awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_3
inst : Category.{u_2, u_3} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
J : GrothendieckTopology.{u_2, u_3} C
ι : Type (max u_3 u_2)
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
hf : Presieve.ofArrows X f ∈ J.toPrecoverage.coverings S
σ : ι → Type (max u_3 u_2)
Y : (i : ι) → σ i → C
g : (i : ι) → (j : σ i) → Y i j ⟶ X i
hg :
  ∀ (i : ι),
    Membership.mem (γ := Set (Presieve.{u_2, u_3} (X i))) (J.toPrecoverage.coverings (X i))
      (Presieve.ofArrows (Y i) (g i))
⊢ Sieve.generate (Presieve.bindOfArrows X f fun (i : ι) ↦ Presieve.ofArrows (Y i) (g i)) ∈
    (J : (a : C) → Set (Sieve a)) S

Tactic:
exact J.bindOfArrows hf hg

Premises used: CategoryTheory.GrothendieckTopology.bindOfArrows
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_3
inst : Category.{u_2, u_3} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
J : GrothendieckTopology.{u_2, u_3} C
ι : Type (max u_3 u_2)
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
hf : Presieve.ofArrows X f ∈ J.toPrecoverage.coverings S
Y : C
g : Y ⟶ S
P : ι → C
p₁ : (i : ι) → P i ⟶ Y
p₂ : (i : ι) → P i ⟶ X i
h : ∀ (i : ι), IsPullback (p₁ i) (p₂ i) g (f i)
rw : Sieve.pullback g (Sieve.ofArrows X f) ∈ (J : (a : C) → Set (Sieve a)) Y
⊢ Presieve.ofArrows P p₁ ∈ J.toPrecoverage.coverings Y

Tactic:
rw [mem_toPrecoverage_iff, ← Sieve.ofArrows, Sieve.ofArrows_eq_pullback_of_isPullback _ h]

Premises used: CategoryTheory.Sieve.ofArrows, CategoryTheory.GrothendieckTopology.mem_toPrecoverage_iff, CategoryTheory.Sieve.ofArrows_eq_pullback_of_isPullback
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_3
inst : Category.{u_2, u_3} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
J : GrothendieckTopology.{u_2, u_3} C
ι : Type (max u_3 u_2)
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
hf : Presieve.ofArrows X f ∈ J.toPrecoverage.coverings S
Y : C
g : Y ⟶ S
P : ι → C
p₁ : (i : ι) → P i ⟶ Y
p₂ : (i : ι) → P i ⟶ X i
h : ∀ (i : ι), IsPullback (p₁ i) (p₂ i) g (f i)
⊢ Sieve.pullback g (Sieve.ofArrows X f) ∈ (J : (a : C) → Set (Sieve a)) Y

Tactic:
J.pullback_stable _ hf

Premises used: CategoryTheory.GrothendieckTopology.pullback_stable
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_3
inst : Category.{u_2, u_3} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
J : GrothendieckTopology.{u_2, u_3} C
ι : Type (max u_3 u_2)
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
hf : Presieve.ofArrows X f ∈ J.toPrecoverage.coverings S
Y : C
g : Y ⟶ S
P : ι → C
p₁ : (i : ι) → P i ⟶ Y
p₂ : (i : ι) → P i ⟶ X i
h : ∀ (i : ι), IsPullback (p₁ i) (p₂ i) g (f i)
⊢ Sieve.pullback g (Sieve.ofArrows X f) ∈ (J : (a : C) → Set (Sieve a)) Y

Tactic:
exact J.pullback_stable _ hf

Premises used: CategoryTheory.GrothendieckTopology.pullback_stable
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_2
inst : Category.{u_3, u_2} C
K : Precoverage.{u_3, u_2} C
J : GrothendieckTopology.{u_3, u_2} C
⊢ K.toGrothendieck ≤ J ↔ K ≤ J.toPrecoverage

