awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Sheaf (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A : Type u₂
inst_1 : Category.{v₂, u₂} A
U : C
R : Presieve.{v₁, u₁} U
P : Functor.{v₁, v₂, u₁, u₂} Cᵒᵖ A
inst_2 : HasProducts A
inst_3 : HasPullbacks.{v₁, u₁} C
Y : C
f : Y ⟶ U
hf : R f
Z : C
g : Z ⟶ U
hg : R g
⊢ Eq (α :=
    P.obj (op U) ⟶
      Functor.obj (C := Discrete (((V : C) × { f : V ⟶ U // R f }) × (W : C) × { g : W ⟶ U // R g }))
        (Discrete.functor fun (fg : ((V : C) × { f : V ⟶ U // R f }) × (W : C) × { g : W ⟶ U // R g }) ↦
          P.obj
            (op
              (Limits.pullback (↑(fg.1.snd (β := fun (V : C) ↦ { f : V ⟶ U // R f })) : fg.1.fst ⟶ U)
                (↑(fg.2.snd (β := fun (W : C) ↦ { g : W ⟶ U // R g })) : fg.2.fst ⟶ U))))
        {
          as :=
            (Sigma.mk (β := fun (V : C) ↦ { f : V ⟶ U // R f }) Y (Subtype.mk (p := fun (f : Y ⟶ U) ↦ R f) f hf),
              Sigma.mk (β := fun (W : C) ↦ { g : W ⟶ U // R g }) Z (Subtype.mk (p := fun (g : Z ⟶ U) ↦ R g) g hg)) })
    (P.map f.op ≫ P.map (Quiver.Hom.op (pullback.fst f g))) (P.map g.op ≫ P.map (Quiver.Hom.op (pullback.snd f g)))
have w₁ : Eq (α :=
  P.obj (op U) ⟶
    Functor.obj (C := Discrete (((V : C) × { f : V ⟶ U // R f }) × (W : C) × { g : W ⟶ U // R g }))
      (Discrete.functor fun (fg : ((V : C) × { f : V ⟶ U // R f }) × (W : C) × { g : W ⟶ U // R g }) ↦
        P.obj
          (op
            (Limits.pullback (↑(fg.1.snd (β := fun (V : C) ↦ { f : V ⟶ U // R f })) : fg.1.fst ⟶ U)
              (↑(fg.2.snd (β := fun (W : C) ↦ { g : W ⟶ U // R g })) : fg.2.fst ⟶ U))))
      {
        as :=
          (Sigma.mk (β := fun (V : C) ↦ { f : V ⟶ U // R f }) Y (Subtype.mk (p := fun (f : Y ⟶ U) ↦ R f) f hf),
            Sigma.mk (β := fun (W : C) ↦ { g : W ⟶ U // R g }) Z (Subtype.mk (p := fun (g : Z ⟶ U) ↦ R g) g hg)) })
  (P.map (Quiver.Hom.op (pullback.snd f g ≫ g))) (P.map g.op ≫ P.map (Quiver.Hom.op (pullback.snd f g)))
Tactic: rw [← P.map_comp, ← op_comp, pullback.condition]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Sheaf (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A : Type u₂
inst_1 : Category.{v₂, u₂} A
A' : Type u₂
inst_2 : Category.{max v₁ u₁, u₂} A'
B : Type u₃
inst_3 : Category.{v₃, u₃} B
J : GrothendieckTopology.{v₁, u₁} C
U_1 : C
R_1 : Presieve.{v₁, u₁} U_1
P_1 : Functor.{v₁, v₂, u₁, u₂} Cᵒᵖ A
P' : Cᵒᵖ ⥤ A'
inst_4 : HasProducts A
inst_5 : HasProducts A'
inst_6 : HasPullbacks.{v₁, u₁} C
P : Functor.{v₁, v₂, u₁, u₂} Cᵒᵖ A
s : A ⥤ Type (max v₁ u₁)
inst_7 : ∀ (J : Type (max v₁ u₁)), PreservesLimitsOfShape (Discrete J) s
U : C
R : Presieve.{v₁, u₁} U
e : parallelPair (X := s.obj (firstObj R P)) (Y := s.obj (secondObj R P)) (s.map (firstMap R P))
    (s.map (secondMap R P)) ≅
