awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Adhesive.Basic
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
W X Y Z_1 : C
f_1 : W ⟶ X
g_1 : W ⟶ Y
h : X ⟶ Z_1
i : Y ⟶ Z_1
inst_2 : Adhesive.{v, u} C
Z A B : C
a : A ⟶ Z
b : B ⟶ Z
inst_3 : Mono a
inst_4 : Mono b
K : C
f g : K ⟶ pushout (X := pullback a b) (fst a b) (snd a b)
w : f ≫ pushout.desc a b ⋯ = g ≫ pushout.desc a b ⋯
u : A ⟶ pushout (X := pullback a b) (fst a b) (snd a b) := inl (fst a b) (snd a b)
v : B ⟶ pushout (X := pullback a b) (fst a b) (snd a b) := inr (fst a b) (snd a b)
this_1 : Mono u := ⋯
this_2 : Mono v := ⋯
sq_f_u : IsPullback (P := pullback f u) (fst f u) (snd f u) f u := ⋯
sq_f_v : IsPullback (P := pullback f v) (fst f v) (snd f v) f v := ⋯
sq_g_u : IsPullback (P := pullback g u) (fst g u) (snd g u) g u := ⋯
sq_g_v : IsPullback (P := pullback g v) (fst g v) (snd g v) g v := ⋯
l₁ : pullback f u ⟶ K := fst f u
f₁ : pullback f u ⟶ A := snd f u
l₂ : pullback f v ⟶ K := fst f v
f₂ : pullback f v ⟶ B := snd f v
m₁ : pullback g u ⟶ K := fst g u
g₁ : pullback g u ⟶ A := snd g u
m₂ : pullback g v ⟶ K := fst g v
g₂ : pullback g v ⟶ B := snd g v
w_1 : C
f' : w_1 ⟶ pullback f u
w_2 : w_1 ⟶ pullback f v
w_3 : w_1 ⟶ pullback a b
p₁ : IsPullback f' w_3 (snd f u) (fst a b)
left : IsPullback w_2 w_3 (snd f v) (snd a b)
h₁ : IsPushout f' w_2 (fst f u) (fst f v)
this : Mono f' := ⋯
sq₁₁ : IsPullback (P := pullback m₁ l₁) (fst m₁ l₁) (snd m₁ l₁) m₁ l₁ := ⋯
sq₁₂ : IsPullback (P := pullback m₁ l₂) (fst m₁ l₂) (snd m₁ l₂) m₁ l₂ := ⋯
h'_w : Eq (α := pullback m₁ l₁ ⟶ A) (snd m₁ l₁ ≫ snd f u) (fst m₁ l₁ ≫ snd g u)
⊢ Eq (α := pullback m₁ l₁ ⟶ pushout (X := pullback a b) (fst a b) (snd a b)) ((snd m₁ l₁ ≫ snd f u) ≫ u)
    ((fst m₁ l₁ ≫ snd g u) ≫ u)

Tactic:
refine ?_ =≫ u
-- Module: Mathlib.CategoryTheory.Adhesive.Basic
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
W X Y Z_1 : C
f_1 : W ⟶ X
g_1 : W ⟶ Y
h : X ⟶ Z_1
i : Y ⟶ Z_1
inst_2 : Adhesive.{v, u} C
Z A B : C
a : A ⟶ Z
b : B ⟶ Z
inst_3 : Mono a
inst_4 : Mono b
K : C
f g : K ⟶ pushout (X := pullback a b) (fst a b) (snd a b)
w : f ≫ pushout.desc a b ⋯ = g ≫ pushout.desc a b ⋯
u : A ⟶ pushout (X := pullback a b) (fst a b) (snd a b) := inl (fst a b) (snd a b)
v : B ⟶ pushout (X := pullback a b) (fst a b) (snd a b) := inr (fst a b) (snd a b)
this_1 : Mono u := ⋯
this_2 : Mono v := ⋯
sq_f_u : IsPullback (P := pullback f u) (fst f u) (snd f u) f u := ⋯
sq_f_v : IsPullback (P := pullback f v) (fst f v) (snd f v) f v := ⋯
sq_g_u : IsPullback (P := pullback g u) (fst g u) (snd g u) g u := ⋯
sq_g_v : IsPullback (P := pullback g v) (fst g v) (snd g v) g v := ⋯
l₁ : pullback f u ⟶ K := fst f u
f₁ : pullback f u ⟶ A := snd f u
l₂ : pullback f v ⟶ K := fst f v
f₂ : pullback f v ⟶ B := snd f v
m₁ : pullback g u ⟶ K := fst g u
g₁ : pullback g u ⟶ A := snd g u
m₂ : pullback g v ⟶ K := fst g v
g₂ : pullback g v ⟶ B := snd g v
w_1 : C
f' : w_1 ⟶ pullback f u
w_2 : w_1 ⟶ pullback f v
w_3 : w_1 ⟶ pullback a b
