awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.SheafComparison (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
F : Functor.{v_1, v_2, u_1, u_2} C D
inst_2 : F.PreservesEffectiveEpis
inst_3 : F.ReflectsEffectiveEpis
inst_4 : F.Full
inst_5 : F.Faithful
inst_6 : F.EffectivelyEnough
inst_7 : Preregular.{v_2, u_2} D
X : C
S : Sieve.{v_1, u_1} X
this : Preregular.{v_1, u_1} C
⊢ S ∈ (regularTopology C : (a : C) → Set (Sieve a)) X ↔ ∃ (Y : C) (π : Y ⟶ X), EffectiveEpi π ∧ S.arrows π
have h : S ∈ (regularTopology C : (a : C) → Set (Sieve a)) X ↔ S ∈ (regularTopology C : (a : C) → Set (Sieve a)) X
Tactic: rw [← mem_sieves_iff_hasEffectiveEpi S]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.SheafComparison (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
F : Functor.{v_1, v_2, u_1, u_2} C D
inst_2 : F.PreservesEffectiveEpis
inst_3 : F.ReflectsEffectiveEpis
inst_4 : F.Full
inst_5 : F.Faithful
inst_6 : F.EffectivelyEnough
inst_7 : Preregular.{v_2, u_2} D
⊢ ∀ {X : C} {S : Sieve.{v_1, u_1} X},
    Membership.mem (γ := Set (Sieve.{v_2, u_2} (F.obj X))) ((regularTopology D : (a : D) → Set (Sieve a)) (F.obj X))
        (Sieve.functorPushforward F S) ↔
      S ∈ (regularTopology C : (a : C) → Set (Sieve a)) X
have rw : ∀ {X : C} {S : Sieve.{v_1, u_1} X},
  Membership.mem (γ := Set (Sieve.{v_2, u_2} (F.obj X))) ((regularTopology D : (a : D) → Set (Sieve a)) (F.obj X))
      (Sieve.functorPushforward F S) ↔
    S ∈ (F.inducedTopology (regularTopology D) : (a : C) → Set (Sieve a)) X
Tactic: rw [eq_induced F]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.SheafComparison (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
F : Functor.{v_1, v₃, u_1, u₃} Cᵒᵖ A
inst_2 : Preregular.{v_1, u_1} C
inst_3 : FinitaryPreExtensive.{v_1, u_1} C
⊢ IsSheaf (coherentTopology C) F ↔ IsSheaf (extensiveTopology C) F ∧ IsSheaf (regularTopology C) F
have rw : IsSheaf (Coverage.toGrothendieck (extensiveCoverage C ⊔ regularCoverage C)) F ↔
  IsSheaf (extensiveTopology C) F ∧ IsSheaf (regularTopology C) F
Tactic: rw [← extensive_regular_generate_coherent]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.SheafComparison (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
F : Functor.{v_1, v₃, u_1, u₃} Cᵒᵖ A
inst_2 : Preregular.{v_1, u_1} C
inst_3 : FinitaryExtensive.{v_1, u_1} C
h : ∀ {Y X : C} (f : Y ⟶ X) [EffectiveEpi f], HasPullback f f
⊢ IsSheaf (coherentTopology C) F ↔ PreservesFiniteProducts F ∧ EqualizerCondition F
have rw : IsSheaf (extensiveTopology C) F ∧ IsSheaf (regularTopology C) F ↔ PreservesFiniteProducts F ∧ EqualizerCondition F
Tactic: rw [isSheaf_coherent_iff_regular_and_extensive]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.SheafComparison (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
F : Functor.{v_1, v₃, u_1, u₃} Cᵒᵖ A
inst_2 : Preregular.{v_1, u_1} C
inst_3 : FinitaryExtensive.{v_1, u_1} C
inst_4 : ∀ (X : C), Projective.{v_1, u_1} X
⊢ IsSheaf (coherentTopology C) F ↔ PreservesFiniteProducts F
have rw : IsSheaf (extensiveTopology C) F ∧ IsSheaf (regularTopology C) F ↔ PreservesFiniteProducts F
Tactic: rw [isSheaf_coherent_iff_regular_and_extensive, and_iff_left (isSheaf_of_projective F),
