awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : C
inst_1 : Projective.{v_1, u_1} X
F : Functor.{v_1, u_4, u_1, u_4 + 1} Cᵒᵖ (Type u_4)
Y : C
f : Y ⟶ X
hf : EffectiveEpi f
inst_2 : regular.{v_1, u_1} (C := C) (X := X) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
refine_1 :
  ∀ (x : (i : Unit) → F.obj (op Y)) (hx : Arrows.Compatible F (B := X) (fun (x : Unit) ↦ f) x) (x_1 : Unit),
    (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f.op) :
          (a : (fun (X : Type u_4) ↦ X) (F.obj (op X))) → (fun (X : Type u_4) ↦ X) (F.obj (op Y)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (Projective.factorThru (𝟙 X) f).op) :
              (a : (fun (X : Type u_4) ↦ X) (F.obj (op Y))) → (fun (X : Type u_4) ↦ X) (F.obj (op X)))
            (x ()) :
          (fun (X : Type u_4) ↦ X) (F.obj (op X))) =
      x x_1
⊢ ∀ (x : (i : Unit) → F.obj (op Y)),
    Arrows.Compatible F (B := X) (fun (x : Unit) ↦ f) x →
      ∃! t : (fun (X : Type u_4) ↦ X) (F.obj (op X)),
        ∀ (i : Unit),
          (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f.op) :
                (a : (fun (X : Type u_4) ↦ X) (F.obj (op X))) → (fun (X : Type u_4) ↦ X) (F.obj (op Y)))
              t =
            x i
have refine_2 : ∀ (x : (i : Unit) → F.obj (op Y)) (hx : Arrows.Compatible F (B := X) (fun (x : Unit) ↦ f) x)
  (y : (fun (X : Type u_4) ↦ X) (F.obj (op X)))
  (h :
    (fun (t : (fun (X : Type u_4) ↦ X) (F.obj (op X))) ↦
        ∀ (i : Unit),
          (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f.op) :
                (a : (fun (X : Type u_4) ↦ X) (F.obj (op X))) → (fun (X : Type u_4) ↦ X) (F.obj (op Y)))
              t =
            x i)
      y),
  y =
    (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (Projective.factorThru (𝟙 X) f).op) :
        (a : (fun (X : Type u_4) ↦ X) (F.obj (op Y))) → (fun (X : Type u_4) ↦ X) (F.obj (op X)))
      (x ())
Tactic: refine fun x hx ↦ ⟨F.map (Projective.factorThru (𝟙 _) f).op <| x (), fun _ ↦ ?_, fun y h ↦ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preregular.{v_1, u_1} C
W : C
⊢ IsSheaf.{v_1, v_1, u_1} (regularTopology C) (obj yoneda W)
have rw : ∀ {X : C} (R : Presieve.{v_1, u_1} X),
  Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
      (Precoverage.coverings (Coverage.toPrecoverage (regularCoverage C)) X) R →
    IsSheafFor (obj yoneda W) R
Tactic: rw [regularTopology, isSheaf_coverage]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preregular.{v_1, u_1} C
W X : C
S : Presieve.{v_1, u_1} X
⊢ Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
      (Precoverage.coverings (Coverage.toPrecoverage (regularCoverage C)) X) S →
    IsSheafFor (obj yoneda W) S
have this : ∀ (w : C) (hS : ∃ (f : w ⟶ X), (S = ofArrows (fun (x : Unit) ↦ w) fun (x : Unit) ↦ f) ∧ EffectiveEpi f),
  IsSheafFor (obj yoneda W) S
Tactic: ⟨_, hS⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preregular.{v_1, u_1} C
W X : C
S : Presieve.{v_1, u_1} X
w : C
hS : ∃ (f : w ⟶ X), (S = ofArrows (fun (x : Unit) ↦ w) fun (x : Unit) ↦ f) ∧ EffectiveEpi f
⊢ IsSheafFor (obj yoneda W) S
have this : S.regular
Tactic: have : S.regular := ⟨_, hS⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preregular.{v_1, u_1} C
W X : C
S : Presieve.{v_1, u_1} X
