awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.LocallyCartesianClosed.ExponentiableMorphism (goal)
Goal:
C : Type u
inst : Category.{v, u} C
I J K : C
f : I ⟶ J
g : J ⟶ K
inst_1 : ChosenPullbacksAlong f
inst_2 : ChosenPullbacksAlong g
inst_3 : ChosenPullbacksAlong (f ≫ g)
inst_4 : ExponentiableMorphism f
inst_5 : ExponentiableMorphism g
inst_6 : ExponentiableMorphism (f ≫ g)
⊢ Eq (α := @pushforward _ _ _ _ (f ≫ g) inst_3 _ ⋙ pullback (f ≫ g) ⟶ 𝟭 (Over.{v, u} I))
    (CategoryStruct.comp (Y := (@pushforward _ _ _ _ f inst_1 _ ⋙ @pushforward _ _ _ _ g inst_2 _) ⋙ pullback (f ≫ g))
      (Functor.whiskerRight (Iso.hom (pushforwardComp f g)) (pullback (f ≫ g)))
      (counit (pullbackPushforwardAdj (f ≫ g))))
    (counit (pullbackPushforwardAdj (f ≫ g)))
have rw : Eq (α := @pushforward _ _ _ _ (f ≫ g) inst_3 _ ⋙ pullback (f ≫ g) ⟶ 𝟭 (Over.{v, u} I))
  (CategoryStruct.comp (Y := (@pushforward _ _ _ _ f inst_1 _ ⋙ @pushforward _ _ _ _ g inst_2 _) ⋙ pullback (f ≫ g))
    (Functor.whiskerRight (Iso.hom (rightAdjointUniq (pullbackPushforwardAdj (f ≫ g)) (pullbackPushforwardAdj (f ≫ g))))
        (pullback (f ≫ g)) :
      @pushforward _ _ _ _ (f ≫ g) inst_3 _ ⋙ pullback (f ≫ g) ⟶
        (@pushforward _ _ _ _ f inst_1 _ ⋙ @pushforward _ _ _ _ g inst_2 _) ⋙ pullback (f ≫ g))
    (counit (pullbackPushforwardAdj (f ≫ g)) :
      @pushforward _ _ _ _ (f ≫ g) inst_3 _ ⋙ pullback (f ≫ g) ⟶ 𝟭 (Over.{v, u} I)))
  (counit (pullbackPushforwardAdj (f ≫ g)))
Tactic: rw [pushforwardComp, Adjunction.rightAdjointUniq_hom_counit]
-- Lean 4 proof step from Mathlib.CategoryTheory.LocallyCartesianClosed.ExponentiableMorphism (goal)
Goal:
C : Type u
inst : Category.{v, u} C
I J K : C
f : I ⟶ J
g : J ⟶ K
inst_1 : ChosenPullbacksAlong f
inst_2 : ChosenPullbacksAlong g
inst_3 : ChosenPullbacksAlong (f ≫ g)
inst_4 : ExponentiableMorphism f
inst_5 : ExponentiableMorphism g
inst_6 : ExponentiableMorphism (f ≫ g)
rw :
  Eq (α := @pushforward _ _ _ _ (f ≫ g) inst_3 _ ⋙ pullback (f ≫ g) ⟶ 𝟭 (Over.{v, u} I))
    (CategoryStruct.comp (Y := (@pushforward _ _ _ _ f inst_1 _ ⋙ @pushforward _ _ _ _ g inst_2 _) ⋙ pullback (f ≫ g))
      (Functor.whiskerRight
          (Iso.hom (rightAdjointUniq (pullbackPushforwardAdj (f ≫ g)) (pullbackPushforwardAdj (f ≫ g))))
          (pullback (f ≫ g)) :
        @pushforward _ _ _ _ (f ≫ g) inst_3 _ ⋙ pullback (f ≫ g) ⟶
          (@pushforward _ _ _ _ f inst_1 _ ⋙ @pushforward _ _ _ _ g inst_2 _) ⋙ pullback (f ≫ g))
      (counit (pullbackPushforwardAdj (f ≫ g)) :
        @pushforward _ _ _ _ (f ≫ g) inst_3 _ ⋙ pullback (f ≫ g) ⟶ 𝟭 (Over.{v, u} I)))
    (counit (pullbackPushforwardAdj (f ≫ g)))
⊢ Eq (α := @pushforward _ _ _ _ (f ≫ g) inst_3 _ ⋙ pullback (f ≫ g) ⟶ 𝟭 (Over.{v, u} I))
    (CategoryStruct.comp (Y := (@pushforward _ _ _ _ f inst_1 _ ⋙ @pushforward _ _ _ _ g inst_2 _) ⋙ pullback (f ≫ g))
      (Functor.whiskerRight (Iso.hom (pushforwardComp f g)) (pullback (f ≫ g)))
      (counit (pullbackPushforwardAdj (f ≫ g))))
    (counit (pullbackPushforwardAdj (f ≫ g)))
have rw₁ : Eq (α := @pushforward _ _ _ _ (f ≫ g) inst_3 _ ⋙ pullback (f ≫ g) ⟶ 𝟭 (Over.{v, u} I))
