l3lab/ntp-mathlib huggingface dataset:
/-- `CompatiblePreserving` functors indeed preserve compatible families. -/
theorem Presieve.FamilyOfElements.Compatible.functorPushforward :
    (x.functorPushforward G).Compatible := by
  rintro Z₁ Z₂ W g₁ g₂ f₁' f₂' H₁ H₂ eq
  unfold FamilyOfElements.functorPushforward
  

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
A : Type u₃
inst✝ : Category.{v₃, u₃} A
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology A
G : C ⥤ D
hG : CompatiblePreserving K G
ℱ : SheafOfTypes K
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
h : Compatible x
Z₁ Z₂ W : D
g₁ : W ⟶ Z₁
g₂ : W ⟶ Z₂
f₁' : Z₁ ⟶ G.obj Z
f₂' : Z₂ ⟶ G.obj Z
H₁ : Presieve.functorPushforward G T f₁'
H₂ : Presieve.functorPushforward G T f₂'
eq : g₁ ≫ f₁' = g₂ ≫ f₂'
⊢ ℱ.val.map g₁.op
      (FunctorPushforwardStructure.casesOn (getFunctorPushforwardStructure H₁) fun Z_1 g h h₁ fac =>
        ℱ.val.map h.op (x g h₁)) =
    ℱ.val.map g₂.op
      (FunctorPushforwardStructure.casesOn (getFunctorPushforwardStructure H₂) fun Z_1 g h h₁ fac =>
        ℱ.val.map h.op (x g h₁))

Next tactic:
rcases getFunctorPushforwardStructure H₁ with ⟨X₁, f₁, h₁, hf₁, rfl⟩
/-- `CompatiblePreserving` functors indeed preserve compatible families. -/
theorem Presieve.FamilyOfElements.Compatible.functorPushforward :
    (x.functorPushforward G).Compatible := by
  rintro Z₁ Z₂ W g₁ g₂ f₁' f₂' H₁ H₂ eq
  unfold FamilyOfElements.functorPushforward
  rcases getFunctorPushforwardStructure H₁ with ⟨X₁, f₁, h₁, hf₁, rfl⟩
  

Proof state:
case mk
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
A : Type u₃
inst✝ : Category.{v₃, u₃} A
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology A
G : C ⥤ D
hG : CompatiblePreserving K G
ℱ : SheafOfTypes K
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
h : Compatible x
Z₁ Z₂ W : D
g₁ : W ⟶ Z₁
g₂ : W ⟶ Z₂
f₂' : Z₂ ⟶ G.obj Z
H₂ : Presieve.functorPushforward G T f₂'
X₁ : C
f₁ : X₁ ⟶ Z
h₁ : Z₁ ⟶ G.obj X₁
hf₁ : T f₁
H₁ : Presieve.functorPushforward G T (h₁ ≫ G.map f₁)
eq : g₁ ≫ h₁ ≫ G.map f₁ = g₂ ≫ f₂'
⊢ ℱ.val.map g₁.op
      (FunctorPushforwardStructure.casesOn
        { preobj := X₁, premap := f₁, lift := h₁, cover := hf₁, fac := (_ : h₁ ≫ G.map f₁ = h₁ ≫ G.map f₁) }
        fun Z_1 g h h₁_1 fac => ℱ.val.map h.op (x g h₁_1)) =
    ℱ.val.map g₂.op
      (FunctorPushforwardStructure.casesOn (getFunctorPushforwardStructure H₂) fun Z_1 g h h₁ fac =>
        ℱ.val.map h.op (x g h₁))

Next tactic:
rcases getFunctorPushforwardStructure H₂ with ⟨X₂, f₂, h₂, hf₂, rfl⟩
/-- `CompatiblePreserving` functors indeed preserve compatible families. -/
theorem Presieve.FamilyOfElements.Compatible.functorPushforward :
    (x.functorPushforward G).Compatible := by
  rintro Z₁ Z₂ W g₁ g₂ f₁' f₂' H₁ H₂ eq
  unfold FamilyOfElements.functorPushforward
  rcases getFunctorPushforwardStructure H₁ with ⟨X₁, f₁, h₁, hf₁, rfl⟩
  rcases getFunctorPushforwardStructure H₂ with ⟨X₂, f₂, h₂, hf₂, rfl⟩
  