  parallelPair (Y := Equalizer.Presieve.SecondObj (P ⋙ s) R) (Equalizer.Presieve.firstMap (P ⋙ s) R)
    (Equalizer.Presieve.secondMap (P ⋙ s) R) :=
  parallelPair.ext
    (PreservesProduct.iso s fun (f : (V : C) × { f : V ⟶ U // R f }) ↦
      P.obj (op (f.fst (β := fun (V : C) ↦ { f : V ⟶ U // R f }))))
    (PreservesProduct.iso s fun (fg : ((V : C) × { f : V ⟶ U // R f }) × (W : C) × { g : W ⟶ U // R g }) ↦
      P.obj
        (op
          (Limits.pullback (↑(fg.1.snd (β := fun (V : C) ↦ { f : V ⟶ U // R f })) : fg.1.fst ⟶ U)
            (↑(fg.2.snd (β := fun (W : C) ↦ { g : W ⟶ U // R g })) : fg.2.fst ⟶ U))))
    ⋯ ⋯
⊢ IsLimit (F := parallelPair (X := firstObj R P) (Y := secondObj R P) (firstMap R P) (secondMap R P) ⋙ s)
      (s.mapCone (Fork.ofι (forkMap R P) ⋯)) ≃
    IsLimit (F :=
      parallelPair (Y := Equalizer.Presieve.SecondObj (P ⋙ s) R) (Equalizer.Presieve.firstMap (P ⋙ s) R)
        (Equalizer.Presieve.secondMap (P ⋙ s) R))
      (Fork.ofι (Equalizer.forkMap (P ⋙ s) R) ⋯)
have refine : IsLimit (F :=
    parallelPair (X := s.obj (firstObj R P)) (Y := s.obj (secondObj R P)) (s.map (firstMap R P))
      (s.map (secondMap R P)))
    (Fork.ofι (s.map (forkMap R P)) sorry) ≃
  IsLimit (F :=
    parallelPair (Y := Equalizer.Presieve.SecondObj (P ⋙ s) R) (Equalizer.Presieve.firstMap (P ⋙ s) R)
      (Equalizer.Presieve.secondMap (P ⋙ s) R))
    (Fork.ofι (Equalizer.forkMap (P ⋙ s) R) sorry)
Tactic: refine Equiv.trans (isLimitMapConeForkEquiv _ _) ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Sheaf (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A : Type u₂
inst_1 : Category.{v₂, u₂} A
A' : Type u₂
inst_2 : Category.{max v₁ u₁, u₂} A'
B : Type u₃
inst_3 : Category.{v₃, u₃} B
J : GrothendieckTopology.{v₁, u₁} C
U_1 : C
R_1 : Presieve.{v₁, u₁} U_1
P_1 : Functor.{v₁, v₂, u₁, u₂} Cᵒᵖ A
P' : Cᵒᵖ ⥤ A'
inst_4 : HasProducts A
inst_5 : HasProducts A'
inst_6 : HasPullbacks.{v₁, u₁} C
P : Functor.{v₁, v₂, u₁, u₂} Cᵒᵖ A
s : A ⥤ Type (max v₁ u₁)
inst_7 : ∀ (J : Type (max v₁ u₁)), PreservesLimitsOfShape (Discrete J) s
U : C
R : Presieve.{v₁, u₁} U
e : parallelPair (X := s.obj (firstObj R P)) (Y := s.obj (secondObj R P)) (s.map (firstMap R P))
    (s.map (secondMap R P)) ≅
  parallelPair (Y := Equalizer.Presieve.SecondObj (P ⋙ s) R) (Equalizer.Presieve.firstMap (P ⋙ s) R)
    (Equalizer.Presieve.secondMap (P ⋙ s) R) :=
  parallelPair.ext
    (PreservesProduct.iso s fun (f : (V : C) × { f : V ⟶ U // R f }) ↦
      P.obj (op (f.fst (β := fun (V : C) ↦ { f : V ⟶ U // R f }))))
    (PreservesProduct.iso s fun (fg : ((V : C) × { f : V ⟶ U // R f }) × (W : C) × { g : W ⟶ U // R g }) ↦
      P.obj
        (op
          (Limits.pullback (↑(fg.1.snd (β := fun (V : C) ↦ { f : V ⟶ U // R f })) : fg.1.fst ⟶ U)
            (↑(fg.2.snd (β := fun (W : C) ↦ { g : W ⟶ U // R g })) : fg.2.fst ⟶ U))))