  (counit (pullbackPushforwardAdj (f ≫ g))) (counit (pullbackPushforwardAdj (f ≫ g)))
Tactic: rw [pushforwardComp, Adjunction.rightAdjointUniq_hom_counit]
-- Lean 4 proof step from Mathlib.CategoryTheory.LocallyCartesianClosed.Over (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
X : C
Y Z : Over.{v₁, u₁} X
inst_1 : ChosenPullbacksAlong Z.hom
T : Over.{v₁, u₁} X
u : T ⟶ Y
v : T ⟶ Z
⊢ Over.Hom.left u ≫ Y.hom = Over.Hom.left v ≫ Z.hom
have rw : T.hom = Over.Hom.left v ≫ Z.hom
Tactic: rw [Over.w u, Over.w v]
-- Lean 4 proof step from Mathlib.CategoryTheory.LocallyCartesianClosed.Over (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
X : C
Y Z : Over.{v₁, u₁} X
inst_1 : ChosenPullbacksAlong Z.hom
T : Over.{v₁, u₁} X
u : T ⟶ Y
v : T ⟶ Z
rw : T.hom = Over.Hom.left v ≫ Z.hom
⊢ Over.Hom.left u ≫ Y.hom = Over.Hom.left v ≫ Z.hom
have rw₁ : T.hom = T.hom
Tactic: rw [Over.w u, Over.w v]
-- Lean 4 proof step from Mathlib.CategoryTheory.LocallyCartesianClosed.Over (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
X : C
Y Z : Over.{v₁, u₁} X
inst_1 : ChosenPullbacksAlong Z.hom
T : Over.{v₁, u₁} X
a : T ⟶ Y
b : T ⟶ Z
m : T ⟶ (binaryFan Y Z).pt
h₁ : m ≫ (binaryFan Y Z).fst = a
h₂ : m ≫ (binaryFan Y Z).snd = b
⊢ m =
    (fun {T : Over.{v₁, u₁} X} (u : T ⟶ Y) (v : T ⟶ Z) ↦
        (Over.homMk (lift (Over.Hom.left u) (Over.Hom.left v) ⋯) ⋯ : T ⟶ (binaryFan Y Z).pt))
      a b
have h : Over.Hom.left m =
  Over.Hom.left
    ((fun {T : Over.{v₁, u₁} X} (u : T ⟶ Y) (v : T ⟶ Z) ↦
        (Over.homMk (lift (Over.Hom.left u) (Over.Hom.left v) sorry) sorry : T ⟶ (binaryFan Y Z).pt))
      a b)
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.LocallyCartesianClosed.Over (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
X : C
Y Z : Over.{v₁, u₁} X
inst_1 : ChosenPullbacksAlong Z.hom
T : Over.{v₁, u₁} X
a : T ⟶ Y
b : T ⟶ Z
m : T ⟶ (binaryFan Y Z).pt
h₁ : m ≫ (binaryFan Y Z).fst = a
h₂ : m ≫ (binaryFan Y Z).snd = b
⊢ Over.Hom.left m =
    Over.Hom.left
      ((fun {T : Over.{v₁, u₁} X} (u : T ⟶ Y) (v : T ⟶ Z) ↦
          (Over.homMk (lift (Over.Hom.left u) (Over.Hom.left v) ⋯) ⋯ : T ⟶ (binaryFan Y Z).pt))
        a b)
have h : Over.Hom.left m = lift (Over.Hom.left a) (Over.Hom.left b) sorry
Tactic: dsimp [Over.map, Comma.mapRight]
-- Lean 4 proof step from Mathlib.CategoryTheory.LocallyCartesianClosed.Over (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : CartesianMonoidalCategory.{v₁, u₁} C
X : C
Z : Over.{v₁, u₁} X
A : C
f : Z ⟶ Functor.obj (toOver X) A
⊢ (Equiv.symm (Adjunction.homEquiv (forgetAdjToOver X) Z A) : (a : Z ⟶ (toOver X).obj A) → (Over.forget X).obj Z ⟶ A)
      f =
    Over.Hom.left f ≫ SemiCartesianMonoidalCategory.fst A X
have rw : Functor.map (Over.forget X) f ≫ SemiCartesianMonoidalCategory.fst A X =
  Over.Hom.left f ≫ SemiCartesianMonoidalCategory.fst A X
Tactic: rw [Adjunction.homEquiv_counit, forgetAdjToOver_counit_app]
-- Lean 4 proof step from Mathlib.CategoryTheory.LocallyCartesianClosed.Sections (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : CartesianMonoidalCategory.{v₁, u₁} C
I : C