w : C
hS : ∃ (f : w ⟶ X), (S = ofArrows (fun (x : Unit) ↦ w) fun (x : Unit) ↦ f) ∧ EffectiveEpi f
this : S.regular
⊢ IsSheafFor (obj yoneda W) S
have obtain : ∀ (Y : C) (f : Y ⟶ X) (hf : EffectiveEpi f)
  (hS :
    ∃ (f_1 : w ⟶ X),
      Eq (α := Presieve.{v_1, u_1} X) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
          (ofArrows (fun (x : Unit) ↦ w) fun (x : Unit) ↦ f_1) ∧
        EffectiveEpi f_1)
  (this : regular.{v_1, u_1} (C := C) (X := X) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)),
  IsSheafFor.{v_1, v_1, u_1} (X := X) (obj yoneda W) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
Tactic: obtain ⟨Y, f, rfl, hf⟩ := Presieve.regular.single_epi (R := S)
Cleared hypotheses: S, hS, this
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preregular.{v_1, u_1} C
W X w Y : C
f : Y ⟶ X
hf : EffectiveEpi f
hS :
  ∃ (f_1 : w ⟶ X),
    Eq (α := Presieve.{v_1, u_1} X) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
        (ofArrows (fun (x : Unit) ↦ w) fun (x : Unit) ↦ f_1) ∧
      EffectiveEpi f_1
this : regular.{v_1, u_1} (C := C) (X := X) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
⊢ IsSheafFor.{v_1, v_1, u_1} (X := X) (obj yoneda W) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
have have : ∀ (h_colim : IsColimit (Sieve.generateSingleton f).arrows.cocone),
  IsSheafFor.{v_1, v_1, u_1} (X := X) (obj yoneda W) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
Tactic: have h_colim := isColimitOfEffectiveEpiStruct f hf.effectiveEpi.some
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preregular.{v_1, u_1} C
W X w Y : C
f : Y ⟶ X
hf : EffectiveEpi f
hS :
  ∃ (f_1 : w ⟶ X),
    Eq (α := Presieve.{v_1, u_1} X) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
        (ofArrows (fun (x : Unit) ↦ w) fun (x : Unit) ↦ f_1) ∧
      EffectiveEpi f_1
this : regular.{v_1, u_1} (C := C) (X := X) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
h_colim : IsColimit (Sieve.generateSingleton f).arrows.cocone
⊢ IsSheafFor.{v_1, v_1, u_1} (X := X) (obj yoneda W) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
have rw : ∀
  (h_colim :
    IsColimit (J :=
      Presieve.category.{v_1, u_1} (C := C) (X := X)
        (Sieve.arrows (Sieve.generate (ofArrows (fun (x : PUnit.{1}) ↦ Y) fun (x : PUnit.{1}) ↦ f))))
      (cocone (Sieve.arrows (Sieve.generate (ofArrows (fun (x : PUnit.{1}) ↦ Y) fun (x : PUnit.{1}) ↦ f))))),
  IsSheafFor.{v_1, v_1, u_1} (X := X) (obj yoneda W) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
Tactic: rw [← Sieve.generateSingleton_eq, ← Presieve.ofArrows_pUnit] at h_colim
Cleared hypotheses: h_colim
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preregular.{v_1, u_1} C
W X w Y : C
f : Y ⟶ X
hf : EffectiveEpi f
hS :
  ∃ (f_1 : w ⟶ X),
    Eq (α := Presieve.{v_1, u_1} X) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
        (ofArrows (fun (x : Unit) ↦ w) fun (x : Unit) ↦ f_1) ∧
      EffectiveEpi f_1
this : regular.{v_1, u_1} (C := C) (X := X) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
h_colim :
  IsColimit (J :=
    Presieve.category.{v_1, u_1} (C := C) (X := X)
      (Sieve.arrows (Sieve.generate (ofArrows (fun (x : PUnit.{1}) ↦ Y) fun (x : PUnit.{1}) ↦ f))))