p₁ : IsPullback f' w_3 (snd f u) (fst a b)
left : IsPullback w_2 w_3 (snd f v) (snd a b)
h₁ : IsPushout f' w_2 (fst f u) (fst f v)
this : Mono f' := ⋯
sq₁₁ : IsPullback (P := pullback m₁ l₁) (fst m₁ l₁) (snd m₁ l₁) m₁ l₁ := ⋯
sq₁₂ : IsPullback (P := pullback m₁ l₂) (fst m₁ l₂) (snd m₁ l₂) m₁ l₂ := ⋯
rwa rwa₁ rwa₂ rwa₃ : Mono a
⊢ Mono (u ≫ pushout.desc a b ⋯)

Tactic:
rwa [pushout.inl_desc]

Premises used: CategoryTheory.Limits.pushout.inl_desc
-- Module: Mathlib.CategoryTheory.Adhesive.Basic
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
W X Y Z_1 : C
f_1 : W ⟶ X
g_1 : W ⟶ Y
h : X ⟶ Z_1
i : Y ⟶ Z_1
inst_2 : Adhesive.{v, u} C
Z A B : C
a : A ⟶ Z
b : B ⟶ Z
inst_3 : Mono a
inst_4 : Mono b
K : C
f g : K ⟶ pushout (X := pullback a b) (fst a b) (snd a b)
w : f ≫ pushout.desc a b ⋯ = g ≫ pushout.desc a b ⋯
u : A ⟶ pushout (X := pullback a b) (fst a b) (snd a b) := inl (fst a b) (snd a b)
v : B ⟶ pushout (X := pullback a b) (fst a b) (snd a b) := inr (fst a b) (snd a b)
this_1 : Mono u := ⋯
this_2 : Mono v := ⋯
sq_f_u : IsPullback (P := pullback f u) (fst f u) (snd f u) f u := ⋯
sq_f_v : IsPullback (P := pullback f v) (fst f v) (snd f v) f v := ⋯
sq_g_u : IsPullback (P := pullback g u) (fst g u) (snd g u) g u := ⋯
sq_g_v : IsPullback (P := pullback g v) (fst g v) (snd g v) g v := ⋯
l₁ : pullback f u ⟶ K := fst f u
f₁ : pullback f u ⟶ A := snd f u
l₂ : pullback f v ⟶ K := fst f v
f₂ : pullback f v ⟶ B := snd f v
m₁ : pullback g u ⟶ K := fst g u
g₁ : pullback g u ⟶ A := snd g u
m₂ : pullback g v ⟶ K := fst g v
g₂ : pullback g v ⟶ B := snd g v
w_1 : C
f' : w_1 ⟶ pullback f u
w_2 : w_1 ⟶ pullback f v
w_3 : w_1 ⟶ pullback a b
p₁ : IsPullback f' w_3 (snd f u) (fst a b)
left : IsPullback w_2 w_3 (snd f v) (snd a b)
h₁ : IsPushout f' w_2 (fst f u) (fst f v)
this_3 : Mono f' := ⋯
sq₁₁ : IsPullback (P := pullback m₁ l₁) (fst m₁ l₁) (snd m₁ l₁) m₁ l₁ := ⋯
sq₁₂ : IsPullback (P := pullback m₁ l₂) (fst m₁ l₂) (snd m₁ l₂) m₁ l₂ := ⋯
this : Mono (u ≫ pushout.desc a b ⋯) := ⋯
h'_w :
  Eq (α := pullback m₁ l₁ ⟶ Z) ((snd m₁ l₁ ≫ snd f u) ≫ u ≫ pushout.desc a b ⋯)
    ((fst m₁ l₁ ≫ snd g u) ≫ u ≫ pushout.desc a b ⋯)
h'_w₁ :
  Eq (α := pullback m₁ l₁ ⟶ Z) (snd m₁ l₁ ≫ snd f u ≫ u ≫ pushout.desc a b ⋯)
    ((fst m₁ l₁ ≫ snd g u) ≫ u ≫ pushout.desc a b ⋯)
h'_w₂ :
  Eq (α := pullback m₁ l₁ ⟶ Z) (snd m₁ l₁ ≫ fst f u ≫ f ≫ pushout.desc a b ⋯)
    ((fst m₁ l₁ ≫ snd g u) ≫ u ≫ pushout.desc a b ⋯)
h'_w₃ :
  Eq (α := pullback m₁ l₁ ⟶ Z) (snd m₁ l₁ ≫ fst f u ≫ g ≫ pushout.desc a b ⋯)
    ((fst m₁ l₁ ≫ snd g u) ≫ u ≫ pushout.desc a b ⋯)
h'_w₄ :
  Eq (α := pullback m₁ l₁ ⟶ Z) (fst m₁ l₁ ≫ m₁ ≫ g ≫ pushout.desc a b ⋯)
    ((fst m₁ l₁ ≫ snd g u) ≫ u ≫ pushout.desc a b ⋯)
h'_w₅ :
  Eq (α := pullback m₁ l₁ ⟶ Z) (fst m₁ l₁ ≫ m₁ ≫ g ≫ pushout.desc a b ⋯) (fst m₁ l₁ ≫ snd g u ≫ u ≫ pushout.desc a b ⋯)
h'_w₆ h'_w₇ :
  Eq (α := pullback m₁ l₁ ⟶ Z) (fst m₁ l₁ ≫ m₁ ≫ g ≫ pushout.desc a b ⋯) (fst m₁ l₁ ≫ fst g u ≫ g ≫ pushout.desc a b ⋯)