    isSheaf_iff_preservesFiniteProducts]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.SheafComparison (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
F : Functor.{v_1, v₃, u_1, u₃} Cᵒᵖ A
inst_2 : Preregular.{v_1, u_1} C
inst_3 : FinitaryExtensive.{v_1, u_1} C
inst_4 : ∀ (X : C), Projective.{v_1, u_1} X
rw : IsSheaf (extensiveTopology C) F ∧ IsSheaf (regularTopology C) F ↔ PreservesFiniteProducts F
⊢ IsSheaf (coherentTopology C) F ↔ PreservesFiniteProducts F
have rw₁ : IsSheaf (extensiveTopology C) F ↔ PreservesFiniteProducts F
Tactic: rw [isSheaf_coherent_iff_regular_and_extensive, and_iff_left (isSheaf_of_projective F),
    isSheaf_iff_preservesFiniteProducts]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.SheafComparison (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
F : Functor.{v_1, v₃, u_1, u₃} Cᵒᵖ A
inst_2 : Preregular.{v_1, u_1} C
inst_3 : FinitaryExtensive.{v_1, u_1} C
inst_4 : ∀ (X : C), Projective.{v_1, u_1} X
rw : IsSheaf (extensiveTopology C) F ∧ IsSheaf (regularTopology C) F ↔ PreservesFiniteProducts F
rw₁ : IsSheaf (extensiveTopology C) F ↔ PreservesFiniteProducts F
⊢ IsSheaf (coherentTopology C) F ↔ PreservesFiniteProducts F
have rw₂ : PreservesFiniteProducts F ↔ PreservesFiniteProducts F
Tactic: rw [isSheaf_coherent_iff_regular_and_extensive, and_iff_left (isSheaf_of_projective F),
    isSheaf_iff_preservesFiniteProducts]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.SheafComparison (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
F : Functor.{v_1, v₃, u_1, u₃} Cᵒᵖ A
inst_2 : Preregular.{v_1, u_1} C
inst_3 : FinitaryExtensive.{v_1, u_1} C
inst_4 : ∀ (X : C), Projective.{v_1, u_1} X
⊢ IsSheaf (coherentTopology C) F ↔ IsSheaf (extensiveTopology C) F
have rw : PreservesFiniteProducts F ↔ IsSheaf (extensiveTopology C) F
Tactic: rw [isSheaf_iff_preservesFiniteProducts_of_projective, isSheaf_iff_preservesFiniteProducts]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.SheafComparison (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
F : Functor.{v_1, v₃, u_1, u₃} Cᵒᵖ A
inst_2 : Preregular.{v_1, u_1} C
inst_3 : FinitaryExtensive.{v_1, u_1} C
inst_4 : ∀ (X : C), Projective.{v_1, u_1} X
rw : PreservesFiniteProducts F ↔ IsSheaf (extensiveTopology C) F
⊢ IsSheaf (coherentTopology C) F ↔ IsSheaf (extensiveTopology C) F
have rw₁ : PreservesFiniteProducts F ↔ PreservesFiniteProducts F
Tactic: rw [isSheaf_iff_preservesFiniteProducts_of_projective, isSheaf_iff_preservesFiniteProducts]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.SheafComparison (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
F : Functor.{v_1, v₃, u_1, u₃} Cᵒᵖ A
B : Type u₄
inst_2 : Category.{v₄, u₄} B
s : Functor.{v₃, v₄, u₃, u₄} A B
inst_3 : Preregular.{v_1, u_1} C
inst_4 : FinitaryExtensive.{v_1, u_1} C
h : ∀ {Y X : C} (f : Y ⟶ X) [EffectiveEpi f], HasPullback f f
inst_5 : PreservesFiniteLimits s
hF : IsSheaf (coherentTopology C) F
⊢ IsSheaf (coherentTopology C) (F ⋙ s)
have rw : ∀ (hF : PreservesFiniteProducts F ∧ EqualizerCondition F), PreservesFiniteProducts (F ⋙ s) ∧ EqualizerCondition (F ⋙ s)