    (cocone (Sieve.arrows (Sieve.generate (ofArrows (fun (x : PUnit.{1}) ↦ Y) fun (x : PUnit.{1}) ↦ f))))
x : FamilyOfElements.{v_1, v_1, u_1} (X := X) (obj yoneda W) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
hx : x.Compatible (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
x_ext : FamilyOfElements.{v_1, v_1, u_1} (X := X) (obj yoneda W)
  (Sieve.arrows (Sieve.generate (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f))) :=
  x.sieveExtend (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
⊢ ∃! t : obj (obj yoneda W) (op X), x.IsAmalgamation (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f) t
have hx_ext : x.sieveExtend (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f).Compatible (R :=
  Sieve.arrows (Sieve.generate (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)))
Tactic: have hx_ext := Presieve.FamilyOfElements.Compatible.sieveExtend hx
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preregular.{v_1, u_1} C
W X w Y : C
f : Y ⟶ X
hf : EffectiveEpi f
hS :
  ∃ (f_1 : w ⟶ X),
    Eq (α := Presieve.{v_1, u_1} X) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
        (ofArrows (fun (x : Unit) ↦ w) fun (x : Unit) ↦ f_1) ∧
      EffectiveEpi f_1
this : regular.{v_1, u_1} (C := C) (X := X) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
h_colim :
  IsColimit (J :=
    Presieve.category.{v_1, u_1} (C := C) (X := X)
      (Sieve.arrows (Sieve.generate (ofArrows (fun (x : PUnit.{1}) ↦ Y) fun (x : PUnit.{1}) ↦ f))))
    (cocone (Sieve.arrows (Sieve.generate (ofArrows (fun (x : PUnit.{1}) ↦ Y) fun (x : PUnit.{1}) ↦ f))))
x : FamilyOfElements.{v_1, v_1, u_1} (X := X) (obj yoneda W) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
hx : x.Compatible (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
x_ext : FamilyOfElements.{v_1, v_1, u_1} (X := X) (obj yoneda W)
  (Sieve.arrows (Sieve.generate (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f))) :=
  x.sieveExtend (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
hx_ext :
  x.sieveExtend (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f).Compatible (R :=
    Sieve.arrows (Sieve.generate (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)))
S : Sieve.{v_1, u_1} X :=
  Sieve.generate
    (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦
      match (motive := (x : Unit) → (fun (x : Unit) ↦ Y) x ⟶ X) x with
      | PUnit.unit => f)
⊢ ∃! t : obj (obj yoneda W) (op X), x.IsAmalgamation (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f) t
have obtain : ∀ (t : obj (obj yoneda W) (op X)) (t_amalg : x_ext.IsAmalgamation (R := S.arrows) t)
  (t_uniq :
    ∀ (y : obj (obj yoneda W) (op X)),
      (fun (t : obj (obj yoneda W) (op X)) ↦ x_ext.IsAmalgamation (R := S.arrows) t) y → y = t),
  ∃! t : obj (obj yoneda W) (op X), x.IsAmalgamation (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f) t
Tactic: obtain ⟨t, t_amalg, t_uniq⟩ :=
    (Sieve.forallYonedaIsSheaf_iff_colimit S).mpr ⟨h_colim⟩ W x_ext hx_ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preregular.{v_1, u_1} C
W X w Y : C
f : Y ⟶ X
hf : EffectiveEpi f
hS :
  ∃ (f_1 : w ⟶ X),
    Eq (α := Presieve.{v_1, u_1} X) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
        (ofArrows (fun (x : Unit) ↦ w) fun (x : Unit) ↦ f_1) ∧