⊢ Eq (α := pullback m₁ l₁ ⟶ A) (snd m₁ l₁ ≫ snd f u) (fst m₁ l₁ ≫ snd g u)

Tactic:
rw [← cancel_mono (u ≫ pushout.desc a b pullback.condition), Category.assoc,
          ← sq_f_u.w_assoc, w, ← pullback.condition_assoc, Category.assoc, ← sq_g_u.w_assoc]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.cancel_mono, CategoryTheory.Limits.pullback.condition_assoc, CategoryTheory.Limits.pushout.desc, CategoryTheory.CategoryStruct.comp, CategoryTheory.IsPullback.toCommSq, CategoryTheory.CommSq.w_assoc, CategoryTheory.Limits.pullback.condition
-- Module: Mathlib.CategoryTheory.Adhesive.Basic
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
W X Y Z_1 : C
f_1 : W ⟶ X
g_1 : W ⟶ Y
h : X ⟶ Z_1
i : Y ⟶ Z_1
inst_2 : Adhesive.{v, u} C
Z A B : C
a : A ⟶ Z
b : B ⟶ Z
inst_3 : Mono a
inst_4 : Mono b
K : C
f g : K ⟶ pushout (X := pullback a b) (fst a b) (snd a b)
w : f ≫ pushout.desc a b ⋯ = g ≫ pushout.desc a b ⋯
u : A ⟶ pushout (X := pullback a b) (fst a b) (snd a b) := inl (fst a b) (snd a b)
v : B ⟶ pushout (X := pullback a b) (fst a b) (snd a b) := inr (fst a b) (snd a b)
this_1 : Mono u := ⋯
this_2 : Mono v := ⋯
sq_f_u : IsPullback (P := pullback f u) (fst f u) (snd f u) f u := ⋯
sq_f_v : IsPullback (P := pullback f v) (fst f v) (snd f v) f v := ⋯
sq_g_u : IsPullback (P := pullback g u) (fst g u) (snd g u) g u := ⋯
sq_g_v : IsPullback (P := pullback g v) (fst g v) (snd g v) g v := ⋯
l₁ : pullback f u ⟶ K := fst f u
f₁ : pullback f u ⟶ A := snd f u
l₂ : pullback f v ⟶ K := fst f v
f₂ : pullback f v ⟶ B := snd f v
m₁ : pullback g u ⟶ K := fst g u
g₁ : pullback g u ⟶ A := snd g u
m₂ : pullback g v ⟶ K := fst g v
g₂ : pullback g v ⟶ B := snd g v
w_1 : C
f' : w_1 ⟶ pullback f u
w_2 : w_1 ⟶ pullback f v
w_3 : w_1 ⟶ pullback a b
p₁ : IsPullback f' w_3 (snd f u) (fst a b)
left : IsPullback w_2 w_3 (snd f v) (snd a b)
h₁ : IsPushout f' w_2 (fst f u) (fst f v)
this : Mono f' := ⋯
sq₁₁ : IsPullback (P := pullback m₁ l₁) (fst m₁ l₁) (snd m₁ l₁) m₁ l₁ := ⋯
sq₁₂ : IsPullback (P := pullback m₁ l₂) (fst m₁ l₂) (snd m₁ l₂) m₁ l₂ := ⋯
rw :
  Eq (α := pullback m₁ l₂ ⟶ Z)
    ((fst m₁ l₂ ≫ g₁) ≫
      CategoryStruct.comp (Y := pushout (X := pullback a b) (fst a b) (snd a b)) (inl (fst a b) (snd a b))
        (pushout.desc a b ⋯))
    ((snd m₁ l₂ ≫ f₂) ≫ b)
rw₁ :
  Eq (α := pullback m₁ l₂ ⟶ Z)
    (fst m₁ l₂ ≫
      g₁ ≫
        CategoryStruct.comp (Y := pushout (X := pullback a b) (fst a b) (snd a b)) (inl (fst a b) (snd a b))
          (pushout.desc a b ⋯))
    ((snd m₁ l₂ ≫ f₂) ≫ b)
rw₂ : Eq (α := pullback m₁ l₂ ⟶ Z) (fst m₁ l₂ ≫ fst g u ≫ g ≫ pushout.desc a b ⋯) ((snd m₁ l₂ ≫ f₂) ≫ b)
rw₃ : Eq (α := pullback m₁ l₂ ⟶ Z) (snd m₁ l₂ ≫ l₂ ≫ g ≫ pushout.desc a b ⋯) ((snd m₁ l₂ ≫ f₂) ≫ b)
rw₄ : Eq (α := pullback m₁ l₂ ⟶ Z) (snd m₁ l₂ ≫ l₂ ≫ f ≫ pushout.desc a b ⋯) ((snd m₁ l₂ ≫ f₂) ≫ b)
rw₅ : Eq (α := pullback m₁ l₂ ⟶ Z) (snd m₁ l₂ ≫ l₂ ≫ f ≫ pushout.desc a b ⋯) (snd m₁ l₂ ≫ f₂ ≫ b)
rw₆ : Eq (α := pullback m₁ l₂ ⟶ Z) (snd m₁ l₂ ≫ snd f v ≫ v ≫ pushout.desc a b ⋯) (snd m₁ l₂ ≫ f₂ ≫ b)
