l3lab/ntp-mathlib huggingface dataset:
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
    let e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl _)
      (by simpa using (Over.w f₁).symm)
    convert congr_arg (F.val.map e.inv.op)
      (hx g₁' g₂' hg₁ hg₂ (by ext; exact (Over.forget _).congr_map h)) using 1
    all_goals
      

Proof state:
case h.e'_3
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left)
⊢ F.val.map f₂.op (x g₂ hg₂) = F.val.map e.inv.op (((Over.map f).op ⋙ F.val).map g₂'.op (x g₂ hg₂))

Next tactic:
dsimp
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
    let e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl _)
      (by simpa using (Over.w f₁).symm)
    convert congr_arg (F.val.map e.inv.op)
      (hx g₁' g₂' hg₁ hg₂ (by ext; exact (Over.forget _).congr_map h)) using 1
    all_goals
      dsimp
      

Proof state:
case h.e'_3
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left)
⊢ F.val.map f₂.op (x g₂ hg₂) =
    F.val.map (Over.isoMk (Iso.refl W.left)).inv.op (F.val.map ((Over.map f).map (Over.homMk f₂.left)).op (x g₂ hg₂))

Next tactic:
rw [← FunctorToTypes.map_comp_apply]
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
    let e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl _)
      (by simpa using (Over.w f₁).symm)
    convert congr_arg (F.val.map e.inv.op)
      (hx g₁' g₂' hg₁ hg₂ (by ext; exact (Over.forget _).congr_map h)) using 1
    all_goals
      dsimp
      rw [← FunctorToTypes.map_comp_apply]
      

Proof state:
case h.e'_3
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left)
⊢ F.val.map f₂.op (x g₂ hg₂) =
    F.val.map (((Over.map f).map (Over.homMk f₂.left)).op ≫ (Over.isoMk (Iso.refl W.left)).inv.op) (x g₂ hg₂)

Next tactic:
apply congr_fun
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
    let e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl _)
      (by simpa using (Over.w f₁).symm)
    convert congr_arg (F.val.map e.inv.op)
      (hx g₁' g₂' hg₁ hg₂ (by ext; exact (Over.forget _).congr_map h)) using 1
    all_goals
      dsimp
      rw [← FunctorToTypes.map_comp_apply]
      apply congr_fun
      

Proof state:
case h.e'_3.h
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left)
⊢ F.val.map f₂.op = F.val.map (((Over.map f).map (Over.homMk f₂.left)).op ≫ (Over.isoMk (Iso.refl W.left)).inv.op)

Next tactic:
congr 1
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
    let e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl _)
      (by simpa using (Over.w f₁).symm)
    convert congr_arg (F.val.map e.inv.op)
      (hx g₁' g₂' hg₁ hg₂ (by ext; exact (Over.forget _).congr_map h)) using 1
    all_goals
      dsimp
      rw [← FunctorToTypes.map_comp_apply]
      apply congr_fun
      congr 1
      

Proof state:
case h.e'_3.h.e_a
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left)
⊢ f₂.op = ((Over.map f).map (Over.homMk f₂.left)).op ≫ (Over.isoMk (Iso.refl W.left)).inv.op

Next tactic:
rw [← op_comp]
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
    let e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl _)
      (by simpa using (Over.w f₁).symm)
    convert congr_arg (F.val.map e.inv.op)
      (hx g₁' g₂' hg₁ hg₂ (by ext; exact (Over.forget _).congr_map h)) using 1
    all_goals
      dsimp
      rw [← FunctorToTypes.map_comp_apply]
      apply congr_fun
      congr 1
      rw [← op_comp]
      

Proof state:
case h.e'_3.h.e_a
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left)
⊢ f₂.op = ((Over.isoMk (Iso.refl W.left)).inv ≫ (Over.map f).map (Over.homMk f₂.left)).op

Next tactic:
congr 1
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
    let e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl _)
      (by simpa using (Over.w f₁).symm)
    convert congr_arg (F.val.map e.inv.op)
      (hx g₁' g₂' hg₁ hg₂ (by ext; exact (Over.forget _).congr_map h)) using 1
    all_goals
      dsimp
      rw [← FunctorToTypes.map_comp_apply]
      apply congr_fun
      congr 1
      rw [← op_comp]
      congr 1
      

Proof state:
case h.e'_3.h.e_a.e_f
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left)
⊢ f₂ = (Over.isoMk (Iso.refl W.left)).inv ≫ (Over.map f).map (Over.homMk f₂.left)

