:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
f' : ℂ → ℂ →L[ℝ] E
z w : ℂ
s : Set ℂ
hs : Set.Countable s
Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])
Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun z => I • (f' z) 1 - (f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])
e : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm
he₁ : (ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1
he₂ : (ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I
he : ℝ → ℝ → True
⊢ (((∫ (x : ℝ) in z.re..w.re, f ((ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -
          ∫ (x : ℝ) in z.re..w.re, f ((ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +
        I • ∫ (y : ℝ) in z.im..w.im, f ((ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -
      I • ∫ (y : ℝ) in z.im..w.im, f ((ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y)) =
    ∫ (x : ℝ) in z.re..w.re,
      ∫ (y : ℝ) in z.im..w.im,
        I • (f' ((ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -
          (f' ((ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I

Next tactic:
set F : ℝ × ℝ → E := f ∘ e
/-- Suppose that a function `f : ℂ → E` is continuous on a closed rectangle with opposite corners at
`z w : ℂ`, is *real* differentiable at all but countably many points of the corresponding open
rectangle, and $\frac{\partial f}{\partial \bar z}$ is integrable on this rectangle. Then the
integral of `f` over the boundary of the rectangle is equal to the integral of
$2i\frac{\partial f}{\partial \bar z}=i\frac{\partial f}{\partial x}-\frac{\partial f}{\partial y}$
over the rectangle. -/
theorem integral_boundary_rect_of_hasFDerivAt_real_off_countable (f : ℂ → E) (f' : ℂ → ℂ →L[ℝ] E)
    (z w : ℂ) (s : Set ℂ) (hs : s.Countable)
    (Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]]))
    (Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s,
      HasFDerivAt f (f' x) x)
    (Hi : IntegrableOn (fun z => I • f' z 1 - f' z I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) :
    (∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) +
      I • (∫ y : ℝ in z.im..w.im, f (re w + y * I)) -
      I • ∫ y : ℝ in z.im..w.im, f (re z + y * I) =
      ∫ x : ℝ in z.re..w.re, ∫ y : ℝ in z.im..w.im, I • f' (x + y * I) 1 - f' (x + y * I) I := by
  set e : (ℝ × ℝ) ≃L[ℝ] ℂ := equivRealProdClm.symm
  have he : ∀ x y : ℝ, ↑x + ↑y * I = e (x, y) := fun x y => (mk_eq_add_mul_I x y).symm
  have he₁ : e (1, 0) = 1 := rfl; have he₂ : e (0, 1) = I := rfl
  simp only [he] at *
  set F : ℝ × ℝ → E := f ∘ e
  

Proof state:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
f' : ℂ → ℂ →L[ℝ] E
z w : ℂ
s : Set ℂ
hs : Set.Countable s
Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])
Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun z => I • (f' z) 1 - (f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])
e : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm
he₁ : (ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1
he₂ : (ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I
he : ℝ → ℝ → True
F : ℝ × ℝ → E := f ∘ ⇑e
⊢ (((∫ (x : ℝ) in z.re..w.re, f ((ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -
          ∫ (x : ℝ) in z.re..w.re, f ((ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +
        I • ∫ (y : ℝ) in z.im..w.im, f ((ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -
      I • ∫ (y : ℝ) in z.im..w.im, f ((ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y)) =
    ∫ (x : ℝ) in z.re..w.re,
      ∫ (y : ℝ) in z.im..w.im,
        I • (f' ((ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -
          (f' ((ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I

Next tactic:
set F' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => (f' (e p)).comp (e : ℝ × ℝ →L[ℝ] ℂ)
/-- Suppose that a function `f : ℂ → E` is continuous on a closed rectangle with opposite corners at
`z w : ℂ`, is *real* differentiable at all but countably many points of the corresponding open
rectangle, and $\frac{\partial f}{\partial \bar z}$ is integrable on this rectangle. Then the
integral of `f` over the boundary of the rectangle is equal to the integral of
$2i\frac{\partial f}{\partial \bar z}=i\frac{\partial f}{\partial x}-\frac{\partial f}{\partial y}$
over the rectangle. -/
theorem integral_boundary_rect_of_hasFDerivAt_real_off_countable (f : ℂ → E) (f' : ℂ → ℂ →L[ℝ] E)
    (z w : ℂ) (s : Set ℂ) (hs : s.Countable)
    (Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]]))
    (Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s,
      HasFDerivAt f (f' x) x)
    (Hi : IntegrableOn (fun z => I • f' z 1 - f' z I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) :
    (∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) +
      I • (∫ y : ℝ in z.im..w.im, f (re w + y * I)) -
      I • ∫ y : ℝ in z.im..w.im, f (re z + y * I) =
      ∫ x : ℝ in z.re..w.re, ∫ y : ℝ in z.im..w.im, I • f' (x + y * I) 1 - f' (x + y * I) I := by
  set e : (ℝ × ℝ) ≃L[ℝ] ℂ := equivRealProdClm.symm
  have he : ∀ x y : ℝ, ↑x + ↑y * I = e (x, y) := fun x y => (mk_eq_add_mul_I x y).symm
  have he₁ : e (1, 0) = 1 := rfl; have he₂ : e (0, 1) = I := rfl
  simp only [he] at *
  set F : ℝ × ℝ → E := f ∘ e
  set F' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => (f' (e p)).comp (e : ℝ × ℝ →L[ℝ] ℂ)
  have hF' : ∀ p : ℝ × ℝ, (-(I • F' p)) (1, 0) + F' p (0, 1) = -(I • f' (e p) 1 - f' (e p) I) := by
    

Proof state:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
f' : ℂ → ℂ →L[ℝ] E
z w : ℂ
s : Set ℂ
hs : Set.Countable s
Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])
Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun z => I • (f' z) 1 - (f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])
e : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm
he₁ : (ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1
he₂ : (ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I
he : ℝ → ℝ → True
F : ℝ × ℝ → E := f ∘ ⇑e
F' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (e p)) ↑e
⊢ ∀ (p : ℝ × ℝ), (-(I • F' p)) (1, 0) + (F' p) (0, 1) = -(I • (f' (e p)) 1 - (f' (e p)) I)

Next tactic:
rintro ⟨x, y⟩