Proof state:
case mk.mk
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
A : Type u₃
inst✝ : Category.{v₃, u₃} A
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology A
G : C ⥤ D
hG : CompatiblePreserving K G
ℱ : SheafOfTypes K
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
h : Compatible x
Z₁ Z₂ W : D
g₁ : W ⟶ Z₁
g₂ : W ⟶ Z₂
X₁ : C
f₁ : X₁ ⟶ Z
h₁ : Z₁ ⟶ G.obj X₁
hf₁ : T f₁
H₁ : Presieve.functorPushforward G T (h₁ ≫ G.map f₁)
X₂ : C
f₂ : X₂ ⟶ Z
h₂ : Z₂ ⟶ G.obj X₂
hf₂ : T f₂
H₂ : Presieve.functorPushforward G T (h₂ ≫ G.map f₂)
eq : g₁ ≫ h₁ ≫ G.map f₁ = g₂ ≫ h₂ ≫ G.map f₂
⊢ ℱ.val.map g₁.op
      (FunctorPushforwardStructure.casesOn
        { preobj := X₁, premap := f₁, lift := h₁, cover := hf₁, fac := (_ : h₁ ≫ G.map f₁ = h₁ ≫ G.map f₁) }
        fun Z_1 g h h₁_1 fac => ℱ.val.map h.op (x g h₁_1)) =
    ℱ.val.map g₂.op
      (FunctorPushforwardStructure.casesOn
        { preobj := X₂, premap := f₂, lift := h₂, cover := hf₂, fac := (_ : h₂ ≫ G.map f₂ = h₂ ≫ G.map f₂) }
        fun Z_1 g h h₁ fac => ℱ.val.map h.op (x g h₁))

Next tactic:
suffices ℱ.val.map (g₁ ≫ h₁).op (x f₁ hf₁) = ℱ.val.map (g₂ ≫ h₂).op (x f₂ hf₂) by
    simpa using this
/-- `CompatiblePreserving` functors indeed preserve compatible families. -/
theorem Presieve.FamilyOfElements.Compatible.functorPushforward :
    (x.functorPushforward G).Compatible := by
  rintro Z₁ Z₂ W g₁ g₂ f₁' f₂' H₁ H₂ eq
  unfold FamilyOfElements.functorPushforward
  rcases getFunctorPushforwardStructure H₁ with ⟨X₁, f₁, h₁, hf₁, rfl⟩
  rcases getFunctorPushforwardStructure H₂ with ⟨X₂, f₂, h₂, hf₂, rfl⟩
  suffices ℱ.val.map (g₁ ≫ h₁).op (x f₁ hf₁) = ℱ.val.map (g₂ ≫ h₂).op (x f₂ hf₂) by
    