    ⋯ ⋯
⊢ IsLimit (F :=
      parallelPair (X := s.obj (firstObj R P)) (Y := s.obj (secondObj R P)) (s.map (firstMap R P))
        (s.map (secondMap R P)))
      (Fork.ofι (s.map (forkMap R P)) ⋯) ≃
    IsLimit (F :=
      parallelPair (Y := Equalizer.Presieve.SecondObj (P ⋙ s) R) (Equalizer.Presieve.firstMap (P ⋙ s) R)
        (Equalizer.Presieve.secondMap (P ⋙ s) R))
      (Fork.ofι (Equalizer.forkMap (P ⋙ s) R) ⋯)
have refine : Iso.hom (Iso.refl ((Cone.postcompose e.hom).obj (Fork.ofι (s.map (forkMap R P)) sorry)).pt) ≫
    Fork.ι (Fork.ofι (Equalizer.forkMap (P ⋙ s) R) sorry) =
  Fork.ι ((Cone.postcompose e.hom).obj (Fork.ofι (s.map (forkMap R P)) sorry))
Tactic: refine (IsLimit.postcomposeHomEquiv e _).symm.trans
    (IsLimit.equivIsoLimit (Fork.ext (Iso.refl _) ?_))
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Sheaf (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A : Type u₂
inst_1 : Category.{v₂, u₂} A
A' : Type u₂
inst_2 : Category.{max v₁ u₁, u₂} A'
B : Type u₃
inst_3 : Category.{v₃, u₃} B
J : GrothendieckTopology.{v₁, u₁} C
U_1 : C
R_1 : Presieve.{v₁, u₁} U_1
P_1 : Functor.{v₁, v₂, u₁, u₂} Cᵒᵖ A
P' : Cᵒᵖ ⥤ A'
inst_4 : HasProducts A
inst_5 : HasProducts A'
inst_6 : HasPullbacks.{v₁, u₁} C
P : Functor.{v₁, v₂, u₁, u₂} Cᵒᵖ A
s : A ⥤ Type (max v₁ u₁)
inst_7 : ∀ (J : Type (max v₁ u₁)), PreservesLimitsOfShape (Discrete J) s
U : C
R : Presieve.{v₁, u₁} U
e : parallelPair (X := s.obj (firstObj R P)) (Y := s.obj (secondObj R P)) (s.map (firstMap R P))
    (s.map (secondMap R P)) ≅
  parallelPair (Y := Equalizer.Presieve.SecondObj (P ⋙ s) R) (Equalizer.Presieve.firstMap (P ⋙ s) R)
    (Equalizer.Presieve.secondMap (P ⋙ s) R) :=
  parallelPair.ext
    (PreservesProduct.iso s fun (f : (V : C) × { f : V ⟶ U // R f }) ↦
      P.obj (op (f.fst (β := fun (V : C) ↦ { f : V ⟶ U // R f }))))
    (PreservesProduct.iso s fun (fg : ((V : C) × { f : V ⟶ U // R f }) × (W : C) × { g : W ⟶ U // R g }) ↦
      P.obj
        (op
          (Limits.pullback (↑(fg.1.snd (β := fun (V : C) ↦ { f : V ⟶ U // R f })) : fg.1.fst ⟶ U)
            (↑(fg.2.snd (β := fun (W : C) ↦ { g : W ⟶ U // R g })) : fg.2.fst ⟶ U))))
    ⋯ ⋯
⊢ Iso.hom (Iso.refl ((Cone.postcompose e.hom).obj (Fork.ofι (s.map (forkMap R P)) ⋯)).pt) ≫
      Fork.ι (Fork.ofι (Equalizer.forkMap (P ⋙ s) R) ⋯) =
    Fork.ι ((Cone.postcompose e.hom).obj (Fork.ofι (s.map (forkMap R P)) ⋯))
have dsimp : Eq (α := s.obj (P.obj (op U)) ⟶ Equalizer.FirstObj (P ⋙ s) R)
  (𝟙 (s.obj (P.obj (op U))) ≫
    Pi.lift (f := fun (f : (Y : C) × { f : Y ⟶ U // R f }) ↦
      s.obj (P.obj (op (f.fst (β := fun (Y : C) ↦ { f : Y ⟶ U // R f }))))) fun (f : (Y : C) × { f : Y ⟶ U // R f }) ↦
      s.map (P.map (↑(f.snd (β := fun (Y : C) ↦ { f : Y ⟶ U // R f })) : f.fst ⟶ U).op))