Next tactic:
ext
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
    let e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl _)
      (by simpa using (Over.w f₁).symm)
    convert congr_arg (F.val.map e.inv.op)
      (hx g₁' g₂' hg₁ hg₂ (by ext; exact (Over.forget _).congr_map h)) using 1
    all_goals
      dsimp
      rw [← FunctorToTypes.map_comp_apply]
      apply congr_fun
      congr 1
      rw [← op_comp]
      congr 1
      ext
      

Proof state:
case h.e'_3.h.e_a.e_f.h
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left)
⊢ f₂.left = ((Over.isoMk (Iso.refl W.left)).inv ≫ (Over.map f).map (Over.homMk f₂.left)).left

Next tactic:
simp
@[ext]
lemma FirstObj.ext (z₁ z₂ : FirstObj P R) (h : ∀ (Y : C) (f : Y ⟶ X)
    (hf : R f), (Pi.π _ ⟨Y, f, hf⟩ : FirstObj P R ⟶ _) z₁ =
      (Pi.π _ ⟨Y, f, hf⟩ : FirstObj P R ⟶ _) z₂) : z₁ = z₂ := by
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
P : Cᵒᵖ ⥤ Type (max v u)
X : C
R : Presieve X
S : Sieve X
z₁ z₂ : FirstObj P R
h :
  ∀ (Y : C) (f : Y ⟶ X) (hf : R f),
    Pi.π (fun f => P.obj (op f.fst)) { fst := Y, snd := { val := f, property := hf } } z₁ =
      Pi.π (fun f => P.obj (op f.fst)) { fst := Y, snd := { val := f, property := hf } } z₂
⊢ z₁ = z₂

Next tactic:
apply Limits.Types.limit_ext
@[ext]
lemma FirstObj.ext (z₁ z₂ : FirstObj P R) (h : ∀ (Y : C) (f : Y ⟶ X)
    (hf : R f), (Pi.π _ ⟨Y, f, hf⟩ : FirstObj P R ⟶ _) z₁ =
      (Pi.π _ ⟨Y, f, hf⟩ : FirstObj P R ⟶ _) z₂) : z₁ = z₂ := by
  apply Limits.Types.limit_ext
  

Proof state:
case w
C : Type u
inst✝ : Category.{v, u} C
P : Cᵒᵖ ⥤ Type (max v u)
X : C
R : Presieve X
S : Sieve X
z₁ z₂ : FirstObj P R
h :
  ∀ (Y : C) (f : Y ⟶ X) (hf : R f),
    Pi.π (fun f => P.obj (op f.fst)) { fst := Y, snd := { val := f, property := hf } } z₁ =
      Pi.π (fun f => P.obj (op f.fst)) { fst := Y, snd := { val := f, property := hf } } z₂
⊢ ∀ (j : Discrete ((Y : C) × { f // R f })),
    limit.π (Discrete.functor fun f => P.obj (op f.fst)) j z₁ =
      limit.π (Discrete.functor fun f => P.obj (op f.fst)) j z₂

Next tactic:
rintro ⟨⟨Y, f, hf⟩⟩
@[ext]
lemma FirstObj.ext (z₁ z₂ : FirstObj P R) (h : ∀ (Y : C) (f : Y ⟶ X)
    (hf : R f), (Pi.π _ ⟨Y, f, hf⟩ : FirstObj P R ⟶ _) z₁ =
      (Pi.π _ ⟨Y, f, hf⟩ : FirstObj P R ⟶ _) z₂) : z₁ = z₂ := by
  apply Limits.Types.limit_ext
  rintro ⟨⟨Y, f, hf⟩⟩
  

Proof state:
case w.mk.mk.mk
C : Type u
inst✝ : Category.{v, u} C
P : Cᵒᵖ ⥤ Type (max v u)
X : C
R : Presieve X
S : Sieve X
z₁ z₂ : FirstObj P R
h :
  ∀ (Y : C) (f : Y ⟶ X) (hf : R f),
    Pi.π (fun f => P.obj (op f.fst)) { fst := Y, snd := { val := f, property := hf } } z₁ =
      Pi.π (fun f => P.obj (op f.fst)) { fst := Y, snd := { val := f, property := hf } } z₂
Y : C
f : Y ⟶ X
hf : R f
⊢ limit.π (Discrete.functor fun f => P.obj (op f.fst)) { as := { fst := Y, snd := { val := f, property := hf } } } z₁ =
    limit.π (Discrete.functor fun f => P.obj (op f.fst)) { as := { fst := Y, snd := { val := f, property := hf } } } z₂