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
A : Type u₃
inst✝ : Category.{v₃, u₃} A
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology A
G : C ⥤ D
hG : CompatiblePreserving K G
ℱ : SheafOfTypes K
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
h : Compatible x
Z₁ Z₂ W : D
g₁ : W ⟶ Z₁
g₂ : W ⟶ Z₂
X₁ : C
f₁ : X₁ ⟶ Z
h₁ : Z₁ ⟶ G.obj X₁
hf₁ : T f₁
H₁ : Presieve.functorPushforward G T (h₁ ≫ G.map f₁)
X₂ : C
f₂ : X₂ ⟶ Z
h₂ : Z₂ ⟶ G.obj X₂
hf₂ : T f₂
H₂ : Presieve.functorPushforward G T (h₂ ≫ G.map f₂)
eq : g₁ ≫ h₁ ≫ G.map f₁ = g₂ ≫ h₂ ≫ G.map f₂
this : ℱ.val.map (g₁ ≫ h₁).op (x f₁ hf₁) = ℱ.val.map (g₂ ≫ h₂).op (x f₂ hf₂)
⊢ ℱ.val.map g₁.op
      (FunctorPushforwardStructure.casesOn
        { preobj := X₁, premap := f₁, lift := h₁, cover := hf₁, fac := (_ : h₁ ≫ G.map f₁ = h₁ ≫ G.map f₁) }
        fun Z_1 g h h₁_1 fac => ℱ.val.map h.op (x g h₁_1)) =
    ℱ.val.map g₂.op
      (FunctorPushforwardStructure.casesOn
        { preobj := X₂, premap := f₂, lift := h₂, cover := hf₂, fac := (_ : h₂ ≫ G.map f₂ = h₂ ≫ G.map f₂) }
        fun Z_1 g h h₁ fac => ℱ.val.map h.op (x g h₁))

Next tactic:
simpa using this
/-- `CompatiblePreserving` functors indeed preserve compatible families. -/
theorem Presieve.FamilyOfElements.Compatible.functorPushforward :
    (x.functorPushforward G).Compatible := by
  rintro Z₁ Z₂ W g₁ g₂ f₁' f₂' H₁ H₂ eq
  unfold FamilyOfElements.functorPushforward
  rcases getFunctorPushforwardStructure H₁ with ⟨X₁, f₁, h₁, hf₁, rfl⟩
  rcases getFunctorPushforwardStructure H₂ with ⟨X₂, f₂, h₂, hf₂, rfl⟩
  suffices ℱ.val.map (g₁ ≫ h₁).op (x f₁ hf₁) = ℱ.val.map (g₂ ≫ h₂).op (x f₂ hf₂) by
    simpa using this
  

Proof state:
case mk.mk
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
A : Type u₃
inst✝ : Category.{v₃, u₃} A
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology A
G : C ⥤ D
hG : CompatiblePreserving K G
ℱ : SheafOfTypes K
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
h : Compatible x
Z₁ Z₂ W : D
g₁ : W ⟶ Z₁
g₂ : W ⟶ Z₂
X₁ : C
f₁ : X₁ ⟶ Z
h₁ : Z₁ ⟶ G.obj X₁
hf₁ : T f₁
H₁ : Presieve.functorPushforward G T (h₁ ≫ G.map f₁)
X₂ : C
f₂ : X₂ ⟶ Z
h₂ : Z₂ ⟶ G.obj X₂
hf₂ : T f₂
H₂ : Presieve.functorPushforward G T (h₂ ≫ G.map f₂)
eq : g₁ ≫ h₁ ≫ G.map f₁ = g₂ ≫ h₂ ≫ G.map f₂
⊢ ℱ.val.map (g₁ ≫ h₁).op (x f₁ hf₁) = ℱ.val.map (g₂ ≫ h₂).op (x f₂ hf₂)

Next tactic:
apply hG.compatible ℱ h _ _ hf₁ hf₂
/-- `CompatiblePreserving` functors indeed preserve compatible families. -/
theorem Presieve.FamilyOfElements.Compatible.functorPushforward :
    (x.functorPushforward G).Compatible := by
  rintro Z₁ Z₂ W g₁ g₂ f₁' f₂' H₁ H₂ eq
  unfold FamilyOfElements.functorPushforward
  rcases getFunctorPushforwardStructure H₁ with ⟨X₁, f₁, h₁, hf₁, rfl⟩
  rcases getFunctorPushforwardStructure H₂ with ⟨X₂, f₂, h₂, hf₂, rfl⟩
  suffices ℱ.val.map (g₁ ≫ h₁).op (x f₁ hf₁) = ℱ.val.map (g₂ ≫ h₂).op (x f₂ hf₂) by
    simpa using this
  apply hG.compatible ℱ h _ _ hf₁ hf₂
  