Tactic: rw [isSheaf_iff_preservesFiniteProducts_and_equalizerCondition (h := h)] at hF ⊢
Cleared hypotheses: hF
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.SheafComparison (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
F : Functor.{v_1, v₃, u_1, u₃} Cᵒᵖ A
B : Type u₄
inst_2 : Category.{v₄, u₄} B
s : Functor.{v₃, v₄, u₃, u₄} A B
inst_3 : Preregular.{v_1, u_1} C
inst_4 : FinitaryExtensive.{v_1, u_1} C
h : ∀ {Y X : C} (f : Y ⟶ X) [EffectiveEpi f], HasPullback f f
inst_5 : PreservesFiniteLimits s
hF : PreservesFiniteProducts F ∧ EqualizerCondition F
⊢ PreservesFiniteProducts (F ⋙ s) ∧ EqualizerCondition (F ⋙ s)
have this : PreservesFiniteProducts F
Tactic: have := hF.1
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.SheafComparison (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
F : Functor.{v_1, v₃, u_1, u₃} Cᵒᵖ A
B : Type u₄
inst_2 : Category.{v₄, u₄} B
s : Functor.{v₃, v₄, u₃, u₄} A B
inst_3 : Preregular.{v_1, u_1} C
inst_4 : FinitaryExtensive.{v_1, u_1} C
h : ∀ {Y X : C} (f : Y ⟶ X) [EffectiveEpi f], HasPullback f f
inst_5 : PreservesFiniteLimits s
hF : PreservesFiniteProducts F ∧ EqualizerCondition F
this : PreservesFiniteProducts F
⊢ PreservesFiniteProducts (F ⋙ s) ∧ EqualizerCondition (F ⋙ s)
have refine : (x x_1 : C) →
  (π : x ⟶ x_1) → EffectiveEpi π → (c : PullbackCone π π) → (hc : IsLimit c) → IsLimit (Fork.ofι ((F ⋙ s).map π.op) sorry)
Tactic: refine ⟨inferInstance, fun _ _ π _ c hc ↦ ⟨?_⟩⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.SheafComparison (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
F : Functor.{v_1, v₃, u_1, u₃} Cᵒᵖ A
B : Type u₄
inst_2 : Category.{v₄, u₄} B
s : Functor.{v₃, v₄, u₃, u₄} A B
inst_3 : Preregular.{v_1, u_1} C
inst_4 : FinitaryExtensive.{v_1, u_1} C
h : ∀ {Y X : C} (f : Y ⟶ X) [EffectiveEpi f], HasPullback f f
inst_5 : ReflectsFiniteLimits s
hF : IsSheaf (coherentTopology C) (F ⋙ s)
⊢ IsSheaf (coherentTopology C) F
have rw : ∀ (hF : PreservesFiniteProducts (F ⋙ s) ∧ EqualizerCondition (F ⋙ s)), PreservesFiniteProducts F ∧ EqualizerCondition F
Tactic: rw [isSheaf_iff_preservesFiniteProducts_and_equalizerCondition (h := h)] at hF ⊢
Cleared hypotheses: hF
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.SheafComparison (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
F : Functor.{v_1, v₃, u_1, u₃} Cᵒᵖ A
B : Type u₄
inst_2 : Category.{v₄, u₄} B
s : Functor.{v₃, v₄, u₃, u₄} A B
inst_3 : Preregular.{v_1, u_1} C
inst_4 : FinitaryExtensive.{v_1, u_1} C
h : ∀ {Y X : C} (f : Y ⟶ X) [EffectiveEpi f], HasPullback f f
inst_5 : ReflectsFiniteLimits s
hF : PreservesFiniteProducts (F ⋙ s) ∧ EqualizerCondition (F ⋙ s)
⊢ PreservesFiniteProducts F ∧ EqualizerCondition F
have left : PreservesFiniteProducts (F ⋙ s)
Tactic: obtain ⟨_, hF₂⟩ := hF
Cleared hypotheses: hF
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.SheafComparison (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
F : Functor.{v_1, v₃, u_1, u₃} Cᵒᵖ A
B : Type u₄
inst_2 : Category.{v₄, u₄} B
s : Functor.{v₃, v₄, u₃, u₄} A B
inst_3 : Preregular.{v_1, u_1} C
inst_4 : FinitaryExtensive.{v_1, u_1} C
h : ∀ {Y X : C} (f : Y ⟶ X) [EffectiveEpi f], HasPullback f f