rw₇ rw₈ : Eq (α := pullback m₁ l₂ ⟶ Z) (snd m₁ l₂ ≫ snd f v ≫ b) (snd m₁ l₂ ≫ f₂ ≫ b)
⊢ Eq (α := pullback m₁ l₂ ⟶ Z) ((fst m₁ l₂ ≫ g₁) ≫ a) ((snd m₁ l₂ ≫ f₂) ≫ b)

Tactic:
rw [← _ ≫= pushout.inl_desc a b pullback.condition, Category.assoc, ← sq_g_u.w_assoc,
            sq₁₂.w_assoc, ← w, Category.assoc, pullback.condition_assoc, pushout.inr_desc]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Limits.pushout.inr_desc, CategoryTheory.Limits.pullback.condition_assoc, CategoryTheory.IsPullback.toCommSq, CategoryTheory.CommSq.w_assoc, CategoryTheory.Limits.pushout.inl_desc, CategoryTheory.Limits.pullback.condition
-- Module: Mathlib.CategoryTheory.Adhesive.Basic
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
W X Y Z_1 : C
f_1 : W ⟶ X
g_1 : W ⟶ Y
h : X ⟶ Z_1
i : Y ⟶ Z_1
inst_2 : Adhesive.{v, u} C
Z A B : C
a : A ⟶ Z
b : B ⟶ Z
inst_3 : Mono a
inst_4 : Mono b
K : C
f g : K ⟶ pushout (X := pullback a b) (fst a b) (snd a b)
w : f ≫ pushout.desc a b ⋯ = g ≫ pushout.desc a b ⋯
u : A ⟶ pushout (X := pullback a b) (fst a b) (snd a b) := inl (fst a b) (snd a b)
v : B ⟶ pushout (X := pullback a b) (fst a b) (snd a b) := inr (fst a b) (snd a b)
this_1 : Mono u := ⋯
this_2 : Mono v := ⋯
sq_f_u : IsPullback (P := pullback f u) (fst f u) (snd f u) f u := ⋯
sq_f_v : IsPullback (P := pullback f v) (fst f v) (snd f v) f v := ⋯
sq_g_u : IsPullback (P := pullback g u) (fst g u) (snd g u) g u := ⋯
sq_g_v : IsPullback (P := pullback g v) (fst g v) (snd g v) g v := ⋯
l₁ : pullback f u ⟶ K := fst f u
f₁ : pullback f u ⟶ A := snd f u
l₂ : pullback f v ⟶ K := fst f v
f₂ : pullback f v ⟶ B := snd f v
m₁ : pullback g u ⟶ K := fst g u
g₁ : pullback g u ⟶ A := snd g u
m₂ : pullback g v ⟶ K := fst g v
g₂ : pullback g v ⟶ B := snd g v
w_1 : C
f' : w_1 ⟶ pullback f u
w_2 : w_1 ⟶ pullback f v
w_3 : w_1 ⟶ pullback a b
p₁ : IsPullback f' w_3 (snd f u) (fst a b)
left : IsPullback w_2 w_3 (snd f v) (snd a b)
h₁ : IsPushout f' w_2 (fst f u) (fst f v)
this : Mono f' := ⋯
sq₁₁ : IsPullback (P := pullback m₁ l₁) (fst m₁ l₁) (snd m₁ l₁) m₁ l₁ := ⋯
sq₁₂ : IsPullback (P := pullback m₁ l₂) (fst m₁ l₂) (snd m₁ l₂) m₁ l₂ := ⋯
⊢ Eq (α := pullback m₁ l₂ ⟶ Z) ((fst m₁ l₂ ≫ g₁) ≫ a) ((snd m₁ l₂ ≫ f₂) ≫ b)

Tactic:
have : (pullback.fst m₁ l₂ ≫ g₁) ≫ a = (pullback.snd m₁ l₂ ≫ f₂) ≫ b := by
          rw [← _ ≫= pushout.inl_desc a b pullback.condition, Category.assoc, ← sq_g_u.w_assoc,
            sq₁₂.w_assoc, ← w, Category.assoc, pullback.condition_assoc, pushout.inr_desc]
-- Module: Mathlib.CategoryTheory.Adhesive.Basic
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
W X Y Z_1 : C
f_1 : W ⟶ X
g_1 : W ⟶ Y
h : X ⟶ Z_1
i : Y ⟶ Z_1
inst_2 : Adhesive.{v, u} C
Z A B : C
a : A ⟶ Z
b : B ⟶ Z
inst_3 : Mono a
inst_4 : Mono b
K : C
f g : K ⟶ pushout (X := pullback a b) (fst a b) (snd a b)
w : f ≫ pushout.desc a b ⋯ = g ≫ pushout.desc a b ⋯
u : A ⟶ pushout (X := pullback a b) (fst a b) (snd a b) := inl (fst a b) (snd a b)