Proof state:
case mk.mk
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
A : Type u₃
inst✝ : Category.{v₃, u₃} A
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology A
G : C ⥤ D
hG : CompatiblePreserving K G
ℱ : SheafOfTypes K
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
h : Compatible x
Z₁ Z₂ W : D
g₁ : W ⟶ Z₁
g₂ : W ⟶ Z₂
X₁ : C
f₁ : X₁ ⟶ Z
h₁ : Z₁ ⟶ G.obj X₁
hf₁ : T f₁
H₁ : Presieve.functorPushforward G T (h₁ ≫ G.map f₁)
X₂ : C
f₂ : X₂ ⟶ Z
h₂ : Z₂ ⟶ G.obj X₂
hf₂ : T f₂
H₂ : Presieve.functorPushforward G T (h₂ ≫ G.map f₂)
eq : g₁ ≫ h₁ ≫ G.map f₁ = g₂ ≫ h₂ ≫ G.map f₂
⊢ (g₁ ≫ h₁) ≫ G.map f₁ = (g₂ ≫ h₂) ≫ G.map f₂

Next tactic:
simpa using eq
@[simp]
theorem CompatiblePreserving.apply_map {Y : C} {f : Y ⟶ Z} (hf : T f) :
    x.functorPushforward G (G.map f) (image_mem_functorPushforward G T hf) = x f hf := by
  

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
A : Type u₃
inst✝ : Category.{v₃, u₃} A
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology A
G : C ⥤ D
hG : CompatiblePreserving K G
ℱ : SheafOfTypes K
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
h : Compatible x
Y : C
f : Y ⟶ Z
hf : T f
⊢ FamilyOfElements.functorPushforward G x (G.map f) (_ : Presieve.functorPushforward G T (G.map f)) = x f hf

Next tactic:
unfold FamilyOfElements.functorPushforward
@[simp]
theorem CompatiblePreserving.apply_map {Y : C} {f : Y ⟶ Z} (hf : T f) :
    x.functorPushforward G (G.map f) (image_mem_functorPushforward G T hf) = x f hf := by
  unfold FamilyOfElements.functorPushforward
  

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
A : Type u₃
inst✝ : Category.{v₃, u₃} A
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology A
G : C ⥤ D
hG : CompatiblePreserving K G
ℱ : SheafOfTypes K
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
h : Compatible x
Y : C
f : Y ⟶ Z
hf : T f
⊢ (FunctorPushforwardStructure.casesOn (getFunctorPushforwardStructure (_ : Presieve.functorPushforward G T (G.map f)))
      fun Z_1 g h h₁ fac => ℱ.val.map h.op (x g h₁)) =
    x f hf

Next tactic:
rcases e₁ : getFunctorPushforwardStructure (image_mem_functorPushforward G T hf) with
    ⟨X, g, f', hg, eq⟩
@[simp]
theorem CompatiblePreserving.apply_map {Y : C} {f : Y ⟶ Z} (hf : T f) :
    x.functorPushforward G (G.map f) (image_mem_functorPushforward G T hf) = x f hf := by
  unfold FamilyOfElements.functorPushforward
  rcases e₁ : getFunctorPushforwardStructure (image_mem_functorPushforward G T hf) with
    ⟨X, g, f', hg, eq⟩
  

Proof state:
case mk
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
A : Type u₃
inst✝ : Category.{v₃, u₃} A
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology A
G : C ⥤ D
hG : CompatiblePreserving K G
ℱ : SheafOfTypes K
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
h : Compatible x
Y : C
f : Y ⟶ Z
hf : T f
X : C
g : X ⟶ Z
f' : G.obj Y ⟶ G.obj X
hg : T g
eq : G.map f = f' ≫ G.map g
e₁ :
  getFunctorPushforwardStructure (_ : Presieve.functorPushforward G T (G.map f)) =
    { preobj := X, premap := g, lift := f', cover := hg, fac := eq }
⊢ (FunctorPushforwardStructure.casesOn { preobj := X, premap := g, lift := f', cover := hg, fac := eq }
      fun Z_1 g h h₁ fac => ℱ.val.map h.op (x g h₁)) =
    x f hf