Tactic:
⟨fun h X R hR ↦ ?_, fun h ↦ ?_⟩

Premises used: Iff.intro
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_2
inst : Category.{u_3, u_2} C
K : Precoverage.{u_3, u_2} C
J : GrothendieckTopology.{u_3, u_2} C
refine_1 :
  ∀ (h : K.toGrothendieck ≤ J) (X : C) (R : Presieve.{u_3, u_2} X) (hR : R ∈ K.coverings X),
    R ∈ J.toPrecoverage.coverings X
refine_2 : ∀ (h : K ≤ J.toPrecoverage), K.toGrothendieck ≤ J
⊢ K.toGrothendieck ≤ J ↔ K ≤ J.toPrecoverage

Tactic:
refine ⟨fun h X R hR ↦ ?_, fun h ↦ ?_⟩
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_2
inst : Category.{u_3, u_2} C
K : Precoverage.{u_3, u_2} C
J : GrothendieckTopology.{u_3, u_2} C
h : K.toGrothendieck ≤ J
X : C
R : Presieve.{u_3, u_2} X
hR : R ∈ K.coverings X
⊢ Sieve.generate R ∈ (J : (a : C) → Set (Sieve a)) X

Tactic:
h _ (generate_mem_toGrothendieck hR)

Premises used: CategoryTheory.Precoverage.generate_mem_toGrothendieck
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_2
inst : Category.{u_3, u_2} C
K : Precoverage.{u_3, u_2} C
J : GrothendieckTopology.{u_3, u_2} C
h : K.toGrothendieck ≤ J
X : C
R : Presieve.{u_3, u_2} X
hR : R ∈ K.coverings X
⊢ Sieve.generate R ∈ (J : (a : C) → Set (Sieve a)) X

Tactic:
exact h _ (generate_mem_toGrothendieck hR)

Premises used: CategoryTheory.Precoverage.generate_mem_toGrothendieck
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_2
inst : Category.{u_3, u_2} C
K : Precoverage.{u_3, u_2} C
J : GrothendieckTopology.{u_3, u_2} C
h : K ≤ J.toPrecoverage
X_1 : C
R : Sieve.{u_3, u_2} X_1
X : C
S : Presieve.{u_3, u_2} X
hS : S ∈ K.coverings X
⊢ Sieve.generate S ∈ (J : (a : C) → Set (Sieve a)) X

Tactic:
h _ hS

Premises used: Set.Mem, setOf
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_2
inst : Category.{u_3, u_2} C
K : Precoverage.{u_3, u_2} C
J : GrothendieckTopology.{u_3, u_2} C
h : K ≤ J.toPrecoverage
X_1 : C
R : Sieve.{u_3, u_2} X_1
X : C
S : Presieve.{u_3, u_2} X
hS : S ∈ K.coverings X
⊢ Sieve.generate S ∈ (J : (a : C) → Set (Sieve a)) X

Tactic:
exact h _ hS
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_2
inst : Category.{u_3, u_2} C
K : Precoverage.{u_3, u_2} C
J : GrothendieckTopology.{u_3, u_2} C
h : K ≤ J.toPrecoverage
X_1 : C
R : Sieve.{u_3, u_2} X_1
X : C
⊢ ⊤ ∈ (J : (a : C) → Set (Sieve a)) X

Tactic:
grind
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_2
inst : Category.{u_3, u_2} C
K : Precoverage.{u_3, u_2} C
J : GrothendieckTopology.{u_3, u_2} C
h : K ≤ J.toPrecoverage
X_1 : C
R : Sieve.{u_3, u_2} X_1
X : C
S : Sieve.{u_3, u_2} X
h_1 : K.Saturate X S
Y : C
f : Y ⟶ X
a_ih : S ∈ (J : (a : C) → Set (Sieve a)) X
⊢ Sieve.pullback f S ∈ (J : (a : C) → Set (Sieve a)) Y