  (CategoryStruct.comp (Y := s.obj (firstObj R P))
    (s.map
      (Pi.lift fun (f : (V : C) × { f : V ⟶ U // R f }) ↦
        P.map (↑(f.snd (β := fun (V : C) ↦ { f : V ⟶ U // R f })) : f.fst ⟶ U).op))
    (piComparison s fun (f : (V : C) × { f : V ⟶ U // R f }) ↦
      P.obj (op (f.fst (β := fun (V : C) ↦ { f : V ⟶ U // R f })))))
Tactic: dsimp [Equalizer.forkMap, forkMap, e, Fork.ι]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Sheaf (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A' : Type u₂
inst_1 : Category.{max v₁ u₁, u₂} A'
J : GrothendieckTopology.{v₁, u₁} C
P' : Cᵒᵖ ⥤ A'
inst_2 : HasProducts A'
inst_3 : HasPullbacks.{v₁, u₁} C
⊢ IsSheaf J P' ↔ IsSheaf' J P'
have mp : IsSheaf J P' → IsSheaf' J P'
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Sheaf (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A' : Type u₂
inst_1 : Category.{max v₁ u₁, u₂} A'
J : GrothendieckTopology.{v₁, u₁} C
P' : Cᵒᵖ ⥤ A'
inst_2 : HasProducts A'
inst_3 : HasPullbacks.{v₁, u₁} C
mp : IsSheaf J P' → IsSheaf' J P'
⊢ IsSheaf J P' ↔ IsSheaf' J P'
have mpr : IsSheaf' J P' → IsSheaf J P'
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Sheaf (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A' : Type u₂
inst_1 : Category.{max v₁ u₁, u₂} A'
J : GrothendieckTopology.{v₁, u₁} C
P' : Cᵒᵖ ⥤ A'
inst_2 : HasProducts A'
inst_3 : HasPullbacks.{v₁, u₁} C
h : IsSheaf J P'
U : C
R : Presieve.{v₁, u₁} U
hR : generate R ∈ (J : (a : C) → Set (Sieve a)) U
⊢ Nonempty
    (IsLimit (F := parallelPair (X := firstObj R P') (Y := secondObj R P') (firstMap R P') (secondMap R P'))
      (Fork.ofι (forkMap R P') ⋯))
have mp : IsLimit (F := parallelPair (X := firstObj R P') (Y := secondObj R P') (firstMap R P') (secondMap R P'))
  (Fork.ofι (forkMap R P') sorry)
Tactic: refine ⟨?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Sheaf (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A' : Type u₂
inst_1 : Category.{max v₁ u₁, u₂} A'
J : GrothendieckTopology.{v₁, u₁} C
P' : Cᵒᵖ ⥤ A'
inst_2 : HasProducts A'
inst_3 : HasPullbacks.{v₁, u₁} C
h : IsSheaf J P'
U : C
R : Presieve.{v₁, u₁} U
hR : generate R ∈ (J : (a : C) → Set (Sieve a)) U
⊢ IsLimit (F := parallelPair (X := firstObj R P') (Y := secondObj R P') (firstMap R P') (secondMap R P'))
    (Fork.ofι (forkMap R P') ⋯)
have mp : (X : A'ᵒᵖ) →
  IsLimit (F :=
    parallelPair (X := firstObj R P') (Y := secondObj R P') (firstMap R P') (secondMap R P') ⋙ Functor.obj coyoneda X)
    (Functor.mapCone (Functor.obj coyoneda X) (Fork.ofι (forkMap R P') sorry))
Tactic: apply coyonedaJointlyReflectsLimits
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Sheaf (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A' : Type u₂
inst_1 : Category.{max v₁ u₁, u₂} A'
J : GrothendieckTopology.{v₁, u₁} C