v : B ⟶ pushout (X := pullback a b) (fst a b) (snd a b) := inr (fst a b) (snd a b)
this_1 : Mono u := ⋯
this_2 : Mono v := ⋯
sq_f_u : IsPullback (P := pullback f u) (fst f u) (snd f u) f u := ⋯
sq_f_v : IsPullback (P := pullback f v) (fst f v) (snd f v) f v := ⋯
sq_g_u : IsPullback (P := pullback g u) (fst g u) (snd g u) g u := ⋯
sq_g_v : IsPullback (P := pullback g v) (fst g v) (snd g v) g v := ⋯
l₁ : pullback f u ⟶ K := fst f u
f₁ : pullback f u ⟶ A := snd f u
l₂ : pullback f v ⟶ K := fst f v
f₂ : pullback f v ⟶ B := snd f v
m₁ : pullback g u ⟶ K := fst g u
g₁ : pullback g u ⟶ A := snd g u
m₂ : pullback g v ⟶ K := fst g v
g₂ : pullback g v ⟶ B := snd g v
w_1 : C
f' : w_1 ⟶ pullback f u
w_2 : w_1 ⟶ pullback f v
w_3 : w_1 ⟶ pullback a b
p₁ : IsPullback f' w_3 (snd f u) (fst a b)
left : IsPullback w_2 w_3 (snd f v) (snd a b)
h₁ : IsPushout f' w_2 (fst f u) (fst f v)
this : Mono f' := ⋯
sq₂₁ : IsPullback (P := pullback m₂ l₁) (fst m₂ l₁) (snd m₂ l₁) m₂ l₁ := ⋯
sq₂₂ : IsPullback (P := pullback m₂ l₂) (fst m₂ l₂) (snd m₂ l₂) m₂ l₂ := ⋯
i'_w :
  Eq (α := pullback m₂ l₁ ⟶ pushout (X := pullback a b) (fst a b) (snd a b)) (fst m₂ l₁ ≫ fst g v ≫ f)
    (fst m₂ l₁ ≫ fst g v ≫ g)
i'_w₁ :
  Eq (α := pullback m₂ l₂ ⟶ pushout (X := pullback a b) (fst a b) (snd a b)) (fst m₂ l₂ ≫ fst g v ≫ f)
    (fst m₂ l₂ ≫ fst g v ≫ g)
⊢ Eq (α := pullback g v ⟶ pushout (X := pullback a b) (fst a b) (snd a b)) (fst g v ≫ f) (fst g v ≫ g)

Tactic:
apply isPushout_isPullback_isPullback_hom_ext h₁ sq₂₁ sq₂₂

Premises used: CategoryTheory.Adhesive.isPushout_isPullback_isPullback_hom_ext
-- Module: Mathlib.CategoryTheory.Adhesive.Basic
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
W X Y Z_1 : C
f_1 : W ⟶ X
g_1 : W ⟶ Y
h : X ⟶ Z_1
i : Y ⟶ Z_1
inst_2 : Adhesive.{v, u} C
Z A B : C
a : A ⟶ Z
b : B ⟶ Z
inst_3 : Mono a
inst_4 : Mono b
K : C
f g : K ⟶ pushout (X := pullback a b) (fst a b) (snd a b)
w : f ≫ pushout.desc a b ⋯ = g ≫ pushout.desc a b ⋯
u : A ⟶ pushout (X := pullback a b) (fst a b) (snd a b) := inl (fst a b) (snd a b)
v : B ⟶ pushout (X := pullback a b) (fst a b) (snd a b) := inr (fst a b) (snd a b)
this_1 : Mono u := ⋯
this_2 : Mono v := ⋯
sq_f_u : IsPullback (P := pullback f u) (fst f u) (snd f u) f u := ⋯
sq_f_v : IsPullback (P := pullback f v) (fst f v) (snd f v) f v := ⋯
sq_g_u : IsPullback (P := pullback g u) (fst g u) (snd g u) g u := ⋯
sq_g_v : IsPullback (P := pullback g v) (fst g v) (snd g v) g v := ⋯
l₁ : pullback f u ⟶ K := fst f u
f₁ : pullback f u ⟶ A := snd f u
l₂ : pullback f v ⟶ K := fst f v
f₂ : pullback f v ⟶ B := snd f v
m₁ : pullback g u ⟶ K := fst g u
g₁ : pullback g u ⟶ A := snd g u
m₂ : pullback g v ⟶ K := fst g v
g₂ : pullback g v ⟶ B := snd g v
w_1 : C
f' : w_1 ⟶ pullback f u
w_2 : w_1 ⟶ pullback f v
w_3 : w_1 ⟶ pullback a b
p₁ : IsPullback f' w_3 (snd f u) (fst a b)
left : IsPullback w_2 w_3 (snd f v) (snd a b)
h₁ : IsPushout f' w_2 (fst f u) (fst f v)
this : Mono f' := ⋯
sq₂₁ : IsPullback (P := pullback m₂ l₁) (fst m₂ l₁) (snd m₂ l₁) m₂ l₁ := ⋯