Tactic:
grind
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_2
inst : Category.{u_3, u_2} C
K : Precoverage.{u_3, u_2} C
J : GrothendieckTopology.{u_3, u_2} C
h : K ≤ J.toPrecoverage
X_1 : C
R_1 : Sieve.{u_3, u_2} X_1
X : C
S R : Sieve.{u_3, u_2} X
h_1 : K.Saturate X S
h_2 : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → K.Saturate Y (Sieve.pullback f R)
a_ih : S ∈ (J : (a : C) → Set (Sieve a)) X
a_ih_1 : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Sieve.pullback f R ∈ (J : (a : C) → Set (Sieve a)) Y
⊢ R ∈ (J : (a : C) → Set (Sieve a)) X

Tactic:
grind
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_2
inst : Category.{u_3, u_2} C
K : Precoverage.{u_3, u_2} C
J : GrothendieckTopology.{u_3, u_2} C
h : K ≤ J.toPrecoverage
X : C
R : Sieve.{u_3, u_2} X
hR : R ∈ (K.toGrothendieck : (a : C) → Set (Sieve a)) X
refine_2 :
  ∀ (X : C) (S : Presieve.{u_3, u_2} X) (hS : S ∈ K.coverings X), Sieve.generate S ∈ (J : (a : C) → Set (Sieve a)) X
refine_2₁ : ∀ (X : C), ⊤ ∈ (J : (a : C) → Set (Sieve a)) X
refine_2₂ :
  ∀ (X : C) (S : Sieve.{u_3, u_2} X),
    K.Saturate X S →
      ∀ (Y : C) (f : Y ⟶ X), S ∈ (J : (a : C) → Set (Sieve a)) X → Sieve.pullback f S ∈ (J : (a : C) → Set (Sieve a)) Y
refine_2₃ :
  ∀ (X : C) (S R : Sieve.{u_3, u_2} X),
    K.Saturate X S →
      (∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → K.Saturate Y (Sieve.pullback f R)) →
        S ∈ (J : (a : C) → Set (Sieve a)) X →
          (∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Sieve.pullback f R ∈ (J : (a : C) → Set (Sieve a)) Y) →
            R ∈ (J : (a : C) → Set (Sieve a)) X
⊢ R ∈ (J : (a : C) → Set (Sieve a)) X

Tactic:
induction hR with
    | of X S hS => exact h _ hS
    | top X => grind
    | pullback X S _ Y f _ => grind
    | transitive X S R _ _ _ _ => grind

Premises used: CategoryTheory.Limits.pullback
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_2
inst : Category.{u_3, u_2} C
⊢ GaloisConnection (α := Precoverage.{u_3, u_2} C) toGrothendieck GrothendieckTopology.toPrecoverage

Tactic:
fun _ _ ↦ toGrothendieck_le_iff_le_toPrecoverage

Premises used: CategoryTheory.Precoverage.toGrothendieck_le_iff_le_toPrecoverage, GaloisConnection
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_2
inst : Category.{u_3, u_2} C
⊢ Eq (α := GrothendieckTopology.{u_3, u_2} C) (toGrothendieck ⊥) ⊥

Tactic:
(galoisConnection_toGrothendieck_toPrecoverage C).l_bot

Premises used: CategoryTheory.Precoverage.galoisConnection_toGrothendieck_toPrecoverage, GaloisConnection.l_bot
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_2
inst : Category.{u_3, u_2} C
⊢ Eq (α := Precoverage.{u_3, u_2} C) (toPrecoverage ⊤) ⊤

Tactic:
(Precoverage.galoisConnection_toGrothendieck_toPrecoverage C).u_top

Premises used: CategoryTheory.Precoverage.galoisConnection_toGrothendieck_toPrecoverage, GaloisConnection.u_top
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_2
inst : Category.{u_3, u_2} C
⊢ Monotone (α := Precoverage.{u_3, u_2} C) toGrothendieck