P' : Cᵒᵖ ⥤ A'
inst_2 : HasProducts A'
inst_3 : HasPullbacks.{v₁, u₁} C
h : IsSheaf J P'
U : C
R : Presieve.{v₁, u₁} U
hR : generate R ∈ (J : (a : C) → Set (Sieve a)) U
X : A'ᵒᵖ
⊢ IsLimit (F :=
    parallelPair (X := firstObj R P') (Y := secondObj R P') (firstMap R P') (secondMap R P') ⋙ Functor.obj coyoneda X)
    (Functor.mapCone (Functor.obj coyoneda X) (Fork.ofι (forkMap R P') ⋯))
have q : Presieve.IsSheafFor (P' ⋙ Functor.obj coyoneda X) (generate R).arrows
Tactic: have q : Presieve.IsSheafFor (P' ⋙ coyoneda.obj X) _ := h X.unop _ hR
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Sheaf (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A' : Type u₂
inst_1 : Category.{max v₁ u₁, u₂} A'
J : GrothendieckTopology.{v₁, u₁} C
P' : Cᵒᵖ ⥤ A'
inst_2 : HasProducts A'
inst_3 : HasPullbacks.{v₁, u₁} C
h : IsSheaf J P'
U : C
R : Presieve.{v₁, u₁} U
hR : generate R ∈ (J : (a : C) → Set (Sieve a)) U
X : A'ᵒᵖ
q : Presieve.IsSheafFor (P' ⋙ Functor.obj coyoneda X) (generate R).arrows
⊢ IsLimit (F :=
    parallelPair (X := firstObj R P') (Y := secondObj R P') (firstMap R P') (secondMap R P') ⋙ Functor.obj coyoneda X)
    (Functor.mapCone (Functor.obj coyoneda X) (Fork.ofι (forkMap R P') ⋯))
have q₁ : Presieve.IsSheafFor (P' ⋙ Functor.obj coyoneda X) R
Tactic: rw [← Presieve.isSheafFor_iff_generate] at q
Cleared hypotheses: q
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Sheaf (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A' : Type u₂
inst_1 : Category.{max v₁ u₁, u₂} A'
J : GrothendieckTopology.{v₁, u₁} C
P' : Cᵒᵖ ⥤ A'
inst_2 : HasProducts A'
inst_3 : HasPullbacks.{v₁, u₁} C
h : IsSheaf J P'
U : C
R : Presieve.{v₁, u₁} U
hR : generate R ∈ (J : (a : C) → Set (Sieve a)) U
X : A'ᵒᵖ
q : Presieve.IsSheafFor (P' ⋙ Functor.obj coyoneda X) R
⊢ IsLimit (F :=
    parallelPair (X := firstObj R P') (Y := secondObj R P') (firstMap R P') (secondMap R P') ⋙ Functor.obj coyoneda X)
    (Functor.mapCone (Functor.obj coyoneda X) (Fork.ofι (forkMap R P') ⋯))
have q₁ : Nonempty
  (IsLimit (F :=
    parallelPair (Y := Equalizer.Presieve.SecondObj (P' ⋙ Functor.obj coyoneda X) R)
      (Equalizer.Presieve.firstMap (P' ⋙ Functor.obj coyoneda X) R)
      (Equalizer.Presieve.secondMap (P' ⋙ Functor.obj coyoneda X) R))
    (Fork.ofι (Equalizer.forkMap (P' ⋙ Functor.obj coyoneda X) R) sorry))
Tactic: rw [Equalizer.Presieve.sheaf_condition] at q
Cleared hypotheses: q
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Sheaf (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A' : Type u₂
inst_1 : Category.{max v₁ u₁, u₂} A'
J : GrothendieckTopology.{v₁, u₁} C
P' : Cᵒᵖ ⥤ A'
inst_2 : HasProducts A'
inst_3 : HasPullbacks.{v₁, u₁} C
h : IsSheaf J P'
U : C
R : Presieve.{v₁, u₁} U
hR : generate R ∈ (J : (a : C) → Set (Sieve a)) U
X : A'ᵒᵖ
q :
  Nonempty