Next tactic:
simpa using hG.compatible ℱ h f' (𝟙 _) hg hf (by simp [eq])
@[simp]
theorem CompatiblePreserving.apply_map {Y : C} {f : Y ⟶ Z} (hf : T f) :
    x.functorPushforward G (G.map f) (image_mem_functorPushforward G T hf) = x f hf := by
  unfold FamilyOfElements.functorPushforward
  rcases e₁ : getFunctorPushforwardStructure (image_mem_functorPushforward G T hf) with
    ⟨X, g, f', hg, eq⟩
  simpa using hG.compatible ℱ h f' (𝟙 _) hg hf (by 

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
A : Type u₃
inst✝ : Category.{v₃, u₃} A
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology A
G : C ⥤ D
hG : CompatiblePreserving K G
ℱ : SheafOfTypes K
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
h : Compatible x
Y : C
f : Y ⟶ Z
hf : T f
X : C
g : X ⟶ Z
f' : G.obj Y ⟶ G.obj X
hg : T g
eq : G.map f = f' ≫ G.map g
e₁ :
  getFunctorPushforwardStructure (_ : Presieve.functorPushforward G T (G.map f)) =
    { preobj := X, premap := g, lift := f', cover := hg, fac := eq }
⊢ f' ≫ G.map g = 𝟙 (G.obj Y) ≫ G.map f

Next tactic:
simp [eq]
theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type u₁} [Category.{v₁} D]
    (K : GrothendieckTopology D) (G : C ⥤ D) [RepresentablyFlat G] : CompatiblePreserving K G := by
  

Proof state:
C✝ : Type u₁
inst✝⁵ : Category.{v₁, u₁} C✝
D✝ : Type u₂
inst✝⁴ : Category.{v₂, u₂} D✝
F : C✝ ⥤ D✝
A : Type u₃
inst✝³ : Category.{v₃, u₃} A
J : GrothendieckTopology C✝
K✝ : GrothendieckTopology D✝
L : GrothendieckTopology A
G✝ : C✝ ⥤ D✝
hG : CompatiblePreserving K✝ G✝
ℱ : SheafOfTypes K✝
Z : C✝
T : Presieve Z
x : FamilyOfElements (G✝.op ⋙ ℱ.val) T
h : Compatible x
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₁
inst✝¹ : Category.{v₁, u₁} D
K : GrothendieckTopology D
G : C ⥤ D
inst✝ : RepresentablyFlat G
⊢ CompatiblePreserving K G

Next tactic:
constructor
theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type u₁} [Category.{v₁} D]
    (K : GrothendieckTopology D) (G : C ⥤ D) [RepresentablyFlat G] : CompatiblePreserving K G := by
  constructor
  

Proof state:
case compatible
C✝ : Type u₁
inst✝⁵ : Category.{v₁, u₁} C✝
D✝ : Type u₂
inst✝⁴ : Category.{v₂, u₂} D✝
F : C✝ ⥤ D✝
A : Type u₃
inst✝³ : Category.{v₃, u₃} A
J : GrothendieckTopology C✝
K✝ : GrothendieckTopology D✝
L : GrothendieckTopology A
G✝ : C✝ ⥤ D✝
hG : CompatiblePreserving K✝ G✝
ℱ : SheafOfTypes K✝
Z : C✝
T : Presieve Z
x : FamilyOfElements (G✝.op ⋙ ℱ.val) T
h : Compatible x
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₁
inst✝¹ : Category.{v₁, u₁} D
K : GrothendieckTopology D
G : C ⥤ D
inst✝ : RepresentablyFlat G
⊢ ∀ (ℱ : SheafOfTypes K) {Z : C} {T : Presieve Z} {x : FamilyOfElements (G.op ⋙ ℱ.val) T},
    Compatible x →
      ∀ {Y₁ Y₂ : C} {X : D} (f₁ : X ⟶ G.obj Y₁) (f₂ : X ⟶ G.obj Y₂) {g₁ : Y₁ ⟶ Z} {g₂ : Y₂ ⟶ Z} (hg₁ : T g₁)
        (hg₂ : T g₂), f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂ → ℱ.val.map f₁.op (x g₁ hg₁) = ℱ.val.map f₂.op (x g₂ hg₂)