Tactic:
(galoisConnection_toGrothendieck_toPrecoverage C).monotone_l

Premises used: CategoryTheory.Precoverage.galoisConnection_toGrothendieck_toPrecoverage, GaloisConnection.monotone_l
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_3
inst : Category.{u_2, u_3} C
J K : Precoverage.{u_2, u_3} C
h : J ≤ K
⊢ J.toGrothendieck ≤ K.toGrothendieck

Tactic:
toGrothendieck_monotone h

Premises used: CategoryTheory.Precoverage.toGrothendieck_monotone
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_2
inst : Category.{u_3, u_2} C
⊢ Monotone (α := GrothendieckTopology.{u_3, u_2} C) toPrecoverage

Tactic:
(Precoverage.galoisConnection_toGrothendieck_toPrecoverage C).monotone_u

Premises used: GaloisConnection.monotone_u, CategoryTheory.Precoverage.galoisConnection_toGrothendieck_toPrecoverage
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_3
inst : Category.{u_2, u_3} C
J : Precoverage.{u_2, u_3} C
⊢ J ≤ J.toGrothendieck.toPrecoverage

Tactic:
(galoisConnection_toGrothendieck_toPrecoverage C).le_u_l _

Premises used: CategoryTheory.Precoverage.galoisConnection_toGrothendieck_toPrecoverage, GaloisConnection.le_u_l
-- Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck
Goal:
C : Type u_3
inst : Category.{u_2, u_3} C
J : GrothendieckTopology.{u_2, u_3} C
⊢ J.toPrecoverage.toGrothendieck ≤ J

Tactic:
(Precoverage.galoisConnection_toGrothendieck_toPrecoverage C).l_u_le _

Premises used: CategoryTheory.Precoverage.galoisConnection_toGrothendieck_toPrecoverage, GaloisConnection.l_u_le
-- Module: Mathlib.CategoryTheory.Sites.Preserves
Goal:
C : Type u
inst : Category.{v, u} C
I : C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
hF :
  IsSheafFor (X := I) F
    (ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
refine : (Y : Type w) → Unique (Y ⟶ F.obj (op I))
⊢ IsTerminal (F.obj (op I))

Tactic:
refine @IsTerminal.ofUnique _ _ _ fun Y ↦ ?_

Premises used: CategoryTheory.Limits.IsTerminal.ofUnique
-- Module: Mathlib.CategoryTheory.Sites.Preserves
Goal:
C : Type u
inst : Category.{v, u} C
I : C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
hF :
  IsSheafFor (X := I) F
    (ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
Y : Type w
tauto tauto₁ tauto₂ tauto₃ :
  ⦃Y : C⦄ →
    (f : Y ⟶ I) →
      ofArrows Empty.elim (fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a) f →
        F.obj (op Y)
⊢ FamilyOfElements (X := I) F
    (ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)

Tactic:
tauto
-- Module: Mathlib.CategoryTheory.Sites.Preserves
Goal:
C : Type u
inst : Category.{v, u} C
I : C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
hF :
  IsSheafFor (X := I) F
    (ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
Y : Type w
tauto tauto₁ tauto₂ tauto₃ tauto₄ tauto₅ tauto₆ tauto₇ tauto₈ tauto₉ tauto₁₀ tauto₁₁ tauto₁₂ :
  ∀ ⦃Y₁ Y₂ Z : C⦄ (g₁ : Z ⟶ Y₁) (g₂ : Z ⟶ Y₂) ⦃f₁ : Y₁ ⟶ I⦄ ⦃f₂ : Y₂ ⟶ I⦄
    (h₁ : ofArrows Empty.elim (fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a) f₁)
    (h₂ : ofArrows Empty.elim (fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a) f₂),
    g₁ ≫ f₁ = g₂ ≫ f₂ →
      Eq (α := (fun (X : Type w) ↦ X) (F.obj (op Z)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map g₁.op) :
              (a : (fun (X : Type w) ↦ X) (F.obj (op Y₁))) → (fun (X : Type w) ↦ X) (F.obj (op Z)))
            ((fun ⦃Y : C⦄ (f : Y ⟶ I)
                  (a :
                    ofArrows Empty.elim (fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
                      f) ↦
                (False.elim ⋯ : F.obj (op Y)))
              f₁ h₁) :
          (fun (X : Type w) ↦ X) (F.obj (op Z)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map g₂.op) :
              (a : (fun (X : Type w) ↦ X) (F.obj (op Y₂))) → (fun (X : Type w) ↦ X) (F.obj (op Z)))
            ((fun ⦃Y : C⦄ (f : Y ⟶ I)
                  (a :
                    ofArrows Empty.elim (fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
                      f) ↦
                (False.elim ⋯ : F.obj (op Y)))
              f₂ h₂) :
          (fun (X : Type w) ↦ X) (F.obj (op Z)))
⊢ FamilyOfElements.Compatible.{w, v, u} (P := F) (R :=
    ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
    fun ⦃Y : C⦄ (f : Y ⟶ I)
      (a : ofArrows Empty.elim (fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a) f) ↦
    False.elim ⋯