sq₂₂ : IsPullback (P := pullback m₂ l₂) (fst m₂ l₂) (snd m₂ l₂) m₂ l₂ := ⋯
rw :
  Eq (α := pullback m₂ l₁ ⟶ Z)
    ((snd m₂ l₁ ≫ f₁) ≫
      CategoryStruct.comp (Y := pushout (X := pullback a b) (fst a b) (snd a b)) (inl (fst a b) (snd a b))
        (pushout.desc a b ⋯))
    ((fst m₂ l₁ ≫ g₂) ≫ b)
rw₁ :
  Eq (α := pullback m₂ l₁ ⟶ Z)
    (snd m₂ l₁ ≫
      f₁ ≫
        CategoryStruct.comp (Y := pushout (X := pullback a b) (fst a b) (snd a b)) (inl (fst a b) (snd a b))
          (pushout.desc a b ⋯))
    ((fst m₂ l₁ ≫ g₂) ≫ b)
rw₂ : Eq (α := pullback m₂ l₁ ⟶ Z) (snd m₂ l₁ ≫ fst f u ≫ f ≫ pushout.desc a b ⋯) ((fst m₂ l₁ ≫ g₂) ≫ b)
rw₃ : Eq (α := pullback m₂ l₁ ⟶ Z) (snd m₂ l₁ ≫ fst f u ≫ g ≫ pushout.desc a b ⋯) ((fst m₂ l₁ ≫ g₂) ≫ b)
rw₄ : Eq (α := pullback m₂ l₁ ⟶ Z) (fst m₂ l₁ ≫ m₂ ≫ g ≫ pushout.desc a b ⋯) ((fst m₂ l₁ ≫ g₂) ≫ b)
rw₅ : Eq (α := pullback m₂ l₁ ⟶ Z) (fst m₂ l₁ ≫ m₂ ≫ g ≫ pushout.desc a b ⋯) (fst m₂ l₁ ≫ g₂ ≫ b)
rw₆ : Eq (α := pullback m₂ l₁ ⟶ Z) (fst m₂ l₁ ≫ snd g v ≫ v ≫ pushout.desc a b ⋯) (fst m₂ l₁ ≫ g₂ ≫ b)
rw₇ rw₈ : Eq (α := pullback m₂ l₁ ⟶ Z) (fst m₂ l₁ ≫ snd g v ≫ b) (fst m₂ l₁ ≫ g₂ ≫ b)
⊢ Eq (α := pullback m₂ l₁ ⟶ Z) ((snd m₂ l₁ ≫ f₁) ≫ a) ((fst m₂ l₁ ≫ g₂) ≫ b)

Tactic:
rw [← _ ≫= pushout.inl_desc a b pullback.condition, Category.assoc, ← sq_f_u.w_assoc,
            w, ← sq₂₁.w_assoc, Category.assoc, sq_g_v.w_assoc, pushout.inr_desc]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Limits.pushout.inr_desc, CategoryTheory.IsPullback.toCommSq, CategoryTheory.CommSq.w_assoc, CategoryTheory.Limits.pushout.inl_desc, CategoryTheory.Limits.pullback.condition
-- Module: Mathlib.CategoryTheory.Adhesive.Basic
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
W X Y Z_1 : C
f_1 : W ⟶ X
g_1 : W ⟶ Y
h : X ⟶ Z_1
i : Y ⟶ Z_1
inst_2 : Adhesive.{v, u} C
Z A B : C
a : A ⟶ Z
b : B ⟶ Z
inst_3 : Mono a
inst_4 : Mono b
K : C
f g : K ⟶ pushout (X := pullback a b) (fst a b) (snd a b)
w : f ≫ pushout.desc a b ⋯ = g ≫ pushout.desc a b ⋯
u : A ⟶ pushout (X := pullback a b) (fst a b) (snd a b) := inl (fst a b) (snd a b)
v : B ⟶ pushout (X := pullback a b) (fst a b) (snd a b) := inr (fst a b) (snd a b)
this_1 : Mono u := ⋯
this_2 : Mono v := ⋯
sq_f_u : IsPullback (P := pullback f u) (fst f u) (snd f u) f u := ⋯
sq_f_v : IsPullback (P := pullback f v) (fst f v) (snd f v) f v := ⋯
sq_g_u : IsPullback (P := pullback g u) (fst g u) (snd g u) g u := ⋯
sq_g_v : IsPullback (P := pullback g v) (fst g v) (snd g v) g v := ⋯
l₁ : pullback f u ⟶ K := fst f u
f₁ : pullback f u ⟶ A := snd f u
l₂ : pullback f v ⟶ K := fst f v
f₂ : pullback f v ⟶ B := snd f v
m₁ : pullback g u ⟶ K := fst g u
g₁ : pullback g u ⟶ A := snd g u
m₂ : pullback g v ⟶ K := fst g v
g₂ : pullback g v ⟶ B := snd g v
w_1 : C
f' : w_1 ⟶ pullback f u
w_2 : w_1 ⟶ pullback f v
w_3 : w_1 ⟶ pullback a b
p₁ : IsPullback f' w_3 (snd f u) (fst a b)
left : IsPullback w_2 w_3 (snd f v) (snd a b)
h₁ : IsPushout f' w_2 (fst f u) (fst f v)
this : Mono f' := ⋯