inst_5 : ReflectsFiniteLimits s
hF : PreservesFiniteProducts (F ⋙ s) ∧ EqualizerCondition (F ⋙ s)
left : PreservesFiniteProducts (F ⋙ s)
⊢ PreservesFiniteProducts F ∧ EqualizerCondition F
have hF₂ : EqualizerCondition (F ⋙ s)
Tactic: obtain ⟨_, hF₂⟩ := hF
Cleared hypotheses: hF
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.SheafComparison (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
F : Functor.{v_1, v₃, u_1, u₃} Cᵒᵖ A
B : Type u₄
inst_2 : Category.{v₄, u₄} B
s : Functor.{v₃, v₄, u₃, u₄} A B
inst_3 : Preregular.{v_1, u_1} C
inst_4 : FinitaryExtensive.{v_1, u_1} C
h : ∀ {Y X : C} (f : Y ⟶ X) [EffectiveEpi f], HasPullback f f
inst_5 : ReflectsFiniteLimits s
left : PreservesFiniteProducts (F ⋙ s)
hF₂ : EqualizerCondition (F ⋙ s)
⊢ PreservesFiniteProducts F ∧ EqualizerCondition F
have refine_1 : ∀ (n : ℕ) {K : Functor.{0, v_1, 0, u_1} (Discrete (Fin n)) Cᵒᵖ} {c : Cone K} (hc : IsLimit c),
  Nonempty (IsLimit (F.mapCone c))
Tactic: refine ⟨⟨fun n ↦ ⟨fun {K} ↦ ⟨fun {c} hc ↦ ?_⟩⟩⟩, fun _ _ π _ c hc ↦ ⟨?_⟩⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.SheafComparison (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
F : Functor.{v_1, v₃, u_1, u₃} Cᵒᵖ A
B : Type u₄
inst_2 : Category.{v₄, u₄} B
s : Functor.{v₃, v₄, u₃, u₄} A B
inst_3 : Preregular.{v_1, u_1} C
inst_4 : FinitaryExtensive.{v_1, u_1} C
h : ∀ {Y X : C} (f : Y ⟶ X) [EffectiveEpi f], HasPullback f f
inst_5 : ReflectsFiniteLimits s
left : PreservesFiniteProducts (F ⋙ s)
hF₂ : EqualizerCondition (F ⋙ s)
refine_1 :
  ∀ (n : ℕ) {K : Functor.{0, v_1, 0, u_1} (Discrete (Fin n)) Cᵒᵖ} {c : Cone K} (hc : IsLimit c),
    Nonempty (IsLimit (F.mapCone c))
⊢ PreservesFiniteProducts F ∧ EqualizerCondition F
have refine_2 : (x x_1 : C) →
  (π : x ⟶ x_1) → EffectiveEpi π → (c : PullbackCone π π) → (hc : IsLimit c) → IsLimit (Fork.ofι (F.map π.op) sorry)
Tactic: refine ⟨⟨fun n ↦ ⟨fun {K} ↦ ⟨fun {c} hc ↦ ?_⟩⟩⟩, fun _ _ π _ c hc ↦ ⟨?_⟩⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.SheafComparison (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
F : Functor.{v_1, v₃, u_1, u₃} Cᵒᵖ A
B : Type u₄
inst_2 : Category.{v₄, u₄} B
s : Functor.{v₃, v₄, u₃, u₄} A B
inst_3 : Preregular.{v_1, u_1} C
inst_4 : FinitaryExtensive.{v_1, u_1} C
inst_5 : ∀ (X : C), Projective.{v_1, u_1} X
inst_6 : PreservesFiniteProducts s
hF : IsSheaf (coherentTopology C) F
⊢ IsSheaf (coherentTopology C) (F ⋙ s)
have rw : ∀ (hF : PreservesFiniteProducts F), PreservesFiniteProducts (F ⋙ s)
Tactic: rw [isSheaf_iff_preservesFiniteProducts_of_projective] at hF ⊢
Cleared hypotheses: hF
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.SheafComparison (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
F : Functor.{v_1, v₃, u_1, u₃} Cᵒᵖ A
B : Type u₄
inst_2 : Category.{v₄, u₄} B
s : Functor.{v₃, v₄, u₃, u₄} A B
inst_3 : Preregular.{v_1, u_1} C
inst_4 : FinitaryExtensive.{v_1, u_1} C
inst_5 : ∀ (X : C), Projective.{v_1, u_1} X
inst_6 : ReflectsFiniteProducts s
hF : IsSheaf (coherentTopology C) (F ⋙ s)
⊢ IsSheaf (coherentTopology C) F
have rw : ∀ (hF : PreservesFiniteProducts (F ⋙ s)), PreservesFiniteProducts F