Tactic:
tauto
-- Module: Mathlib.CategoryTheory.Sites.Preserves
Goal:
C : Type u
inst : Category.{v, u} C
I : C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
hF :
  IsSheafFor (X := I) F
    (ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
Y : Type w
choose :
  (t : F.obj (op I)) →
    (h :
        (fun (t : F.obj (op I)) ↦
              FamilyOfElements.IsAmalgamation (R :=
                ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
                (fun ⦃Y : C⦄ (f : Y ⟶ I)
                    (a :
                      ofArrows Empty.elim
                        (fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a) f) ↦
                  False.elim ⋯)
                t)
            t ∧
          ∀ (y : F.obj (op I)),
            (fun (t : F.obj (op I)) ↦
                  FamilyOfElements.IsAmalgamation (R :=
                    ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
                    (fun ⦃Y : C⦄ (f : Y ⟶ I)
                        (a :
                          ofArrows Empty.elim
                            (fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a) f) ↦
                      False.elim ⋯)
                    t)
                y →
              y = t) →
      Unique (Y ⟶ F.obj (op I))
⊢ Unique (Y ⟶ F.obj (op I))

Tactic:
choose t h using hF (by tauto) (by tauto)
-- Module: Mathlib.CategoryTheory.Sites.Preserves
Goal:
C : Type u
inst : Category.{v, u} C
I : C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
hF :
  IsSheafFor (X := I) F
    (ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
Y : Type w
t : F.obj (op I)
h :
  (fun (t : F.obj (op I)) ↦
        FamilyOfElements.IsAmalgamation (R :=
          ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
          (fun ⦃Y : C⦄ (f : Y ⟶ I)
              (a :
                ofArrows Empty.elim (fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
                  f) ↦
            False.elim ⋯)
          t)
      t ∧
    ∀ (y : F.obj (op I)),
      (fun (t : F.obj (op I)) ↦
            FamilyOfElements.IsAmalgamation (R :=
              ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
              (fun ⦃Y : C⦄ (f : Y ⟶ I)
                  (a :
                    ofArrows Empty.elim (fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
                      f) ↦
                False.elim ⋯)
              t)
          y →
        y = t
a : Y ⟶ F.obj (op I)
h₁ :
  ∀ (x : Y),
    (ConcreteCategory.hom (FC := TypeCat.Fun) a).toFun x =
      TypeCat.Fun.toFun (@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _ default) x
⊢ a = default