sq₂₁ : IsPullback (P := pullback m₂ l₁) (fst m₂ l₁) (snd m₂ l₁) m₂ l₁ := ⋯
sq₂₂ : IsPullback (P := pullback m₂ l₂) (fst m₂ l₂) (snd m₂ l₂) m₂ l₂ := ⋯
⊢ Eq (α := pullback m₂ l₁ ⟶ Z) ((snd m₂ l₁ ≫ f₁) ≫ a) ((fst m₂ l₁ ≫ g₂) ≫ b)

Tactic:
have : (pullback.snd m₂ l₁ ≫ f₁) ≫ a = (pullback.fst m₂ l₁ ≫ g₂) ≫ b := by
          rw [← _ ≫= pushout.inl_desc a b pullback.condition, Category.assoc, ← sq_f_u.w_assoc,
            w, ← sq₂₁.w_assoc, Category.assoc, sq_g_v.w_assoc, pushout.inr_desc]
-- Module: Mathlib.CategoryTheory.Adhesive.Basic
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
W X Y Z_1 : C
f_1 : W ⟶ X
g_1 : W ⟶ Y
h : X ⟶ Z_1
i : Y ⟶ Z_1
inst_2 : Adhesive.{v, u} C
Z A B : C
a : A ⟶ Z
b : B ⟶ Z
inst_3 : Mono a
inst_4 : Mono b
K : C
f g : K ⟶ pushout (X := pullback a b) (fst a b) (snd a b)
w : f ≫ pushout.desc a b ⋯ = g ≫ pushout.desc a b ⋯
u : A ⟶ pushout (X := pullback a b) (fst a b) (snd a b) := inl (fst a b) (snd a b)
v : B ⟶ pushout (X := pullback a b) (fst a b) (snd a b) := inr (fst a b) (snd a b)
this_1 : Mono u := ⋯
this_2 : Mono v := ⋯
sq_f_u : IsPullback (P := pullback f u) (fst f u) (snd f u) f u := ⋯
sq_f_v : IsPullback (P := pullback f v) (fst f v) (snd f v) f v := ⋯
sq_g_u : IsPullback (P := pullback g u) (fst g u) (snd g u) g u := ⋯
sq_g_v : IsPullback (P := pullback g v) (fst g v) (snd g v) g v := ⋯
l₁ : pullback f u ⟶ K := fst f u
f₁ : pullback f u ⟶ A := snd f u
l₂ : pullback f v ⟶ K := fst f v
f₂ : pullback f v ⟶ B := snd f v
m₁ : pullback g u ⟶ K := fst g u
g₁ : pullback g u ⟶ A := snd g u
m₂ : pullback g v ⟶ K := fst g v
g₂ : pullback g v ⟶ B := snd g v
w_1 : C
f' : w_1 ⟶ pullback f u
w_2 : w_1 ⟶ pullback f v
w_3 : w_1 ⟶ pullback a b
p₁ : IsPullback f' w_3 (snd f u) (fst a b)
left : IsPullback w_2 w_3 (snd f v) (snd a b)
h₁ : IsPushout f' w_2 (fst f u) (fst f v)
this : Mono f' := ⋯
sq₂₁ : IsPullback (P := pullback m₂ l₁) (fst m₂ l₁) (snd m₂ l₁) m₂ l₁ := ⋯
sq₂₂ : IsPullback (P := pullback m₂ l₂) (fst m₂ l₂) (snd m₂ l₂) m₂ l₂ := ⋯
i'_w :
  Eq (α := pullback m₂ l₂ ⟶ pushout (X := pullback a b) (fst a b) (snd a b)) ((snd m₂ l₂ ≫ snd f v) ≫ v)
    ((fst m₂ l₂ ≫ snd g v) ≫ v)
⊢ Eq (α := pullback m₂ l₂ ⟶ pushout (X := pullback a b) (fst a b) (snd a b)) (fst m₂ l₂ ≫ fst g v ≫ f)
    (fst m₂ l₂ ≫ fst g v ≫ g)

Tactic:
rw [sq₂₂.w_assoc, sq_f_v.w, sq_g_v.w, ← Category.assoc, ← Category.assoc]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.IsPullback.toCommSq, CategoryTheory.CommSq.w, CategoryTheory.CommSq.w_assoc
-- Module: Mathlib.CategoryTheory.Adhesive.Basic
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
W X Y Z_1 : C
f_1 : W ⟶ X
g_1 : W ⟶ Y
h : X ⟶ Z_1
i : Y ⟶ Z_1
inst_2 : Adhesive.{v, u} C
Z A B : C
a : A ⟶ Z
b : B ⟶ Z
inst_3 : Mono a
inst_4 : Mono b
K : C
f g : K ⟶ pushout (X := pullback a b) (fst a b) (snd a b)
w : f ≫ pushout.desc a b ⋯ = g ≫ pushout.desc a b ⋯
u : A ⟶ pushout (X := pullback a b) (fst a b) (snd a b) := inl (fst a b) (snd a b)
v : B ⟶ pushout (X := pullback a b) (fst a b) (snd a b) := inr (fst a b) (snd a b)