Next tactic:
exact h Y f hf
@[ext]
lemma SecondObj.ext (z₁ z₂ : SecondObj P S) (h : ∀ (Y Z : C) (g : Z ⟶ Y) (f : Y ⟶ X)
    (hf : S.arrows f), (Pi.π _ ⟨Y, Z, g, f, hf⟩ : SecondObj P S ⟶ _) z₁ =
      (Pi.π _ ⟨Y, Z, g, f, hf⟩ : SecondObj P S ⟶ _) z₂) : z₁ = z₂ := by
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
P : Cᵒᵖ ⥤ Type (max v u)
X : C
R : Presieve X
S : Sieve X
z₁ z₂ : SecondObj P S
h :
  ∀ (Y Z : C) (g : Z ⟶ Y) (f : Y ⟶ X) (hf : S.arrows f),
    Pi.π (fun f => P.obj (op f.snd.fst))
        { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } z₁ =
      Pi.π (fun f => P.obj (op f.snd.fst))
        { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } z₂
⊢ z₁ = z₂

Next tactic:
apply Limits.Types.limit_ext
@[ext]
lemma SecondObj.ext (z₁ z₂ : SecondObj P S) (h : ∀ (Y Z : C) (g : Z ⟶ Y) (f : Y ⟶ X)
    (hf : S.arrows f), (Pi.π _ ⟨Y, Z, g, f, hf⟩ : SecondObj P S ⟶ _) z₁ =
      (Pi.π _ ⟨Y, Z, g, f, hf⟩ : SecondObj P S ⟶ _) z₂) : z₁ = z₂ := by
  apply Limits.Types.limit_ext
  

Proof state:
case w
C : Type u
inst✝ : Category.{v, u} C
P : Cᵒᵖ ⥤ Type (max v u)
X : C
R : Presieve X
S : Sieve X
z₁ z₂ : SecondObj P S
h :
  ∀ (Y Z : C) (g : Z ⟶ Y) (f : Y ⟶ X) (hf : S.arrows f),
    Pi.π (fun f => P.obj (op f.snd.fst))
        { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } z₁ =
      Pi.π (fun f => P.obj (op f.snd.fst))
        { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } z₂
⊢ ∀ (j : Discrete ((Y : C) × (Z : C) × (_ : Z ⟶ Y) × { f' // S.arrows f' })),
    limit.π (Discrete.functor fun f => P.obj (op f.snd.fst)) j z₁ =
      limit.π (Discrete.functor fun f => P.obj (op f.snd.fst)) j z₂

Next tactic:
rintro ⟨⟨Y, Z, g, f, hf⟩⟩
@[ext]
lemma SecondObj.ext (z₁ z₂ : SecondObj P S) (h : ∀ (Y Z : C) (g : Z ⟶ Y) (f : Y ⟶ X)
    (hf : S.arrows f), (Pi.π _ ⟨Y, Z, g, f, hf⟩ : SecondObj P S ⟶ _) z₁ =
      (Pi.π _ ⟨Y, Z, g, f, hf⟩ : SecondObj P S ⟶ _) z₂) : z₁ = z₂ := by
  apply Limits.Types.limit_ext
  rintro ⟨⟨Y, Z, g, f, hf⟩⟩
  

Proof state:
case w.mk.mk.mk.mk.mk
C : Type u
inst✝ : Category.{v, u} C
P : Cᵒᵖ ⥤ Type (max v u)
X : C
R : Presieve X
S : Sieve X
z₁ z₂ : SecondObj P S
h :
  ∀ (Y Z : C) (g : Z ⟶ Y) (f : Y ⟶ X) (hf : S.arrows f),
    Pi.π (fun f => P.obj (op f.snd.fst))
        { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } z₁ =
      Pi.π (fun f => P.obj (op f.snd.fst))
        { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } z₂
Y Z : C
g : Z ⟶ Y
f : Y ⟶ X
hf : S.arrows f
⊢ limit.π (Discrete.functor fun f => P.obj (op f.snd.fst))
      { as := { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } } z₁ =
    limit.π (Discrete.functor fun f => P.obj (op f.snd.fst))
      { as := { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } } z₂