Next tactic:
intro ℱ Z T x hx Y₁ Y₂ X f₁ f₂ g₁ g₂ hg₁ hg₂ e
theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type u₁} [Category.{v₁} D]
    (K : GrothendieckTopology D) (G : C ⥤ D) [RepresentablyFlat G] : CompatiblePreserving K G := by
  constructor
  intro ℱ Z T x hx Y₁ Y₂ X f₁ f₂ g₁ g₂ hg₁ hg₂ e
  -- First, `f₁` and `f₂` form a cone over `cospan g₁ g₂ ⋙ u`.
  

Proof state:
case compatible
C✝ : Type u₁
inst✝⁵ : Category.{v₁, u₁} C✝
D✝ : Type u₂
inst✝⁴ : Category.{v₂, u₂} D✝
F : C✝ ⥤ D✝
A : Type u₃
inst✝³ : Category.{v₃, u₃} A
J : GrothendieckTopology C✝
K✝ : GrothendieckTopology D✝
L : GrothendieckTopology A
G✝ : C✝ ⥤ D✝
hG : CompatiblePreserving K✝ G✝
ℱ✝ : SheafOfTypes K✝
Z✝ : C✝
T✝ : Presieve Z✝
x✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝
h : Compatible x✝
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₁
inst✝¹ : Category.{v₁, u₁} D
K : GrothendieckTopology D
G : C ⥤ D
inst✝ : RepresentablyFlat G
ℱ : SheafOfTypes K
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
hx : Compatible x
Y₁ Y₂ : C
X : D
f₁ : X ⟶ G.obj Y₁
f₂ : X ⟶ G.obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
e : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂
⊢ ℱ.val.map f₁.op (x g₁ hg₁) = ℱ.val.map f₂.op (x g₂ hg₂)

Next tactic:
let c : Cone (cospan g₁ g₂ ⋙ G) :=
    (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)
theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type u₁} [Category.{v₁} D]
    (K : GrothendieckTopology D) (G : C ⥤ D) [RepresentablyFlat G] : CompatiblePreserving K G := by
  constructor
  intro ℱ Z T x hx Y₁ Y₂ X f₁ f₂ g₁ g₂ hg₁ hg₂ e
  -- First, `f₁` and `f₂` form a cone over `cospan g₁ g₂ ⋙ u`.
  let c : Cone (cospan g₁ g₂ ⋙ G) :=
    (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)
  /-
    This can then be viewed as a cospan of structured arrows, and we may obtain an arbitrary cone
    over it since `StructuredArrow W u` is cofiltered.
    Then, it suffices to prove that it is compatible when restricted onto `u(c'.X.right)`.
    -/
  

Proof state:
case compatible
C✝ : Type u₁
inst✝⁵ : Category.{v₁, u₁} C✝
D✝ : Type u₂
inst✝⁴ : Category.{v₂, u₂} D✝
F : C✝ ⥤ D✝
A : Type u₃
inst✝³ : Category.{v₃, u₃} A
J : GrothendieckTopology C✝
K✝ : GrothendieckTopology D✝
L : GrothendieckTopology A
G✝ : C✝ ⥤ D✝
hG : CompatiblePreserving K✝ G✝
ℱ✝ : SheafOfTypes K✝
Z✝ : C✝
T✝ : Presieve Z✝
x✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝
h : Compatible x✝
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₁
inst✝¹ : Category.{v₁, u₁} D
K : GrothendieckTopology D
G : C ⥤ D
inst✝ : RepresentablyFlat G
ℱ : SheafOfTypes K
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
hx : Compatible x
Y₁ Y₂ : C
X : D
f₁ : X ⟶ G.obj Y₁
f₂ : X ⟶ G.obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
e : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂
c : Cone (cospan g₁ g₂ ⋙ G) :=
  (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)
⊢ ℱ.val.map f₁.op (x g₁ hg₁) = ℱ.val.map f₂.op (x g₂ hg₂)