this_1 : Mono u := ⋯
this_2 : Mono v := ⋯
sq_f_u : IsPullback (P := pullback f u) (fst f u) (snd f u) f u := ⋯
sq_f_v : IsPullback (P := pullback f v) (fst f v) (snd f v) f v := ⋯
sq_g_u : IsPullback (P := pullback g u) (fst g u) (snd g u) g u := ⋯
sq_g_v : IsPullback (P := pullback g v) (fst g v) (snd g v) g v := ⋯
l₁ : pullback f u ⟶ K := fst f u
f₁ : pullback f u ⟶ A := snd f u
l₂ : pullback f v ⟶ K := fst f v
f₂ : pullback f v ⟶ B := snd f v
m₁ : pullback g u ⟶ K := fst g u
g₁ : pullback g u ⟶ A := snd g u
m₂ : pullback g v ⟶ K := fst g v
g₂ : pullback g v ⟶ B := snd g v
w_1 : C
f' : w_1 ⟶ pullback f u
w_2 : w_1 ⟶ pullback f v
w_3 : w_1 ⟶ pullback a b
p₁ : IsPullback f' w_3 (snd f u) (fst a b)
left : IsPullback w_2 w_3 (snd f v) (snd a b)
h₁ : IsPushout f' w_2 (fst f u) (fst f v)
this : Mono f' := ⋯
sq₂₁ : IsPullback (P := pullback m₂ l₁) (fst m₂ l₁) (snd m₂ l₁) m₂ l₁ := ⋯
sq₂₂ : IsPullback (P := pullback m₂ l₂) (fst m₂ l₂) (snd m₂ l₂) m₂ l₂ := ⋯
⊢ Eq (α := pullback m₂ l₂ ⟶ pushout (X := pullback a b) (fst a b) (snd a b)) ((snd m₂ l₂ ≫ snd f v) ≫ v)
    ((fst m₂ l₂ ≫ snd g v) ≫ v)

Tactic:
?_ =≫ v

Premises used: CategoryTheory.eq_whisker
-- Module: Mathlib.CategoryTheory.Adhesive.Basic
Goal:
J : Type v'
inst : Category.{u', v'} J
C : Type u
inst_1 : Category.{v, u} C
W X Y Z_1 : C
f_1 : W ⟶ X
g_1 : W ⟶ Y
h : X ⟶ Z_1
i : Y ⟶ Z_1
inst_2 : Adhesive.{v, u} C
Z A B : C
a : A ⟶ Z
b : B ⟶ Z
inst_3 : Mono a
inst_4 : Mono b
K : C
f g : K ⟶ pushout (X := pullback a b) (fst a b) (snd a b)
w : f ≫ pushout.desc a b ⋯ = g ≫ pushout.desc a b ⋯
u : A ⟶ pushout (X := pullback a b) (fst a b) (snd a b) := inl (fst a b) (snd a b)
v : B ⟶ pushout (X := pullback a b) (fst a b) (snd a b) := inr (fst a b) (snd a b)
this_1 : Mono u := ⋯
this_2 : Mono v := ⋯
sq_f_u : IsPullback (P := pullback f u) (fst f u) (snd f u) f u := ⋯
sq_f_v : IsPullback (P := pullback f v) (fst f v) (snd f v) f v := ⋯
sq_g_u : IsPullback (P := pullback g u) (fst g u) (snd g u) g u := ⋯
sq_g_v : IsPullback (P := pullback g v) (fst g v) (snd g v) g v := ⋯
l₁ : pullback f u ⟶ K := fst f u
f₁ : pullback f u ⟶ A := snd f u
l₂ : pullback f v ⟶ K := fst f v
f₂ : pullback f v ⟶ B := snd f v
m₁ : pullback g u ⟶ K := fst g u
g₁ : pullback g u ⟶ A := snd g u
m₂ : pullback g v ⟶ K := fst g v
g₂ : pullback g v ⟶ B := snd g v
w_1 : C
f' : w_1 ⟶ pullback f u
w_2 : w_1 ⟶ pullback f v
w_3 : w_1 ⟶ pullback a b
p₁ : IsPullback f' w_3 (snd f u) (fst a b)
left : IsPullback w_2 w_3 (snd f v) (snd a b)
h₁ : IsPushout f' w_2 (fst f u) (fst f v)
this : Mono f' := ⋯
sq₂₁ : IsPullback (P := pullback m₂ l₁) (fst m₂ l₁) (snd m₂ l₁) m₂ l₁ := ⋯
sq₂₂ : IsPullback (P := pullback m₂ l₂) (fst m₂ l₂) (snd m₂ l₂) m₂ l₂ := ⋯
i'_w : Eq (α := pullback m₂ l₂ ⟶ B) (snd m₂ l₂ ≫ snd f v) (fst m₂ l₂ ≫ snd g v)
⊢ Eq (α := pullback m₂ l₂ ⟶ pushout (X := pullback a b) (fst a b) (snd a b)) ((snd m₂ l₂ ≫ snd f v) ≫ v)
    ((fst m₂ l₂ ≫ snd g v) ≫ v)

Tactic:
refine ?_ =≫ v