Tactic:
ext
-- Module: Mathlib.CategoryTheory.Sites.Preserves
Goal:
C : Type u
inst : Category.{v, u} C
I : C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
hF :
  IsSheafFor (X := I) F
    (ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
Y : Type w
t : F.obj (op I)
h :
  (fun (t : F.obj (op I)) ↦
        FamilyOfElements.IsAmalgamation (R :=
          ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
          (fun ⦃Y : C⦄ (f : Y ⟶ I)
              (a :
                ofArrows Empty.elim (fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
                  f) ↦
            False.elim ⋯)
          t)
      t ∧
    ∀ (y : F.obj (op I)),
      (fun (t : F.obj (op I)) ↦
            FamilyOfElements.IsAmalgamation (R :=
              ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
              (fun ⦃Y : C⦄ (f : Y ⟶ I)
                  (a :
                    ofArrows Empty.elim (fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
                      f) ↦
                False.elim ⋯)
              t)
          y →
        y = t
a : Y ⟶ F.obj (op I)
x : Y
⊢ (ConcreteCategory.hom (FC := TypeCat.Fun) a).toFun x =
    TypeCat.Fun.toFun (@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _ default) x

Tactic:
exact h.2 _ (by tauto)

Premises used: And.right
-- Module: Mathlib.CategoryTheory.Sites.Preserves
Goal:
C : Type u
inst : Category.{v, u} C
I : C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
hF :
  IsSheafFor (X := I) F
    (ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
Y : Type w
t : F.obj (op I)
h :
  (fun (t : F.obj (op I)) ↦
        FamilyOfElements.IsAmalgamation (R :=
          ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
          (fun ⦃Y : C⦄ (f : Y ⟶ I)
              (a :
                ofArrows Empty.elim (fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
                  f) ↦
            False.elim ⋯)
          t)
      t ∧
    ∀ (y : F.obj (op I)),
      (fun (t : F.obj (op I)) ↦
            FamilyOfElements.IsAmalgamation (R :=
              ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
              (fun ⦃Y : C⦄ (f : Y ⟶ I)
                  (a :
                    ofArrows Empty.elim (fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
                      f) ↦
                False.elim ⋯)
              t)
          y →
        y = t
⊢ Unique (Y ⟶ F.obj (op I))

Tactic:
exact ⟨⟨↾fun _ ↦ t⟩, fun a ↦ by ext; exact h.2 _ (by tauto)⟩
-- Module: Mathlib.CategoryTheory.Sites.Preserves
Goal:
C : Type u
inst : Category.{v, u} C
I : C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
hF :
  IsSheafFor (X := I) F
    (ofArrows Empty.elim fun (a : Empty) ↦ Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ I) a)
hI : IsInitial.{v, u} I
⊢ PreservesLimit (Functor.empty Cᵒᵖ) F

Tactic:
have := hI.hasInitial
  (preservesTerminal_of_iso F
    ((F.mapIso (terminalIsoIsTerminal (terminalOpOfInitial initialIsInitial)) ≪≫
    (F.mapIso (initialIsoIsInitial hI).symm.op) ≪≫
    (terminalIsoIsTerminal (isTerminal_of_isSheafFor_empty_presieve I F hF)).symm)))

Premises used: CategoryTheory.Presieve.isTerminal_of_isSheafFor_empty_presieve, CategoryTheory.Iso.op, CategoryTheory.Limits.initialIsInitial, CategoryTheory.Limits.initialIsoIsInitial, CategoryTheory.Limits.IsInitial.hasInitial, CategoryTheory.Limits.preservesTerminal_of_iso, CategoryTheory.Iso.trans, CategoryTheory.Limits.terminalIsoIsTerminal, CategoryTheory.Iso.symm, CategoryTheory.Limits.terminalOpOfInitial, CategoryTheory.Functor.mapIso
-- Module: Mathlib.CategoryTheory.Sites.Preserves
Goal:
C : Type u
inst : Category.{v, u} C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
α : Type u_1
inst_1 : Small.{w, u_1} α
X : α → C
c : Cofan.{u_1, v, u} X
hc : IsColimit c
⊢ HasCoproduct.{u_1, v, u} X