Tactic: rw [isSheaf_iff_preservesFiniteProducts_of_projective] at hF ⊢
Cleared hypotheses: hF
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.CompatiblePlus (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
E : Type u_2
inst_2 : Category.{v_2, u_2} E
F : Functor.{v_1, v_2, u_1, u_2} D E
inst_3 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst_4 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst_5 : ∀ (X : C) (W : J.Cover X) (P : Functor.{v, v_1, u, u_1} Cᵒᵖ D), PreservesLimit (W.index P).multicospan F
P : Functor.{v, v_1, u, u_1} Cᵒᵖ D
X : C
W : (J.Cover X)ᵒᵖ
⊢ (J.diagram P X ⋙ F).obj W ≅ (J.diagram (P ⋙ F) X).obj W
have refine : (J.diagram P X ⋙ F).obj W ≅ limit (((unop W).index P).multicospan ⋙ F)
Tactic: refine ?_ ≪≫ HasLimit.isoOfNatIso (W.unop.multicospanComp _ _).symm
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.CompatiblePlus (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
E : Type u_2
inst_2 : Category.{v_2, u_2} E
F : Functor.{v_1, v_2, u_1, u_2} D E
inst_3 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst_4 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst_5 : ∀ (X : C) (W : J.Cover X) (P : Functor.{v, v_1, u, u_1} Cᵒᵖ D), PreservesLimit (W.index P).multicospan F
P : Functor.{v, v_1, u, u_1} Cᵒᵖ D
X : C
A B : (J.Cover X)ᵒᵖ
f : A ⟶ B
⊢ (J.diagram P X ⋙ F).map f ≫
      Iso.hom
        ((fun (W : (J.Cover X)ᵒᵖ) ↦
            (Iso.trans (Y := limit (((unop W).index P).multicospan ⋙ F))
                (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves F (limit.isLimit ((unop W).index P).multicospan))
                  (limit.isLimit (((unop W).index P).multicospan ⋙ F)))
                (HasLimit.isoOfNatIso (Cover.multicospanComp F P (unop W)).symm) :
              (J.diagram P X ⋙ F).obj W ≅ (J.diagram (P ⋙ F) X).obj W))
          B) =
    Iso.hom
        ((fun (W : (J.Cover X)ᵒᵖ) ↦
            (Iso.trans (Y := limit (((unop W).index P).multicospan ⋙ F))
                (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves F (limit.isLimit ((unop W).index P).multicospan))
                  (limit.isLimit (((unop W).index P).multicospan ⋙ F)))
                (HasLimit.isoOfNatIso (Cover.multicospanComp F P (unop W)).symm) :
              (J.diagram P X ⋙ F).obj W ≅ (J.diagram (P ⋙ F) X).obj W))
          A) ≫
      (J.diagram (P ⋙ F) X).map f
have dsimp : Eq (α := F.obj (multiequalizer ((unop A).index P)) ⟶ multiequalizer ((unop B).index (P ⋙ F)))
  (CategoryStruct.comp (Y := F.obj (multiequalizer ((unop B).index P)))
    (F.map
      (Multiequalizer.lift ((unop B).index P) (multiequalizer ((unop A).index P))
        (fun (I : (unop B).Arrow) ↦ Multiequalizer.ι ((unop A).index P) (I.map f.unop)) sorry))
    (CategoryStruct.comp (Y := limit (((unop B).index P).multicospan ⋙ F))
      (Iso.hom
        (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves F (limit.isLimit ((unop B).index P).multicospan))
          (limit.isLimit (((unop B).index P).multicospan ⋙ F))))
      (Iso.hom (HasLimit.isoOfNatIso (Cover.multicospanComp F P (unop B)).symm))))
  (CategoryStruct.comp (Y := multiequalizer ((unop A).index (P ⋙ F)))