inst_2 : Closed.{v₁, u₁} I
A : C
⊢ Eq.{v₁ + 1} (α := A ⟶ Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) I) (toUnit A ≫ curryRightUnitorHom I) (curry (fst I A))
have a : Eq.{v₁ + 1} (α := tensorObj.{v₁, u₁} (𝒞 := inst) I A ⟶ I) (uncurry (toUnit A ≫ curryRightUnitorHom I))
  (uncurry (curry (fst I A)))
Tactic: apply uncurry_injective
-- Lean 4 proof step from Mathlib.CategoryTheory.LocallyCartesianClosed.Sections (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : CartesianMonoidalCategory.{v₁, u₁} C
I : C
inst_2 : Closed.{v₁, u₁} I
inst_3 :
  ChosenPullbacksAlong.{v₁, u₁} (Y := tensorUnit.{v₁, u₁} C (𝒞 := inst)) (X := Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) I)
    (curryRightUnitorHom I)
inst_4 : BraidedCategory.{v₁, u₁} C
X : Over.{v₁, u₁} I
A : C
u : Functor.obj (toOver I) A ⟶ X
⊢ Eq (α := A ⟶ Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) I)
    (curry (Iso.hom (β_ I A) ≫ Hom.left u) ≫ Functor.map (ihom I) X.hom) (toUnit A ≫ curryRightUnitorHom I)
have rw : Eq.{v₁ + 1} (α := A ⟶ Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) I)
  (curry (Iso.hom (β_ I A) ≫ hom (Functor.obj (toOver I) A))) (curry (SemiCartesianMonoidalCategory.fst I A))
Tactic: rw [curry_natural_right, Category.assoc, ← Functor.map_comp, w,
      ← curry_natural_right, toUnit_comp_curryRightUnitorHom]
-- Lean 4 proof step from Mathlib.CategoryTheory.LocallyCartesianClosed.Sections (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : CartesianMonoidalCategory.{v₁, u₁} C
I : C
inst_2 : Closed.{v₁, u₁} I
inst_3 :
  ChosenPullbacksAlong.{v₁, u₁} (Y := tensorUnit.{v₁, u₁} C (𝒞 := inst)) (X := Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) I)
    (curryRightUnitorHom I)
inst_4 : BraidedCategory.{v₁, u₁} C
X : Over.{v₁, u₁} I
A : C
u : Functor.obj (toOver I) A ⟶ X
⊢ Eq.{v₁ + 1} (α := A ⟶ Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) I)
    (curry (Iso.hom (β_ I A) ≫ hom (Functor.obj (toOver I) A))) (curry (SemiCartesianMonoidalCategory.fst I A))
have e_a : Eq.{v₁ + 1} (α := tensorObj.{v₁, u₁} (𝒞 := inst) I A ⟶ I) (Iso.hom (β_ I A) ≫ hom (Functor.obj (toOver I) A))
  (SemiCartesianMonoidalCategory.fst I A)
Tactic: congr
-- Lean 4 proof step from Mathlib.CategoryTheory.LocallyCartesianClosed.Sections (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : CartesianMonoidalCategory.{v₁, u₁} C
I : C
inst_2 : Closed.{v₁, u₁} I
inst_3 :
  ChosenPullbacksAlong.{v₁, u₁} (Y := tensorUnit.{v₁, u₁} C (𝒞 := inst)) (X := Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) I)
    (curryRightUnitorHom I)
inst_4 : BraidedCategory.{v₁, u₁} C
X : Over.{v₁, u₁} I
A : C
v : A ⟶ Functor.obj (sections I) X
v₂ : A ⟶ Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) X.left :=
  v ≫ ChosenPullbacksAlong.fst (Functor.map (ihom I) X.hom) (curryRightUnitorHom I)
⊢ toUnit A ≫ curryRightUnitorHom I = v₂ ≫ Functor.map (ihom I) X.hom
have rw : (v ≫ ChosenPullbacksAlong.snd (Functor.map (ihom I) X.hom) (curryRightUnitorHom I)) ≫ curryRightUnitorHom I =
  v₂ ≫ Functor.map (ihom I) X.hom
Tactic: rw [IsTerminal.hom_ext isTerminalTensorUnit (toUnit A) (v ≫ snd ..)]