      EffectiveEpi f_1
this : regular.{v_1, u_1} (C := C) (X := X) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
h_colim :
  IsColimit (J :=
    Presieve.category.{v_1, u_1} (C := C) (X := X)
      (Sieve.arrows (Sieve.generate (ofArrows (fun (x : PUnit.{1}) ↦ Y) fun (x : PUnit.{1}) ↦ f))))
    (cocone (Sieve.arrows (Sieve.generate (ofArrows (fun (x : PUnit.{1}) ↦ Y) fun (x : PUnit.{1}) ↦ f))))
x : FamilyOfElements.{v_1, v_1, u_1} (X := X) (obj yoneda W) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
hx : x.Compatible (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
x_ext : FamilyOfElements.{v_1, v_1, u_1} (X := X) (obj yoneda W)
  (Sieve.arrows (Sieve.generate (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f))) :=
  x.sieveExtend (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
hx_ext :
  x.sieveExtend (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f).Compatible (R :=
    Sieve.arrows (Sieve.generate (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)))
S : Sieve.{v_1, u_1} X :=
  Sieve.generate
    (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦
      match (motive := (x : Unit) → (fun (x : Unit) ↦ Y) x ⟶ X) x with
      | PUnit.unit => f)
t : obj (obj yoneda W) (op X)
t_amalg : x_ext.IsAmalgamation (R := S.arrows) t
t_uniq :
  ∀ (y : obj (obj yoneda W) (op X)),
    (fun (t : obj (obj yoneda W) (op X)) ↦ x_ext.IsAmalgamation (R := S.arrows) t) y → y = t
⊢ ∃! t : obj (obj yoneda W) (op X), x.IsAmalgamation (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f) t
have refine_1 : (fun (t : obj (obj yoneda W) (op X)) ↦ x.IsAmalgamation (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f) t) t
Tactic: refine ⟨t, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preregular.{v_1, u_1} C
W X w Y : C
f : Y ⟶ X
hf : EffectiveEpi f
hS :
  ∃ (f_1 : w ⟶ X),
    Eq (α := Presieve.{v_1, u_1} X) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
        (ofArrows (fun (x : Unit) ↦ w) fun (x : Unit) ↦ f_1) ∧
      EffectiveEpi f_1
this : regular.{v_1, u_1} (C := C) (X := X) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
h_colim :
  IsColimit (J :=
    Presieve.category.{v_1, u_1} (C := C) (X := X)
      (Sieve.arrows (Sieve.generate (ofArrows (fun (x : PUnit.{1}) ↦ Y) fun (x : PUnit.{1}) ↦ f))))
    (cocone (Sieve.arrows (Sieve.generate (ofArrows (fun (x : PUnit.{1}) ↦ Y) fun (x : PUnit.{1}) ↦ f))))
x : FamilyOfElements.{v_1, v_1, u_1} (X := X) (obj yoneda W) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
hx : x.Compatible (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
x_ext : FamilyOfElements.{v_1, v_1, u_1} (X := X) (obj yoneda W)
  (Sieve.arrows (Sieve.generate (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f))) :=
  x.sieveExtend (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
hx_ext :
  x.sieveExtend (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f).Compatible (R :=
    Sieve.arrows (Sieve.generate (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)))
S : Sieve.{v_1, u_1} X :=
  Sieve.generate
    (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦
      match (motive := (x : Unit) → (fun (x : Unit) ↦ Y) x ⟶ X) x with
      | PUnit.unit => f)
t : obj (obj yoneda W) (op X)