Next tactic:
apply h
theorem w : forkMap P (S : Presieve X) ≫ firstMap P S = forkMap P S ≫ secondMap P S := by
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
P : Cᵒᵖ ⥤ Type (max v u)
X : C
R : Presieve X
S : Sieve X
⊢ forkMap P S.arrows ≫ firstMap P S = forkMap P S.arrows ≫ secondMap P S

Next tactic:
ext
theorem w : forkMap P (S : Presieve X) ≫ firstMap P S = forkMap P S ≫ secondMap P S := by
  ext
  

Proof state:
case h.h
C : Type u
inst✝ : Category.{v, u} C
P : Cᵒᵖ ⥤ Type (max v u)
X : C
R : Presieve X
S : Sieve X
a✝ : P.obj (op X)
Y✝ Z✝ : C
g✝ : Z✝ ⟶ Y✝
f✝ : Y✝ ⟶ X
hf✝ : S.arrows f✝
⊢ Pi.π (fun f => P.obj (op f.snd.fst))
      { fst := Y✝, snd := { fst := Z✝, snd := { fst := g✝, snd := { val := f✝, property := hf✝ } } } }
      ((forkMap P S.arrows ≫ firstMap P S) a✝) =
    Pi.π (fun f => P.obj (op f.snd.fst))
      { fst := Y✝, snd := { fst := Z✝, snd := { fst := g✝, snd := { val := f✝, property := hf✝ } } } }
      ((forkMap P S.arrows ≫ secondMap P S) a✝)

Next tactic:
simp [firstMap, secondMap, forkMap]
/--
The family of elements given by `x : FirstObj P S` is compatible iff `firstMap` and `secondMap`
map it to the same point.
-/
theorem compatible_iff (x : FirstObj P S) :
    ((firstObjEqFamily P S).hom x).Compatible ↔ firstMap P S x = secondMap P S x := by
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
P : Cᵒᵖ ⥤ Type (max v u)
X : C
R : Presieve X
S : Sieve X
x : FirstObj P S.arrows
⊢ Presieve.FamilyOfElements.Compatible ((firstObjEqFamily P S.arrows).hom x) ↔ firstMap P S x = secondMap P S x

Next tactic:
rw [Presieve.compatible_iff_sieveCompatible]
/--
The family of elements given by `x : FirstObj P S` is compatible iff `firstMap` and `secondMap`
map it to the same point.
-/
theorem compatible_iff (x : FirstObj P S) :
    ((firstObjEqFamily P S).hom x).Compatible ↔ firstMap P S x = secondMap P S x := by
  rw [Presieve.compatible_iff_sieveCompatible]
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
P : Cᵒᵖ ⥤ Type (max v u)
X : C
R : Presieve X
S : Sieve X
x : FirstObj P S.arrows
⊢ Presieve.FamilyOfElements.SieveCompatible ((firstObjEqFamily P S.arrows).hom x) ↔ firstMap P S x = secondMap P S x

Next tactic:
constructor
/--
The family of elements given by `x : FirstObj P S` is compatible iff `firstMap` and `secondMap`
map it to the same point.
-/
theorem compatible_iff (x : FirstObj P S) :
    ((firstObjEqFamily P S).hom x).Compatible ↔ firstMap P S x = secondMap P S x := by
  rw [Presieve.compatible_iff_sieveCompatible]
  constructor
  · 

Proof state:
case mp
C : Type u
inst✝ : Category.{v, u} C
P : Cᵒᵖ ⥤ Type (max v u)
X : C
R : Presieve X
S : Sieve X
x : FirstObj P S.arrows
⊢ Presieve.FamilyOfElements.SieveCompatible ((firstObjEqFamily P S.arrows).hom x) → firstMap P S x = secondMap P S x

Next tactic:
intro t
/--
The family of elements given by `x : FirstObj P S` is compatible iff `firstMap` and `secondMap`
map it to the same point.
-/
theorem compatible_iff (x : FirstObj P S) :
    ((firstObjEqFamily P S).hom x).Compatible ↔ firstMap P S x = secondMap P S x := by
  rw [Presieve.compatible_iff_sieveCompatible]
  constructor
  · intro t
    

Proof state:
case mp
C : Type u
inst✝ : Category.{v, u} C
P : Cᵒᵖ ⥤ Type (max v u)
X : C
R : Presieve X
S : Sieve X
x : FirstObj P S.arrows
t : Presieve.FamilyOfElements.SieveCompatible ((firstObjEqFamily P S.arrows).hom x)
⊢ firstMap P S x = secondMap P S x