-- Lean 4 proof step from Mathlib.CategoryTheory.LocallyCartesianClosed.Sections (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : CartesianMonoidalCategory.{v₁, u₁} C
I : C
inst_2 : Closed.{v₁, u₁} I
inst_3 :
  ChosenPullbacksAlong.{v₁, u₁} (Y := tensorUnit.{v₁, u₁} C (𝒞 := inst)) (X := Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) I)
    (curryRightUnitorHom I)
inst_4 : BraidedCategory.{v₁, u₁} C
X : Over.{v₁, u₁} I
A : C
v : A ⟶ Functor.obj (sections I) X
v₂ : A ⟶ Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) X.left :=
  v ≫ ChosenPullbacksAlong.fst (Functor.map (ihom I) X.hom) (curryRightUnitorHom I)
⊢ (Iso.hom (β_ A I) ≫ uncurry v₂) ≫ X.hom = hom (Functor.obj (toOver I) A)
have comm : toUnit A ≫ curryRightUnitorHom I = v₂ ≫ Functor.map (ihom I) X.hom
Tactic: have comm : toUnit A ≫ (curryRightUnitorHom I) = v₂ ≫ (ihom I).map X.hom := by
      rw [IsTerminal.hom_ext isTerminalTensorUnit (toUnit A) (v ≫ snd ..)]
      simp [v₂, condition]
-- Lean 4 proof step from Mathlib.CategoryTheory.LocallyCartesianClosed.Sections (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : CartesianMonoidalCategory.{v₁, u₁} C
I : C
inst_2 : Closed.{v₁, u₁} I
inst_3 :
  ChosenPullbacksAlong.{v₁, u₁} (Y := tensorUnit.{v₁, u₁} C (𝒞 := inst)) (X := Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) I)
    (curryRightUnitorHom I)
inst_4 : BraidedCategory.{v₁, u₁} C
X : Over.{v₁, u₁} I
A : C
v : A ⟶ Functor.obj (sections I) X
v₂ : A ⟶ Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) X.left :=
  v ≫ ChosenPullbacksAlong.fst (Functor.map (ihom I) X.hom) (curryRightUnitorHom I)
comm : toUnit A ≫ curry (Iso.hom (ρ_ I)) = v₂ ≫ Functor.map (ihom I) X.hom
⊢ (Iso.hom (β_ A I) ≫ uncurry v₂) ≫ X.hom = hom (Functor.obj (toOver I) A)
have w' : Eq.{v₁ + 1} (α := Functor.obj.{v₁, v₁, u₁, u₁} (tensorLeft I) A ⟶ I)
  (CategoryStruct.comp (Y := Functor.obj.{v₁, v₁, u₁, u₁} (tensorLeft I) (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
    (Functor.map (tensorLeft I) (toUnit A))
    ((Equiv.symm (Adjunction.homEquiv (ihom.adjunction I) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) I) :
        (a : 𝟙_ C ⟶ (ihom I).obj I) → (tensorLeft I).obj (𝟙_ C) ⟶ I)
      (curry (Iso.hom (ρ_ I)))))
  ((Equiv.symm (Adjunction.homEquiv (ihom.adjunction I) A X.left) :
        (a : A ⟶ (ihom I).obj X.left) → (tensorLeft I).obj A ⟶ X.left)
      v₂ ≫
    X.hom)
Tactic: have w' := (ihom.adjunction I).homEquiv_naturality_right_square _ _ _ _ comm
-- Lean 4 proof step from Mathlib.CategoryTheory.LocallyCartesianClosed.Sections (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : CartesianMonoidalCategory.{v₁, u₁} C
I : C
inst_2 : Closed.{v₁, u₁} I
inst_3 :
  ChosenPullbacksAlong.{v₁, u₁} (Y := tensorUnit.{v₁, u₁} C (𝒞 := inst)) (X := Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) I)
    (curryRightUnitorHom I)
inst_4 : BraidedCategory.{v₁, u₁} C
X : Over.{v₁, u₁} I
A : C
v : A ⟶ Functor.obj (sections I) X