t_amalg : x_ext.IsAmalgamation (R := S.arrows) t
t_uniq :
  ∀ (y : obj (obj yoneda W) (op X)),
    (fun (t : obj (obj yoneda W) (op X)) ↦ x_ext.IsAmalgamation (R := S.arrows) t) y → y = t
refine_1 :
  (fun (t : obj (obj yoneda W) (op X)) ↦ x.IsAmalgamation (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f) t) t
⊢ ∃! t : obj (obj yoneda W) (op X), x.IsAmalgamation (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f) t
have refine_2 : ∀ (y : obj (obj yoneda W) (op X)),
  (fun (t : obj (obj yoneda W) (op X)) ↦ x.IsAmalgamation (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f) t) y →
    y = t
Tactic: refine ⟨t, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preregular.{v_1, u_1} C
W X w Y : C
f : Y ⟶ X
hf : EffectiveEpi f
hS :
  ∃ (f_1 : w ⟶ X),
    Eq (α := Presieve.{v_1, u_1} X) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
        (ofArrows (fun (x : Unit) ↦ w) fun (x : Unit) ↦ f_1) ∧
      EffectiveEpi f_1
this : regular.{v_1, u_1} (C := C) (X := X) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
h_colim :
  IsColimit (J :=
    Presieve.category.{v_1, u_1} (C := C) (X := X)
      (Sieve.arrows (Sieve.generate (ofArrows (fun (x : PUnit.{1}) ↦ Y) fun (x : PUnit.{1}) ↦ f))))
    (cocone (Sieve.arrows (Sieve.generate (ofArrows (fun (x : PUnit.{1}) ↦ Y) fun (x : PUnit.{1}) ↦ f))))
x : FamilyOfElements.{v_1, v_1, u_1} (X := X) (obj yoneda W) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
hx : x.Compatible (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
x_ext : FamilyOfElements.{v_1, v_1, u_1} (X := X) (obj yoneda W)
  (Sieve.arrows (Sieve.generate (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f))) :=
  x.sieveExtend (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
hx_ext :
  x.sieveExtend (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f).Compatible (R :=
    Sieve.arrows (Sieve.generate (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)))
S : Sieve.{v_1, u_1} X :=
  Sieve.generate
    (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦
      match (motive := (x : Unit) → (fun (x : Unit) ↦ Y) x ⟶ X) x with
      | PUnit.unit => f)
t : obj (obj yoneda W) (op X)
t_amalg : x_ext.IsAmalgamation (R := S.arrows) t
t_uniq :
  ∀ (y : obj (obj yoneda W) (op X)),
    (fun (t : obj (obj yoneda W) (op X)) ↦ x_ext.IsAmalgamation (R := S.arrows) t) y → y = t
⊢ (fun (t : obj (obj yoneda W) (op X)) ↦ x.IsAmalgamation (R := ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f) t) t
have h : HEq (α := Presieve.{v_1, u_1} X) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f) (β := Presieve.{v_1, u_1} X)
    (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦
      match (motive := (x : Unit) → (fun (x : Unit) ↦ Y) x ⟶ X) x with
      | PUnit.unit => f) →
  x =
    FamilyOfElements.restrict (R₁ :=
      ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦
        match (motive := (x : Unit) → (fun (x : Unit) ↦ Y) x ⟶ X) x with
        | PUnit.unit => f)
      (R₂ :=
      Sieve.arrows
        (Sieve.generate
          (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦
            match (motive := (x : Unit) → (fun (x : Unit) ↦ Y) x ⟶ X) x with
            | PUnit.unit => f)))
      sorry x_ext
Tactic: convert!