    (IsLimit (F :=
      parallelPair (Y := Equalizer.Presieve.SecondObj (P' ⋙ Functor.obj coyoneda X) R)
        (Equalizer.Presieve.firstMap (P' ⋙ Functor.obj coyoneda X) R)
        (Equalizer.Presieve.secondMap (P' ⋙ Functor.obj coyoneda X) R))
      (Fork.ofι (Equalizer.forkMap (P' ⋙ Functor.obj coyoneda X) R) ⋯))
⊢ IsLimit (F :=
    parallelPair (X := firstObj R P') (Y := secondObj R P') (firstMap R P') (secondMap R P') ⋙ Functor.obj coyoneda X)
    (Functor.mapCone (Functor.obj coyoneda X) (Fork.ofι (forkMap R P') ⋯))
have mp : (q :
    IsLimit (F :=
      parallelPair (Y := Equalizer.Presieve.SecondObj (P' ⋙ Functor.obj coyoneda X) R)
        (Equalizer.Presieve.firstMap (P' ⋙ Functor.obj coyoneda X) R)
        (Equalizer.Presieve.secondMap (P' ⋙ Functor.obj coyoneda X) R))
      (Fork.ofι (Equalizer.forkMap (P' ⋙ Functor.obj coyoneda X) R) sorry)) →
  IsLimit (F :=
    parallelPair (X := firstObj R P') (Y := secondObj R P') (firstMap R P') (secondMap R P') ⋙ Functor.obj coyoneda X)
    (Functor.mapCone (Functor.obj coyoneda X) (Fork.ofι (forkMap R P') sorry))
Tactic: replace q := Classical.choice q
Cleared hypotheses: q
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Sheaf (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A' : Type u₂
inst_1 : Category.{max v₁ u₁, u₂} A'
J : GrothendieckTopology.{v₁, u₁} C
P' : Cᵒᵖ ⥤ A'
inst_2 : HasProducts A'
inst_3 : HasPullbacks.{v₁, u₁} C
h : IsSheaf' J P'
U : A'
X : C
S : Sieve.{v₁, u₁} X
hS : S ∈ (J : (a : C) → Set (Sieve a)) X
⊢ Presieve.IsSheafFor (P' ⋙ Functor.obj coyoneda (op U)) S.arrows
have mpr : Nonempty
  (IsLimit (F :=
    parallelPair (Y := Equalizer.Presieve.SecondObj (P' ⋙ Functor.obj coyoneda (op U)) S.arrows)
      (Equalizer.Presieve.firstMap (P' ⋙ Functor.obj coyoneda (op U)) S.arrows)
      (Equalizer.Presieve.secondMap (P' ⋙ Functor.obj coyoneda (op U)) S.arrows))
    (Fork.ofι (Equalizer.forkMap (P' ⋙ Functor.obj coyoneda (op U)) S.arrows) sorry))
Tactic: rw [Equalizer.Presieve.sheaf_condition]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Sheaf (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A' : Type u₂
inst_1 : Category.{max v₁ u₁, u₂} A'
J : GrothendieckTopology.{v₁, u₁} C
P' : Cᵒᵖ ⥤ A'
inst_2 : HasProducts A'
inst_3 : HasPullbacks.{v₁, u₁} C
h : IsSheaf' J P'
U : A'
X : C
S : Sieve.{v₁, u₁} X
hS : S ∈ (J : (a : C) → Set (Sieve a)) X
⊢ Nonempty
    (IsLimit (F :=
      parallelPair (Y := Equalizer.Presieve.SecondObj (P' ⋙ Functor.obj coyoneda (op U)) S.arrows)
        (Equalizer.Presieve.firstMap (P' ⋙ Functor.obj coyoneda (op U)) S.arrows)
        (Equalizer.Presieve.secondMap (P' ⋙ Functor.obj coyoneda (op U)) S.arrows))
      (Fork.ofι (Equalizer.forkMap (P' ⋙ Functor.obj coyoneda (op U)) S.arrows) ⋯))
have mpr : IsLimit (F :=