v₂ : A ⟶ Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) X.left :=
  v ≫ ChosenPullbacksAlong.fst (Functor.map (ihom I) X.hom) (curryRightUnitorHom I)
comm : toUnit A ≫ curry (Iso.hom (ρ_ I)) = v₂ ≫ Functor.map (ihom I) X.hom
w' :
  Eq.{v₁ + 1} (α := Functor.obj.{v₁, v₁, u₁, u₁} (tensorLeft I) A ⟶ I)
    (CategoryStruct.comp (Y := Functor.obj.{v₁, v₁, u₁, u₁} (tensorLeft I) (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
      (Functor.map (tensorLeft I) (toUnit A))
      ((Equiv.symm (Adjunction.homEquiv (ihom.adjunction I) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) I) :
          (a : 𝟙_ C ⟶ (ihom I).obj I) → (tensorLeft I).obj (𝟙_ C) ⟶ I)
        (curry (Iso.hom (ρ_ I)))))
    ((Equiv.symm (Adjunction.homEquiv (ihom.adjunction I) A X.left) :
          (a : A ⟶ (ihom I).obj X.left) → (tensorLeft I).obj A ⟶ X.left)
        v₂ ≫
      X.hom)
⊢ (Iso.hom (β_ A I) ≫ uncurry v₂) ≫ X.hom = hom (Functor.obj (toOver I) A)
have w'₁ : Eq.{v₁ + 1} (α := tensorObj.{v₁, u₁} (𝒞 := inst) I A ⟶ I) (I ◁ toUnit A ≫ Iso.hom (ρ_ I))
  ((Equiv.symm (Adjunction.homEquiv (ihom.adjunction I) A X.left) : (a : A ⟶ (ihom I).obj X.left) → I ⊗ A ⟶ X.left) v₂ ≫
    X.hom)
Tactic: simp only [curriedTensor_obj_obj, curriedTensor_obj_map, curry,
      Equiv.symm_apply_apply] at w'
Cleared hypotheses: w'
-- Lean 4 proof step from Mathlib.CategoryTheory.LocallyCartesianClosed.Sections (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : CartesianMonoidalCategory.{v₁, u₁} C
I : C
inst_2 : Closed.{v₁, u₁} I
inst_3 :
  ChosenPullbacksAlong.{v₁, u₁} (Y := tensorUnit.{v₁, u₁} C (𝒞 := inst)) (X := Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) I)
    (curryRightUnitorHom I)
inst_4 : BraidedCategory.{v₁, u₁} C
X : Over.{v₁, u₁} I
A : C
v : A ⟶ Functor.obj (sections I) X
v₂ : A ⟶ Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) X.left :=
  v ≫ ChosenPullbacksAlong.fst (Functor.map (ihom I) X.hom) (curryRightUnitorHom I)
comm : toUnit A ≫ curry (Iso.hom (ρ_ I)) = v₂ ≫ Functor.map (ihom I) X.hom
w' :
  Eq.{v₁ + 1} (α := tensorObj.{v₁, u₁} (𝒞 := inst) I A ⟶ I) (I ◁ toUnit A ≫ Iso.hom (ρ_ I))
    ((Equiv.symm (Adjunction.homEquiv (ihom.adjunction I) A X.left) : (a : A ⟶ (ihom I).obj X.left) → I ⊗ A ⟶ X.left)
        v₂ ≫
      X.hom)
⊢ (Iso.hom (β_ A I) ≫ uncurry v₂) ≫ X.hom = hom (Functor.obj (toOver I) A)
have dsimp : Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) A I ⟶ I)
  ((Iso.hom (β_ A I) ≫
      (Equiv.symm (Adjunction.homEquiv (ihom.adjunction I) A X.left) : (a : A ⟶ (ihom I).obj X.left) → I ⊗ A ⟶ X.left)
        v₂) ≫
    X.hom)
  (SemiCartesianMonoidalCategory.snd A I)
Tactic: dsimp [uncurry] at *
-- Lean 4 proof step from Mathlib.CategoryTheory.LocallyCartesianClosed.Sections (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : CartesianMonoidalCategory.{v₁, u₁} C
I : C
inst_2 : Closed.{v₁, u₁} I
inst_3 :
  ChosenPullbacksAlong.{v₁, u₁} (Y := tensorUnit.{v₁, u₁} C (𝒞 := inst)) (X := Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) I)