    Presieve.isAmalgamation_restrict
      (Sieve.le_generate (Presieve.ofArrows (fun () ↦ Y) (fun () ↦ f))) _ _ t_amalg
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularTopology (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preregular.{v_1, u_1} C
X : C
S : Sieve.{v_1, u_1} X
Y : C
π : Y ⟶ X
h : EffectiveEpi π ∧ S.arrows π
⊢ Sieve.generate (Presieve.ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ π) ≤ S
have rw : (Presieve.ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ π) ≤ S.arrows
Tactic: rw [Sieve.generate_le_iff (Presieve.ofArrows _ _) S]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularTopology (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preregular.{v_1, u_1} C
X : C
S : Sieve.{v_1, u_1} X
Y : C
π : Y ⟶ X
h : EffectiveEpi π ∧ S.arrows π
⊢ (Presieve.ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ π) ≤ S.arrows
have apply : Presieve.FactorsThru (Presieve.ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ π) S.arrows
Tactic: apply Presieve.le_of_factorsThru_sieve (Presieve.ofArrows _ _) S _
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularTopology (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preregular.{v_1, u_1} C
X : C
S : Sieve.{v_1, u_1} X
Y : C
π : Y ⟶ X
h : EffectiveEpi π ∧ S.arrows π
W : C
g : W ⟶ X
f : Presieve.ofArrows (fun (x : Unit) ↦ Y) (fun (x : Unit) ↦ π) g
⊢ ∃ (W_1 : C) (i : W ⟶ W_1) (e : W_1 ⟶ X), S.arrows e ∧ i ≫ e = g
have refine : ∃ (e : W ⟶ X), S.arrows e ∧ 𝟙 W ≫ e = g
Tactic: refine ⟨W, 𝟙 W, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularTopology (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preregular.{v_1, u_1} C
X : C
S : Sieve.{v_1, u_1} X
Y : C
π : Y ⟶ X
h : EffectiveEpi π ∧ S.arrows π
W : C
g : W ⟶ X
f : Presieve.ofArrows (fun (x : Unit) ↦ Y) (fun (x : Unit) ↦ π) g
⊢ ∃ (e : W ⟶ X), S.arrows e ∧ 𝟙 W ≫ e = g
have mk : ∀ (i : Unit), ∃ (e : Y ⟶ X), S.arrows e ∧ 𝟙 Y ≫ e = π
Tactic: cases f
Cleared hypotheses: W, g, f
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularTopology (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preregular.{v_1, u_1} C
X : C
S : Sieve.{v_1, u_1} X
Y : C
π : Y ⟶ X
h : EffectiveEpi π ∧ S.arrows π
⊢ S ∈ (regularTopology C : (a : C) → Set (Sieve a)) X
have h_le : Sieve.generate (Presieve.ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ π) ≤ S
Tactic: have h_le : Sieve.generate (Presieve.ofArrows (fun () ↦ Y) (fun _ ↦ π)) ≤ S := by
    rw [Sieve.generate_le_iff (Presieve.ofArrows _ _) S]
    apply Presieve.le_of_factorsThru_sieve (Presieve.ofArrows _ _) S _
    intro W g f
    refine ⟨W, 𝟙 W, ?_⟩
    cases f
    exact ⟨π, ⟨h.2, Category.id_comp π⟩⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularTopology (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preregular.{v_1, u_1} C
X : C
S : Sieve.{v_1, u_1} X
Y : C
π : Y ⟶ X
h : EffectiveEpi π ∧ S.arrows π
h_le : Sieve.generate (Presieve.ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ π) ≤ S
⊢ S ∈ (regularTopology C : (a : C) → Set (Sieve a)) X
have apply : Coverage.Saturate.{v_1, u_1} (regularCoverage C) X
  (Sieve.generate (Presieve.ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ π))
Tactic: apply Coverage.saturate_of_superset (regularCoverage C) h_le
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularTopology (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preregular.{v_1, u_1} C
X Y Y' : C
π : Y ⟶ X
inst_2 : EffectiveEpi π
π' : Y' ⟶ Y
inst_3 : EffectiveEpi π'
⊢ EffectiveEpi (π' ≫ π)