  parallelPair (Y := Equalizer.Presieve.SecondObj (P' ⋙ Functor.obj coyoneda (op U)) S.arrows)
    (Equalizer.Presieve.firstMap (P' ⋙ Functor.obj coyoneda (op U)) S.arrows)
    (Equalizer.Presieve.secondMap (P' ⋙ Functor.obj coyoneda (op U)) S.arrows))
  (Fork.ofι (Equalizer.forkMap (P' ⋙ Functor.obj coyoneda (op U)) S.arrows) sorry)
Tactic: refine ⟨?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Sheaf (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A' : Type u₂
inst_1 : Category.{max v₁ u₁, u₂} A'
J : GrothendieckTopology.{v₁, u₁} C
P' : Cᵒᵖ ⥤ A'
inst_2 : HasProducts A'
inst_3 : HasPullbacks.{v₁, u₁} C
h : IsSheaf' J P'
U : A'
X : C
S : Sieve.{v₁, u₁} X
hS : S ∈ (J : (a : C) → Set (Sieve a)) X
⊢ IsLimit (F :=
    parallelPair (Y := Equalizer.Presieve.SecondObj (P' ⋙ Functor.obj coyoneda (op U)) S.arrows)
      (Equalizer.Presieve.firstMap (P' ⋙ Functor.obj coyoneda (op U)) S.arrows)
      (Equalizer.Presieve.secondMap (P' ⋙ Functor.obj coyoneda (op U)) S.arrows))
    (Fork.ofι (Equalizer.forkMap (P' ⋙ Functor.obj coyoneda (op U)) S.arrows) ⋯)
have mpr : IsLimit (F :=
  parallelPair (X := firstObj S.arrows P') (Y := secondObj S.arrows P') (firstMap S.arrows P') (secondMap S.arrows P') ⋙
    Functor.obj coyoneda (op U))
  (Functor.mapCone (Functor.obj coyoneda (op U)) (Fork.ofι (forkMap S.arrows P') sorry))
Tactic: refine isSheafForIsSheafFor' _ _ _ _ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Sheaf (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A' : Type u₂
inst_1 : Category.{max v₁ u₁, u₂} A'
J : GrothendieckTopology.{v₁, u₁} C
P' : Cᵒᵖ ⥤ A'
inst_2 : HasProducts A'
inst_3 : HasPullbacks.{v₁, u₁} C
h : IsSheaf' J P'
U : A'
X : C
S : Sieve.{v₁, u₁} X
hS : S ∈ (J : (a : C) → Set (Sieve a)) X
this : PreservesLimitsOfSize.{0, 0, max u₁ v₁, max u₁ v₁, u₂, (max u₁ v₁) + 1} (Functor.obj coyoneda (op U)) := ⋯
⊢ IsLimit (F :=
    parallelPair (X := firstObj S.arrows P') (Y := secondObj S.arrows P') (firstMap S.arrows P')
        (secondMap S.arrows P') ⋙
      Functor.obj coyoneda (op U))
    (Functor.mapCone (Functor.obj coyoneda (op U)) (Fork.ofι (forkMap S.arrows P') ⋯))
have mpr : IsLimit (F :=
  parallelPair (X := firstObj S.arrows P') (Y := secondObj S.arrows P') (firstMap S.arrows P') (secondMap S.arrows P'))
  (Fork.ofι (forkMap S.arrows P') sorry)
Tactic: apply isLimitOfPreserves
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Sheaf (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A' : Type u₂
inst_1 : Category.{max v₁ u₁, u₂} A'
J : GrothendieckTopology.{v₁, u₁} C
P' : Cᵒᵖ ⥤ A'
inst_2 : HasProducts A'
inst_3 : HasPullbacks.{v₁, u₁} C
h : IsSheaf' J P'
U : A'
X : C
S : Sieve.{v₁, u₁} X
hS : S ∈ (J : (a : C) → Set (Sieve a)) X
this : PreservesLimitsOfSize.{0, 0, max u₁ v₁, max u₁ v₁, u₂, (max u₁ v₁) + 1} (Functor.obj coyoneda (op U)) := ⋯