    (CategoryStruct.comp (Y := limit (((unop A).index P).multicospan ⋙ F))
      (Iso.hom
        (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves F (limit.isLimit ((unop A).index P).multicospan))
          (limit.isLimit (((unop A).index P).multicospan ⋙ F))))
      (Iso.hom (HasLimit.isoOfNatIso (Cover.multicospanComp F P (unop A)).symm)))
    (Multiequalizer.lift ((unop B).index (P ⋙ F)) (multiequalizer ((unop A).index (P ⋙ F)))
      (fun (I : (unop B).Arrow) ↦ Multiequalizer.ι ((unop A).index (P ⋙ F)) (I.map f.unop)) sorry))
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.CompatiblePlus (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
E : Type u_2
inst_2 : Category.{v_2, u_2} E
F : Functor.{v_1, v_2, u_1, u_2} D E
inst_3 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst_4 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst_5 : ∀ (X : C) (W : J.Cover X) (P : Functor.{v, v_1, u, u_1} Cᵒᵖ D), PreservesLimit (W.index P).multicospan F
P : Functor.{v, v_1, u, u_1} Cᵒᵖ D
X : C
W : (J.Cover X)ᵒᵖ
i : (unop W).Arrow
⊢ Eq (α := (J.diagram P X ⋙ F).obj W ⟶ ((unop W).index (P ⋙ F)).left i)
    (CategoryStruct.comp (Y := (J.diagram (P ⋙ F) X).obj W)
        (NatTrans.app (Iso.hom (J.diagramCompIso F P X)) W : (J.diagram P X ⋙ F).obj W ⟶ (J.diagram (P ⋙ F) X).obj W)
        (Multiequalizer.ι ((unop W).index (P ⋙ F)) i :
          multiequalizer ((unop W).index (P ⋙ F)) ⟶ ((unop W).index (P ⋙ F)).left i) :
      (J.diagram P X ⋙ F).obj W ⟶ ((unop W).index (P ⋙ F)).left i)
    (F.map (Multiequalizer.ι ((unop W).index P) i) :
      F.obj (multiequalizer ((unop W).index P)) ⟶ F.obj (((unop W).index P).left i))
have delta : Eq (α := (J.diagram P X ⋙ F).obj W ⟶ ((unop W).index (P ⋙ F)).left i)
  (CategoryStruct.comp (Y := (J.diagram (P ⋙ F) X).obj W)
      (NatTrans.app
          (Iso.hom
            (NatIso.ofComponents
              (fun (W : (J.Cover X)ᵒᵖ) ↦
                Iso.trans (Y := limit (((unop W).index P).multicospan ⋙ F))
                  (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves F (limit.isLimit ((unop W).index P).multicospan))
                    (limit.isLimit (((unop W).index P).multicospan ⋙ F)))
                  (HasLimit.isoOfNatIso (Cover.multicospanComp F P (unop W)).symm))
              sorry))
          W :
        (J.diagram P X ⋙ F).obj W ⟶ (J.diagram (P ⋙ F) X).obj W)
      (Multiequalizer.ι ((unop W).index (P ⋙ F)) i :
        multiequalizer ((unop W).index (P ⋙ F)) ⟶ ((unop W).index (P ⋙ F)).left i) :
    (J.diagram P X ⋙ F).obj W ⟶ ((unop W).index (P ⋙ F)).left i)
  (F.map (Multiequalizer.ι ((unop W).index P) i) :
    F.obj (multiequalizer ((unop W).index P)) ⟶ F.obj (((unop W).index P).left i))
Tactic: delta diagramCompIso
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.CompatiblePlus (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
E : Type u_2
inst_2 : Category.{v_2, u_2} E
F : Functor.{v_1, v_2, u_1, u_2} D E
inst_3 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst_4 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst_5 : ∀ (X : C) (W : J.Cover X) (P : Functor.{v, v_1, u, u_1} Cᵒᵖ D), PreservesLimit (W.index P).multicospan F