    (curryRightUnitorHom I)
inst_4 : BraidedCategory.{v₁, u₁} C
X : Over.{v₁, u₁} I
A : C
v : A ⟶ Functor.obj (sections I) X
v₂ : A ⟶ Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) X.left :=
  v ≫ ChosenPullbacksAlong.fst (Functor.map (ihom I) X.hom) (curryRightUnitorHom I)
comm : toUnit A ≫ curry (Iso.hom (ρ_ I)) = v₂ ≫ Functor.map (ihom I) X.hom
w' :
  Eq.{v₁ + 1} (α := tensorObj.{v₁, u₁} (𝒞 := inst) I A ⟶ I) (I ◁ toUnit A ≫ Iso.hom (ρ_ I))
    ((Equiv.symm (Adjunction.homEquiv (ihom.adjunction I) A X.left) : (a : A ⟶ (ihom I).obj X.left) → I ⊗ A ⟶ X.left)
        v₂ ≫
      X.hom)
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) A I ⟶ I)
    ((Iso.hom (β_ A I) ≫
        (Equiv.symm (Adjunction.homEquiv (ihom.adjunction I) A X.left) : (a : A ⟶ (ihom I).obj X.left) → I ⊗ A ⟶ X.left)
          v₂) ≫
      X.hom)
    (SemiCartesianMonoidalCategory.snd A I)
have rw : Eq.{v₁ + 1} (α := tensorObj.{v₁, u₁} (𝒞 := inst) A I ⟶ I)
  (Iso.hom (β_ A I) ≫
    (Equiv.symm (Adjunction.homEquiv (ihom.adjunction I) A X.left) : (a : A ⟶ (ihom I).obj X.left) → I ⊗ A ⟶ X.left)
        v₂ ≫
      X.hom)
  (SemiCartesianMonoidalCategory.snd A I)
Tactic: rw [Category.assoc, ← w', whiskerLeft_toUnit_comp_rightUnitor_hom, braiding_hom_fst]
-- Lean 4 proof step from Mathlib.CategoryTheory.LocallyCartesianClosed.Sections (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : CartesianMonoidalCategory.{v₁, u₁} C
I : C
inst_2 : Closed.{v₁, u₁} I
inst_3 :
  ChosenPullbacksAlong.{v₁, u₁} (Y := tensorUnit.{v₁, u₁} C (𝒞 := inst)) (X := Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) I)
    (curryRightUnitorHom I)
inst_4 : BraidedCategory.{v₁, u₁} C
X : Over.{v₁, u₁} I
A : C
v : A ⟶ Functor.obj (sections I) X
v₂ : A ⟶ Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) X.left :=
  v ≫ ChosenPullbacksAlong.fst (Functor.map (ihom I) X.hom) (curryRightUnitorHom I)
comm : toUnit A ≫ curry (Iso.hom (ρ_ I)) = v₂ ≫ Functor.map (ihom I) X.hom
w' :
  Eq.{v₁ + 1} (α := tensorObj.{v₁, u₁} (𝒞 := inst) I A ⟶ I) (I ◁ toUnit A ≫ Iso.hom (ρ_ I))
    ((Equiv.symm (Adjunction.homEquiv (ihom.adjunction I) A X.left) : (a : A ⟶ (ihom I).obj X.left) → I ⊗ A ⟶ X.left)
        v₂ ≫
      X.hom)
rw :
  Eq.{v₁ + 1} (α := tensorObj.{v₁, u₁} (𝒞 := inst) A I ⟶ I)
    (Iso.hom (β_ A I) ≫
      (Equiv.symm (Adjunction.homEquiv (ihom.adjunction I) A X.left) : (a : A ⟶ (ihom I).obj X.left) → I ⊗ A ⟶ X.left)
          v₂ ≫
        X.hom)
    (SemiCartesianMonoidalCategory.snd A I)
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) A I ⟶ I)
    ((Iso.hom (β_ A I) ≫
        (Equiv.symm (Adjunction.homEquiv (ihom.adjunction I) A X.left) : (a : A ⟶ (ihom I).obj X.left) → I ⊗ A ⟶ X.left)
          v₂) ≫
      X.hom)
    (SemiCartesianMonoidalCategory.snd A I)
have rw₁ : Eq.{v₁ + 1} (α := tensorObj.{v₁, u₁} (𝒞 := inst) A I ⟶ I) (Iso.hom (β_ A I) ≫ I ◁ toUnit A ≫ Iso.hom (ρ_ I))
  (SemiCartesianMonoidalCategory.snd A I)