Next tactic:
apply SecondObj.ext
/--
The family of elements given by `x : FirstObj P S` is compatible iff `firstMap` and `secondMap`
map it to the same point.
-/
theorem compatible_iff (x : FirstObj P S) :
    ((firstObjEqFamily P S).hom x).Compatible ↔ firstMap P S x = secondMap P S x := by
  rw [Presieve.compatible_iff_sieveCompatible]
  constructor
  · intro t
    apply SecondObj.ext
    

Proof state:
case mp.h
C : Type u
inst✝ : Category.{v, u} C
P : Cᵒᵖ ⥤ Type (max v u)
X : C
R : Presieve X
S : Sieve X
x : FirstObj P S.arrows
t : Presieve.FamilyOfElements.SieveCompatible ((firstObjEqFamily P S.arrows).hom x)
⊢ ∀ (Y Z : C) (g : Z ⟶ Y) (f : Y ⟶ X) (hf : S.arrows f),
    Pi.π (fun f => P.obj (op f.snd.fst))
        { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } (firstMap P S x) =
      Pi.π (fun f => P.obj (op f.snd.fst))
        { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } (secondMap P S x)

Next tactic:
intros Y Z g f hf
/--
The family of elements given by `x : FirstObj P S` is compatible iff `firstMap` and `secondMap`
map it to the same point.
-/
theorem compatible_iff (x : FirstObj P S) :
    ((firstObjEqFamily P S).hom x).Compatible ↔ firstMap P S x = secondMap P S x := by
  rw [Presieve.compatible_iff_sieveCompatible]
  constructor
  · intro t
    apply SecondObj.ext
    intros Y Z g f hf
    

Proof state:
case mp.h
C : Type u
inst✝ : Category.{v, u} C
P : Cᵒᵖ ⥤ Type (max v u)
X : C
R : Presieve X
S : Sieve X
x : FirstObj P S.arrows
t : Presieve.FamilyOfElements.SieveCompatible ((firstObjEqFamily P S.arrows).hom x)
Y Z : C
g : Z ⟶ Y
f : Y ⟶ X
hf : S.arrows f
⊢ Pi.π (fun f => P.obj (op f.snd.fst))
      { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } (firstMap P S x) =
    Pi.π (fun f => P.obj (op f.snd.fst))
      { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } (secondMap P S x)

Next tactic:
simpa [firstMap, secondMap] using t _ g hf
/--
The family of elements given by `x : FirstObj P S` is compatible iff `firstMap` and `secondMap`
map it to the same point.
-/
theorem compatible_iff (x : FirstObj P S) :
    ((firstObjEqFamily P S).hom x).Compatible ↔ firstMap P S x = secondMap P S x := by
  rw [Presieve.compatible_iff_sieveCompatible]
  constructor
  · intro t
    apply SecondObj.ext
    intros Y Z g f hf
    simpa [firstMap, secondMap] using t _ g hf
  · 

Proof state:
case mpr
C : Type u
inst✝ : Category.{v, u} C
P : Cᵒᵖ ⥤ Type (max v u)
X : C
R : Presieve X
S : Sieve X
x : FirstObj P S.arrows
⊢ firstMap P S x = secondMap P S x → Presieve.FamilyOfElements.SieveCompatible ((firstObjEqFamily P S.arrows).hom x)

Next tactic:
intro t Y Z f g hf
/--
The family of elements given by `x : FirstObj P S` is compatible iff `firstMap` and `secondMap`
map it to the same point.
-/
theorem compatible_iff (x : FirstObj P S) :
    ((firstObjEqFamily P S).hom x).Compatible ↔ firstMap P S x = secondMap P S x := by
  rw [Presieve.compatible_iff_sieveCompatible]
  constructor
  · intro t
    apply SecondObj.ext
    intros Y Z g f hf
    simpa [firstMap, secondMap] using t _ g hf
  · intro t Y Z f g hf
    

Proof state:
case mpr
C : Type u
inst✝ : Category.{v, u} C
P : Cᵒᵖ ⥤ Type (max v u)
X : C
R : Presieve X
S : Sieve X
x : FirstObj P S.arrows
t : firstMap P S x = secondMap P S x
Y Z : C
f : Y ⟶ X
g : Z ⟶ Y
hf : S.arrows f
⊢ (firstObjEqFamily P S.arrows).hom x (g ≫ f) (_ : S.arrows (g ≫ f)) =
    P.map g.op ((firstObjEqFamily P S.arrows).hom x f hf)

Next tactic:
rw [Types.limit_ext_iff'] at t