⊢ IsLimit (F :=
    parallelPair (X := firstObj S.arrows P') (Y := secondObj S.arrows P') (firstMap S.arrows P')
      (secondMap S.arrows P'))
    (Fork.ofι (forkMap S.arrows P') ⋯)
have apply : generate S.arrows ∈ (J : (a : C) → Set (Sieve a)) X
Tactic: apply Classical.choice (h _ S.arrows _)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Sheaf (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A : Type u₂
inst_1 : Category.{v₂, u₂} A
B : Type u₃
inst_2 : Category.{v₃, u₃} B
J : GrothendieckTopology.{v₁, u₁} C
P : Functor.{v₁, v₂, u₁, u₂} Cᵒᵖ A
s : Functor.{v₂, v₃, u₂, u₃} A B
inst_3 : ReflectsLimitsOfSize.{v₁, max v₁ u₁, v₂, v₃, u₂, u₃} s
h : IsSheaf J (P ⋙ s)
⊢ IsSheaf J P
have rw : ∀ (h : ∀ ⦃X : C⦄, ∀ S ∈ (J : (a : C) → Set (Sieve a)) X, Nonempty (IsLimit ((P ⋙ s).mapCone S.arrows.cocone.op)))
  ⦃X : C⦄, ∀ S ∈ (J : (a : C) → Set (Sieve a)) X, Nonempty (IsLimit (P.mapCone S.arrows.cocone.op))
Tactic: rw [isSheaf_iff_isLimit] at h ⊢
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Sheaf (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A : Type u₂
inst_1 : Category.{v₂, u₂} A
B : Type u₃
inst_2 : Category.{v₃, u₃} B
J : GrothendieckTopology.{v₁, u₁} C
P : Functor.{v₁, v₂, u₁, u₂} Cᵒᵖ A
s : Functor.{v₂, v₃, u₂, u₃} A B
inst_3 : PreservesLimitsOfSize.{v₁, max v₁ u₁, v₂, v₃, u₂, u₃} s
h : IsSheaf J P
⊢ IsSheaf J (P ⋙ s)
have rw : ∀ (h : ∀ ⦃X : C⦄, ∀ S ∈ (J : (a : C) → Set (Sieve a)) X, Nonempty (IsLimit (P.mapCone S.arrows.cocone.op))) ⦃X : C⦄,
  ∀ S ∈ (J : (a : C) → Set (Sieve a)) X, Nonempty (IsLimit ((P ⋙ s).mapCone S.arrows.cocone.op))
Tactic: rw [isSheaf_iff_isLimit] at h ⊢
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Sheaf (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A' : Type u₂
inst_1 : Category.{max v₁ u₁, u₂} A'
J : GrothendieckTopology.{v₁, u₁} C
P' : Cᵒᵖ ⥤ A'
s : A' ⥤ Type (max v₁ u₁)
inst_2 : HasLimits A'
inst_3 : PreservesLimits s
inst_4 : s.ReflectsIsomorphisms
⊢ IsSheaf J P' ↔ IsSheaf J (P' ⋙ s)
have this : HasLimitsOfSize.{v₁, max v₁ u₁, max u₁ v₁, u₂} A'
Tactic: have : HasLimitsOfSize.{v₁, max v₁ u₁} A' := hasLimitsOfSizeShrink.{_, _, u₁, 0} A'
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Sheaf (hypothesis)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A' : Type u₂
inst_1 : Category.{max v₁ u₁, u₂} A'
J : GrothendieckTopology.{v₁, u₁} C
P' : Cᵒᵖ ⥤ A'
s : A' ⥤ Type (max v₁ u₁)
inst_2 : HasLimits A'
inst_3 : PreservesLimits s
inst_4 : s.ReflectsIsomorphisms
this : HasLimitsOfSize.{v₁, max v₁ u₁, max u₁ v₁, u₂} A'
⊢ IsSheaf J P' ↔ IsSheaf J (P' ⋙ s)
have this₁ : PreservesLimitsOfSize.{v₁, max v₁ u₁, max u₁ v₁, max u₁ v₁, u₂, max (u₁ + 1) (v₁ + 1)} s
Tactic: have : PreservesLimitsOfSize.{v₁, max v₁ u₁} s := preservesLimitsOfSize_shrink.{_, 0, _, u₁} s