Tactic: rw [Category.assoc, ← w', whiskerLeft_toUnit_comp_rightUnitor_hom, braiding_hom_fst]
-- Lean 4 proof step from Mathlib.CategoryTheory.LocallyCartesianClosed.Sections (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : CartesianMonoidalCategory.{v₁, u₁} C
I : C
inst_2 : Closed.{v₁, u₁} I
inst_3 :
  ChosenPullbacksAlong.{v₁, u₁} (Y := tensorUnit.{v₁, u₁} C (𝒞 := inst)) (X := Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) I)
    (curryRightUnitorHom I)
inst_4 : BraidedCategory.{v₁, u₁} C
X : Over.{v₁, u₁} I
A : C
v : A ⟶ Functor.obj (sections I) X
v₂ : A ⟶ Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) X.left :=
  v ≫ ChosenPullbacksAlong.fst (Functor.map (ihom I) X.hom) (curryRightUnitorHom I)
comm : toUnit A ≫ curry (Iso.hom (ρ_ I)) = v₂ ≫ Functor.map (ihom I) X.hom
w' :
  Eq.{v₁ + 1} (α := tensorObj.{v₁, u₁} (𝒞 := inst) I A ⟶ I) (I ◁ toUnit A ≫ Iso.hom (ρ_ I))
    ((Equiv.symm (Adjunction.homEquiv (ihom.adjunction I) A X.left) : (a : A ⟶ (ihom I).obj X.left) → I ⊗ A ⟶ X.left)
        v₂ ≫
      X.hom)
rw :
  Eq.{v₁ + 1} (α := tensorObj.{v₁, u₁} (𝒞 := inst) A I ⟶ I)
    (Iso.hom (β_ A I) ≫
      (Equiv.symm (Adjunction.homEquiv (ihom.adjunction I) A X.left) : (a : A ⟶ (ihom I).obj X.left) → I ⊗ A ⟶ X.left)
          v₂ ≫
        X.hom)
    (SemiCartesianMonoidalCategory.snd A I)
rw₁ :
  Eq.{v₁ + 1} (α := tensorObj.{v₁, u₁} (𝒞 := inst) A I ⟶ I) (Iso.hom (β_ A I) ≫ I ◁ toUnit A ≫ Iso.hom (ρ_ I))
    (SemiCartesianMonoidalCategory.snd A I)
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) A I ⟶ I)
    ((Iso.hom (β_ A I) ≫
        (Equiv.symm (Adjunction.homEquiv (ihom.adjunction I) A X.left) : (a : A ⟶ (ihom I).obj X.left) → I ⊗ A ⟶ X.left)
          v₂) ≫
      X.hom)
    (SemiCartesianMonoidalCategory.snd A I)
have rw₂ : Eq.{v₁ + 1} (α := tensorObj.{v₁, u₁} (𝒞 := inst) A I ⟶ I) (Iso.hom (β_ A I) ≫ SemiCartesianMonoidalCategory.fst I A)
  (SemiCartesianMonoidalCategory.snd A I)
Tactic: rw [Category.assoc, ← w', whiskerLeft_toUnit_comp_rightUnitor_hom, braiding_hom_fst]
-- Lean 4 proof step from Mathlib.CategoryTheory.LocallyCartesianClosed.Sections (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : CartesianMonoidalCategory.{v₁, u₁} C
I : C
inst_2 : Closed.{v₁, u₁} I
inst_3 :
  ChosenPullbacksAlong.{v₁, u₁} (Y := tensorUnit.{v₁, u₁} C (𝒞 := inst)) (X := Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) I)
    (curryRightUnitorHom I)
inst_4 : BraidedCategory.{v₁, u₁} C
X : Over.{v₁, u₁} I
A : C
v : A ⟶ Functor.obj (sections I) X
v₂ : A ⟶ Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) X.left :=
  v ≫ ChosenPullbacksAlong.fst (Functor.map (ihom I) X.hom) (curryRightUnitorHom I)
comm : toUnit A ≫ curry (Iso.hom (ρ_ I)) = v₂ ≫ Functor.map (ihom I) X.hom
w' :
  Eq.{v₁ + 1} (α := tensorObj.{v₁, u₁} (𝒞 := inst) I A ⟶ I) (I ◁ toUnit A ≫ Iso.hom (ρ_ I))
    ((Equiv.symm (Adjunction.homEquiv (ihom.adjunction I) A X.left) : (a : A ⟶ (ihom I).obj X.left) → I ⊗ A ⟶ X.left)