P : Functor.{v, v_1, u, u_1} Cᵒᵖ D
inst_6 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst_7 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst_8 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
X : Cᵒᵖ
⊢ (J.plusObj P ⋙ F).obj X ≅ (J.plusObj (P ⋙ F)).obj X
have refine : (J.plusObj P ⋙ F).obj X ≅ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X) ⋙ F)
Tactic: refine ?_ ≪≫ HasColimit.isoOfNatIso (J.diagramCompIso F P X.unop)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.CompatiblePlus (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
E : Type u_2
inst_2 : Category.{v_2, u_2} E
F : Functor.{v_1, v_2, u_1, u_2} D E
inst_3 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst_4 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst_5 : ∀ (X : C) (W : J.Cover X) (P : Functor.{v, v_1, u, u_1} Cᵒᵖ D), PreservesLimit (W.index P).multicospan F
P : Functor.{v, v_1, u, u_1} Cᵒᵖ D
inst_6 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst_7 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst_8 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
X Y : Cᵒᵖ
f : X ⟶ Y
W : (J.Cover (unop X))ᵒᵖ
| (CategoryStruct.comp (Y := F.obj (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))))
      (F.map (colimit.ι (J.diagram P (unop X)) W))
      (Iso.hom
        (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (J.diagram P (unop X))))
          (colimit.isColimit (J.diagram P (unop X) ⋙ F)))) :
    F.obj (multiequalizer ((unop W).index P)) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X) ⋙ F))
have a : NatTrans.app (Cocone.ι (colimit.cocone (J.diagram P (unop X) ⋙ F))) W
Tactic: erw [(isColimitOfPreserves F (colimit.isColimit (J.diagram P X.unop))).fac]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.CompatiblePlus (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
E : Type u_2
inst_2 : Category.{v_2, u_2} E
F : Functor.{v_1, v_2, u_1, u_2} D E
inst_3 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst_4 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst_5 : ∀ (X : C) (W : J.Cover X) (P : Functor.{v, v_1, u, u_1} Cᵒᵖ D), PreservesLimit (W.index P).multicospan F
P : Functor.{v, v_1, u, u_1} Cᵒᵖ D
inst_6 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst_7 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst_8 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
X Y : Cᵒᵖ
f : X ⟶ Y
W : (J.Cover (unop X))ᵒᵖ
| CategoryStruct.comp (Y := F.obj (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))))
    (F.map (colimit.ι (J.diagram P (unop X)) W))
    (CategoryStruct.comp (Y := F.obj (colimit ((J.pullback f.unop).op ⋙ J.diagram P (unop Y))))
      (F.map (colimMap (J.diagramPullback P f.unop)))
      (F.map (colimit.pre (J.diagram P (unop Y)) (J.pullback f.unop).op)))
have a : F.map
  (CategoryStruct.comp (Y := colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
    (colimit.ι (J.diagram P (unop X)) W)
    (CategoryStruct.comp (Y := colimit ((J.pullback f.unop).op ⋙ J.diagram P (unop Y)))
      (colimMap (J.diagramPullback P f.unop)) (colimit.pre (J.diagram P (unop Y)) (J.pullback f.unop).op)))
Tactic: simp only [← F.map_comp]