Next tactic:
let c' := IsCofiltered.cone (c.toStructuredArrow ⋙ StructuredArrow.pre _ _ _)
theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type u₁} [Category.{v₁} D]
    (K : GrothendieckTopology D) (G : C ⥤ D) [RepresentablyFlat G] : CompatiblePreserving K G := by
  constructor
  intro ℱ Z T x hx Y₁ Y₂ X f₁ f₂ g₁ g₂ hg₁ hg₂ e
  -- First, `f₁` and `f₂` form a cone over `cospan g₁ g₂ ⋙ u`.
  let c : Cone (cospan g₁ g₂ ⋙ G) :=
    (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)
  /-
    This can then be viewed as a cospan of structured arrows, and we may obtain an arbitrary cone
    over it since `StructuredArrow W u` is cofiltered.
    Then, it suffices to prove that it is compatible when restricted onto `u(c'.X.right)`.
    -/
  let c' := IsCofiltered.cone (c.toStructuredArrow ⋙ StructuredArrow.pre _ _ _)
  

Proof state:
case compatible
C✝ : Type u₁
inst✝⁵ : Category.{v₁, u₁} C✝
D✝ : Type u₂
inst✝⁴ : Category.{v₂, u₂} D✝
F : C✝ ⥤ D✝
A : Type u₃
inst✝³ : Category.{v₃, u₃} A
J : GrothendieckTopology C✝
K✝ : GrothendieckTopology D✝
L : GrothendieckTopology A
G✝ : C✝ ⥤ D✝
hG : CompatiblePreserving K✝ G✝
ℱ✝ : SheafOfTypes K✝
Z✝ : C✝
T✝ : Presieve Z✝
x✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝
h : Compatible x✝
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₁
inst✝¹ : Category.{v₁, u₁} D
K : GrothendieckTopology D
G : C ⥤ D
inst✝ : RepresentablyFlat G
ℱ : SheafOfTypes K
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
hx : Compatible x
Y₁ Y₂ : C
X : D
f₁ : X ⟶ G.obj Y₁
f₂ : X ⟶ G.obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
e : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂
c : Cone (cospan g₁ g₂ ⋙ G) :=
  (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)
c' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=
  IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)
⊢ ℱ.val.map f₁.op (x g₁ hg₁) = ℱ.val.map f₂.op (x g₂ hg₂)

Next tactic:
have eq₁ : f₁ = (c'.pt.hom ≫ G.map (c'.π.app left).right) ≫ eqToHom (by simp) := by
    erw [← (c'.π.app left).w]
    dsimp
    simp
theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type u₁} [Category.{v₁} D]
    (K : GrothendieckTopology D) (G : C ⥤ D) [RepresentablyFlat G] : CompatiblePreserving K G := by
  constructor
  intro ℱ Z T x hx Y₁ Y₂ X f₁ f₂ g₁ g₂ hg₁ hg₂ e
  -- First, `f₁` and `f₂` form a cone over `cospan g₁ g₂ ⋙ u`.
  let c : Cone (cospan g₁ g₂ ⋙ G) :=
    (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)
  /-
    This can then be viewed as a cospan of structured arrows, and we may obtain an arbitrary cone
    over it since `StructuredArrow W u` is cofiltered.
    Then, it suffices to prove that it is compatible when restricted onto `u(c'.X.right)`.
    -/
  let c' := IsCofiltered.cone (c.toStructuredArrow ⋙ StructuredArrow.pre _ _ _)
  have eq₁ : f₁ = (c'.pt.hom ≫ G.map (c'.π.app left).right) ≫ eqToHom (by 