        v₂ ≫
      X.hom)
rw :
  Eq.{v₁ + 1} (α := tensorObj.{v₁, u₁} (𝒞 := inst) A I ⟶ I)
    (Iso.hom (β_ A I) ≫
      (Equiv.symm (Adjunction.homEquiv (ihom.adjunction I) A X.left) : (a : A ⟶ (ihom I).obj X.left) → I ⊗ A ⟶ X.left)
          v₂ ≫
        X.hom)
    (SemiCartesianMonoidalCategory.snd A I)
rw₁ :
  Eq.{v₁ + 1} (α := tensorObj.{v₁, u₁} (𝒞 := inst) A I ⟶ I) (Iso.hom (β_ A I) ≫ I ◁ toUnit A ≫ Iso.hom (ρ_ I))
    (SemiCartesianMonoidalCategory.snd A I)
rw₂ :
  Eq.{v₁ + 1} (α := tensorObj.{v₁, u₁} (𝒞 := inst) A I ⟶ I) (Iso.hom (β_ A I) ≫ SemiCartesianMonoidalCategory.fst I A)
    (SemiCartesianMonoidalCategory.snd A I)
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) A I ⟶ I)
    ((Iso.hom (β_ A I) ≫
        (Equiv.symm (Adjunction.homEquiv (ihom.adjunction I) A X.left) : (a : A ⟶ (ihom I).obj X.left) → I ⊗ A ⟶ X.left)
          v₂) ≫
      X.hom)
    (SemiCartesianMonoidalCategory.snd A I)
have rw₃ : Eq.{v₁ + 1} (α := tensorObj.{v₁, u₁} (𝒞 := inst) A I ⟶ I) (SemiCartesianMonoidalCategory.snd A I)
  (SemiCartesianMonoidalCategory.snd A I)
Tactic: rw [Category.assoc, ← w', whiskerLeft_toUnit_comp_rightUnitor_hom, braiding_hom_fst]
-- Lean 4 proof step from Mathlib.CategoryTheory.LocallyCartesianClosed.Sections (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : CartesianMonoidalCategory.{v₁, u₁} C
I : C
inst_2 : Closed.{v₁, u₁} I
inst_3 :
  ChosenPullbacksAlong.{v₁, u₁} (Y := tensorUnit.{v₁, u₁} C (𝒞 := inst)) (X := Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) I)
    (curryRightUnitorHom I)
inst_4 : BraidedCategory.{v₁, u₁} C
X : Over.{v₁, u₁} I
A : C
v : A ⟶ Functor.obj (sections I) X
⊢ sectionsCurry (sectionsUncurry v) = v
have dsimp : ChosenPullbacksAlong.lift
    (curry
      (Iso.hom (β_ I A) ≫
        Iso.hom (β_ A I) ≫ uncurry (v ≫ ChosenPullbacksAlong.fst (Functor.map (ihom I) X.hom) (curryRightUnitorHom I))))
    (toUnit A) sorry =
  v
Tactic: dsimp [sectionsCurry, sectionsUncurry]
-- Lean 4 proof step from Mathlib.CategoryTheory.LocallyCartesianClosed.Sections (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : CartesianMonoidalCategory.{v₁, u₁} C
I : C
inst_2 : Closed.{v₁, u₁} I
inst_3 :
  ChosenPullbacksAlong.{v₁, u₁} (Y := tensorUnit.{v₁, u₁} C (𝒞 := inst)) (X := Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) I)
    (curryRightUnitorHom I)
inst_4 : BraidedCategory.{v₁, u₁} C
X : Over.{v₁, u₁} I
A : C
u : Functor.obj (toOver I) A ⟶ X
⊢ sectionsUncurry (sectionsCurry u) = u
have dsimp : homMk
    (Iso.hom (β_ A I) ≫
      uncurry
        (CategoryStruct.comp (Y :=
          pullbackObj (Y := Functor.obj.{v₁, v₁, u₁, u₁} (ihom I) X.left) (Functor.map (ihom I) X.hom)
            (curryRightUnitorHom I))
          (ChosenPullbacksAlong.lift (curry (Iso.hom (β_ I A) ≫ Hom.left u)) (toUnit A) sorry)
          (ChosenPullbacksAlong.fst (Functor.map (ihom I) X.hom) (curryRightUnitorHom I))))
    sorry =
  u
Tactic: dsimp [sectionsCurry, sectionsUncurry]