Tactic:
have : HasCoproduct X := ⟨⟨c, hc⟩⟩
-- Module: Mathlib.CategoryTheory.Sites.Preserves
Goal:
C : Type u
inst : Category.{v, u} C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
α : Type u_1
inst_1 : Small.{w, u_1} α
X : α → C
c : Cofan.{u_1, v, u} X
hc : IsColimit c
this : HasCoproduct.{u_1, v, u} X
⊢ Eq (α := F.obj (op c.pt) ⟶ ∏ᶜ fun (i : α) ↦ F.obj (op (X i)))
    (Pi.lift (f := fun (i : α) ↦ F.obj (op (X i))) fun (i : α) ↦ F.map (c.inj i).op)
    (CategoryStruct.comp (Y := F.obj (piObj.{u_1, v, u} fun (i : α) ↦ op (X i)))
      (F.map (Pi.lift fun (i : α) ↦ (c.inj i).op)) (piComparison F fun (i : α) ↦ op (X i)))

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Sites.Preserves
Goal:
C : Type u
inst : Category.{v, u} C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
α : Type u_1
inst_1 : Small.{w, u_1} α
X : α → C
c : Cofan.{u_1, v, u} X
hc : IsColimit c
this : HasCoproduct.{u_1, v, u} X
⊢ Eq (α := F.obj (op c.pt) ⟶ ∏ᶜ fun (i : α) ↦ F.obj (op (X i)))
    (Pi.lift (f := fun (i : α) ↦ F.obj (op (X i))) fun (i : α) ↦ F.map (c.inj i).op)
    (CategoryStruct.comp (Y := F.obj (piObj.{u_1, v, u} fun (i : α) ↦ op (X i)))
      (F.map (Pi.lift fun (i : α) ↦ (c.inj i).op)) (piComparison F fun (i : α) ↦ op (X i)))

Tactic:
have h : Pi.lift (fun i ↦ F.map (c.inj i).op) =
      F.map (Pi.lift (fun i ↦ (c.inj i).op)) ≫ piComparison F _ := by simp
-- Module: Mathlib.CategoryTheory.Sites.Preserves
Goal:
C : Type u
inst : Category.{v, u} C
F : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
α : Type u_1
inst_1 : Small.{w, u_1} α
X : α → C
c : Cofan.{u_1, v, u} X
hc : IsColimit c
this : HasCoproduct.{u_1, v, u} X
h :
  Eq (α := F.obj (op c.pt) ⟶ ∏ᶜ fun (i : α) ↦ F.obj (op (X i)))
    (Pi.lift (f := fun (i : α) ↦ F.obj (op (X i))) fun (i : α) ↦ F.map (c.inj i).op)
    (CategoryStruct.comp (Y := F.obj (piObj.{u_1, v, u} fun (i : α) ↦ op (X i)))
      (F.map (Pi.lift fun (i : α) ↦ (c.inj i).op)) (piComparison F fun (i : α) ↦ op (X i)))
rw :
  Eq (α := F.obj (piObj.{u_1, v, u} fun (x : α) ↦ op (X x)) ⟶ ∏ᶜ fun (b : α) ↦ F.obj (op (X b)))
    (piComparison F fun (x : α) ↦ op (X x))
    (CategoryStruct.comp (Y := F.obj (piObj.{u_1, v, u} fun (i : α) ↦ op (X i)))
      (F.map
        (Iso.inv (opCoproductIsoProduct' hc (productIsProduct fun (x : α) ↦ op (X x))) ≫
          Pi.lift fun (i : α) ↦ (c.inj i).op))
      (piComparison F fun (i : α) ↦ op (X i)))
⊢ Eq (α := F.obj (piObj.{u_1, v, u} fun (x : α) ↦ op (X x)) ⟶ ∏ᶜ fun (b : α) ↦ F.obj (op (X b)))
    (piComparison F fun (x : α) ↦ op (X x))
    (F.map (Iso.inv (opCoproductIsoProduct' hc (productIsProduct fun (x : α) ↦ op (X x)))) ≫
      Pi.lift (f := fun (i : α) ↦ F.obj (op (X i))) fun (i : α) ↦ F.map (c.inj i).op)

Tactic:
rw [h, ← Category.assoc, ← Functor.map_comp]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Functor.map_comp