have rw : Presieve.EffectiveEpimorphic.{v_1, u_1} (C := C) (X := X)
  (Presieve.ofArrows (fun (x : Unit) ↦ Y') fun (x : Unit) ↦
    match (motive := (x : Unit) → (fun (x : Unit) ↦ Y') x ⟶ X) x with
    | PUnit.unit => π' ≫ π)
Tactic: rw [effectiveEpi_iff_effectiveEpiFamily, ← Sieve.effectiveEpimorphic_family]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularTopology (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preregular.{v_1, u_1} C
X Y Y' : C
π : Y ⟶ X
inst_2 : EffectiveEpi π
π' : Y' ⟶ Y
inst_3 : EffectiveEpi π'
h₂ :
  Sieve.generate
      (Presieve.ofArrows (fun (x : Unit) ↦ Y') fun (x : Unit) ↦
        match (motive := (x : Unit) → (fun (x : Unit) ↦ Y') x ⟶ X) x with
        | PUnit.unit => π' ≫ π) ∈
    (regularTopology C : (a : C) → Set (Sieve a)) X
⊢ Nonempty
    (IsColimit (J :=
      Presieve.category.{v_1, u_1} (C := C) (X := X)
        (Sieve.arrows
          (Sieve.generate
            (Presieve.ofArrows (fun (x : Unit) ↦ Y') fun (x : Unit) ↦
              match (motive := (x : Unit) → (fun (x : Unit) ↦ Y') x ⟶ X) x with
              | PUnit.unit => π' ≫ π))))
      (Presieve.cocone
        (Sieve.arrows
          (Sieve.generate
            (Presieve.ofArrows (fun (x : Unit) ↦ Y') fun (x : Unit) ↦
              match (motive := (x : Unit) → (fun (x : Unit) ↦ Y') x ⟶ X) x with
              | PUnit.unit => π' ≫ π)))))
have rw : ∀ (W : C),
  Presieve.IsSheafFor.{v_1, v_1, u_1} (X := X) (Functor.obj yoneda W)
    (Sieve.arrows
      (Sieve.generate
        (Presieve.ofArrows (fun (x : Unit) ↦ Y') fun (x : Unit) ↦
          match (motive := (x : Unit) → (fun (x : Unit) ↦ Y') x ⟶ X) x with
          | PUnit.unit => π' ≫ π)))
Tactic: rw [← Sieve.forallYonedaIsSheaf_iff_colimit]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularTopology (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preregular.{v_1, u_1} C
X Y Y' : C
π : Y ⟶ X
inst_2 : EffectiveEpi π
π' : Y' ⟶ Y
inst_3 : EffectiveEpi π'
⊢ Presieve.EffectiveEpimorphic.{v_1, u_1} (C := C) (X := X)
    (Presieve.ofArrows (fun (x : Unit) ↦ Y') fun (x : Unit) ↦
      match (motive := (x : Unit) → (fun (x : Unit) ↦ Y') x ⟶ X) x with
      | PUnit.unit => π' ≫ π)
have suffices : Sieve.generate
    (Presieve.ofArrows (fun (x : Unit) ↦ Y') fun (x : Unit) ↦
      match (motive := (x : Unit) → (fun (x : Unit) ↦ Y') x ⟶ X) x with
      | PUnit.unit => π' ≫ π) ∈
  (regularTopology C : (a : C) → Set (Sieve a)) X
Tactic: suffices h₂ : (Sieve.generate (Presieve.ofArrows _ _)) ∈ (regularTopology C) X by
    change Nonempty _
    rw [← Sieve.forallYonedaIsSheaf_iff_colimit]
    exact fun W => regularTopology.isSheaf_yoneda_obj W _ h₂
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularTopology (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preregular.{v_1, u_1} C
X Y Y' : C
π : Y ⟶ X
inst_2 : EffectiveEpi π
π' : Y' ⟶ Y
inst_3 : EffectiveEpi π'
⊢ Sieve.generate
      (Presieve.ofArrows (fun (x : Unit) ↦ Y') fun (x : Unit) ↦
        match (motive := (x : Unit) → (fun (x : Unit) ↦ Y') x ⟶ X) x with
        | PUnit.unit => π' ≫ π) ∈
    (regularTopology C : (a : C) → Set (Sieve a)) X
have a : Coverage.Saturate.{v_1, u_1} (regularCoverage C) X
  (Sieve.generate
    (Presieve.ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦
      match (motive := (x : Unit) → (fun (x : Unit) ↦ Y) x ⟶ X) x with
      | PUnit.unit => π))
Tactic: apply Coverage.Saturate.transitive X (Sieve.generate (Presieve.ofArrows (fun () ↦ Y)
      (fun () ↦ π)))