Proof state:
C✝ : Type u₁
inst✝⁵ : Category.{v₁, u₁} C✝
D✝ : Type u₂
inst✝⁴ : Category.{v₂, u₂} D✝
F : C✝ ⥤ D✝
A : Type u₃
inst✝³ : Category.{v₃, u₃} A
J : GrothendieckTopology C✝
K✝ : GrothendieckTopology D✝
L : GrothendieckTopology A
G✝ : C✝ ⥤ D✝
hG : CompatiblePreserving K✝ G✝
ℱ✝ : SheafOfTypes K✝
Z✝ : C✝
T✝ : Presieve Z✝
x✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝
h : Compatible x✝
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₁
inst✝¹ : Category.{v₁, u₁} D
K : GrothendieckTopology D
G : C ⥤ D
inst✝ : RepresentablyFlat G
ℱ : SheafOfTypes K
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
hx : Compatible x
Y₁ Y₂ : C
X : D
f₁ : X ⟶ G.obj Y₁
f₂ : X ⟶ G.obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
e : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂
c : Cone (cospan g₁ g₂ ⋙ G) :=
  (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)
c' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=
  IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)
⊢ G.obj ((Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G).obj left).right = G.obj Y₁

Next tactic:
simp
theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type u₁} [Category.{v₁} D]
    (K : GrothendieckTopology D) (G : C ⥤ D) [RepresentablyFlat G] : CompatiblePreserving K G := by
  constructor
  intro ℱ Z T x hx Y₁ Y₂ X f₁ f₂ g₁ g₂ hg₁ hg₂ e
  -- First, `f₁` and `f₂` form a cone over `cospan g₁ g₂ ⋙ u`.
  let c : Cone (cospan g₁ g₂ ⋙ G) :=
    (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)
  /-
    This can then be viewed as a cospan of structured arrows, and we may obtain an arbitrary cone
    over it since `StructuredArrow W u` is cofiltered.
    Then, it suffices to prove that it is compatible when restricted onto `u(c'.X.right)`.
    -/
  let c' := IsCofiltered.cone (c.toStructuredArrow ⋙ StructuredArrow.pre _ _ _)
  have eq₁ : f₁ = (c'.pt.hom ≫ G.map (c'.π.app left).right) ≫ eqToHom (by simp) := by
    

Proof state:
C✝ : Type u₁
inst✝⁵ : Category.{v₁, u₁} C✝
D✝ : Type u₂
inst✝⁴ : Category.{v₂, u₂} D✝
F : C✝ ⥤ D✝
A : Type u₃
inst✝³ : Category.{v₃, u₃} A
J : GrothendieckTopology C✝
K✝ : GrothendieckTopology D✝
L : GrothendieckTopology A
G✝ : C✝ ⥤ D✝
hG : CompatiblePreserving K✝ G✝
ℱ✝ : SheafOfTypes K✝
Z✝ : C✝
T✝ : Presieve Z✝
x✝ : FamilyOfElements (G✝.op ⋙ ℱ✝.val) T✝
h : Compatible x✝
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₁
inst✝¹ : Category.{v₁, u₁} D
K : GrothendieckTopology D
G : C ⥤ D
inst✝ : RepresentablyFlat G
ℱ : SheafOfTypes K
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
hx : Compatible x
Y₁ Y₂ : C
X : D
f₁ : X ⟶ G.obj Y₁
f₂ : X ⟶ G.obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
e : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂
c : Cone (cospan g₁ g₂ ⋙ G) :=
  (Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)
c' : Cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=
  IsCofiltered.cone (Cone.toStructuredArrow c ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)
⊢ f₁ =
    (c'.pt.hom ≫ G.map (c'.π.app left).right) ≫
      eqToHom
        (_ :
          G.obj ((StructuredArrow.pre X (cospan g₁ g₂) G).obj (StructuredArrow.mk (f₁ ≫ 𝟙 (G.obj Y₁)))).right =
            G.obj Y₁)

Next tactic:
erw [← (c'.π.app left).w]