awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.EffectiveEpimorphic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : C
f : Y ⟶ X
Hf : IsColimit (Sieve.generateSingleton f).arrows.cocone
W : C
e : Y ⟶ W
h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e
A : Over.{v, u} X
hA : (Sieve.generateSingleton f).arrows A.hom
B : Over.{v, u} X
hB : (Sieve.generateSingleton f).arrows B.hom
q : A ⟶ B
a :
  Over.Hom.left q ≫ Exists.choose (p := fun (e : B.left ⟶ Y) ↦ e ≫ f = B.hom) hB ≫ f =
    Exists.choose (p := fun (e : A.left ⟶ Y) ↦ e ≫ f = A.hom) hA ≫ f
a₁ : Over.Hom.left q ≫ B.hom = Exists.choose (p := fun (e : A.left ⟶ Y) ↦ e ≫ f = A.hom) hA ≫ f
a₂ : Over.Hom.left q ≫ B.hom = A.hom
⊢ (Over.Hom.left q ≫ Exists.choose (p := fun (e : B.left ⟶ Y) ↦ e ≫ f = B.hom) hB) ≫ f =
    Exists.choose (p := fun (e : A.left ⟶ Y) ↦ e ≫ f = A.hom) hA ≫ f
have a₃ : A.hom = A.hom
Tactic: rw [Category.assoc, hB.choose_spec, hA.choose_spec, Over.w]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.EffectiveEpimorphic (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : C
f : Y ⟶ X
Hf : IsColimit (Sieve.generateSingleton f).arrows.cocone
aux : {W : C} →
  (e : Y ⟶ W) →
    (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) →
      Cocone (Sieve.generateSingleton f).arrows.diagram :=
  fun {W : C} (e : Y ⟶ W) (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) ↦
    { pt := W,
      ι :=
        {
          app := fun (x : (Sieve.generateSingleton f).arrows.category) ↦
            match (motive :=
              (x : (Sieve.generateSingleton f).arrows.category) →
                (Sieve.generateSingleton f).arrows.diagram.obj x ⟶
                  Functor.obj (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) x)
              x with
            | { obj := obj, property := hT } => Exists.choose (p := fun (e : obj.left ⟶ Y) ↦ e ≫ f = obj.hom) hT ≫ e,
          naturality := ⋯ :
          NatTrans (Sieve.generateSingleton f).arrows.diagram
            (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) } }
W : C
e : Y ⟶ W
h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e
⊢ f ≫ Hf.desc (aux e ⋯) = e
have this : Eq (α :=
  (Sieve.generateSingleton f).arrows.diagram.obj
      (ObjectProperty.FullSubcategory.mk (P := fun (f_1 : Over.{v, u} X) ↦ (Sieve.generateSingleton f).arrows f_1.hom)
        (Over.mk f) sorry) ⟶
    (aux e sorry).pt)
  ((Sieve.generateSingleton f).arrows.cocone.ι.app
      (ObjectProperty.FullSubcategory.mk (P := fun (f_1 : Over.{v, u} X) ↦ (Sieve.generateSingleton f).arrows f_1.hom)
        (Over.mk f) sorry) ≫
    Hf.desc (aux e sorry))
  ((aux e sorry).ι.app
    (ObjectProperty.FullSubcategory.mk (P := fun (f_1 : Over.{v, u} X) ↦ (Sieve.generateSingleton f).arrows f_1.hom)
      (Over.mk f) sorry))
Tactic: have := Hf.fac (aux e h) ⟨Over.mk f, 𝟙 _, by simp⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.EffectiveEpimorphic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : C
f : Y ⟶ X
Hf : IsColimit (Sieve.generateSingleton f).arrows.cocone
aux : {W : C} →
  (e : Y ⟶ W) →
    (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) →
      Cocone (Sieve.generateSingleton f).arrows.diagram :=
  fun {W : C} (e : Y ⟶ W) (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) ↦
    { pt := W,
      ι :=
        {
          app := fun (x : (Sieve.generateSingleton f).arrows.category) ↦
            match (motive :=
              (x : (Sieve.generateSingleton f).arrows.category) →
                (Sieve.generateSingleton f).arrows.diagram.obj x ⟶
                  Functor.obj (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) x)
              x with
            | { obj := obj, property := hT } => Exists.choose (p := fun (e : obj.left ⟶ Y) ↦ e ≫ f = obj.hom) hT ≫ e,
          naturality := ⋯ :
          NatTrans (Sieve.generateSingleton f).arrows.diagram
            (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) } }
W : C
e : Y ⟶ W
h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e
this :
  Eq (α := Y ⟶ W)
    (f ≫
        Hf.desc
          { pt := W,
            ι :=
              {
                app := fun (x : (Sieve.generateSingleton f).arrows.category) ↦
                  Exists.choose (p :=
                      fun
                        (e :
                          x.obj (P := fun (f_1 : Over.{v, u} X) ↦ (Sieve.generateSingleton f).arrows f_1.hom).left ⟶
                            Y) ↦
                      e ≫ f = x.obj (P := fun (f_1 : Over.{v, u} X) ↦ (Sieve.generateSingleton f).arrows f_1.hom).hom)
                      ⋯ ≫
                    e,
                naturality := ⋯ :
                NatTrans (Sieve.generateSingleton f).arrows.diagram
                  (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) } } :
      Y ⟶ W)
    (Exists.choose (p := fun (e : Y ⟶ Y) ↦ e ≫ f = f) ⋯ ≫ e : Y ⟶ W)
⊢ f ≫ Hf.desc (aux e ⋯) = e
have rw : Exists.choose (p := fun (e : Y ⟶ Y) ↦ e ≫ f = f) sorry ≫ e = e
Tactic: rw [this]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.EffectiveEpimorphic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : C
f : Y ⟶ X
Hf : IsColimit (Sieve.generateSingleton f).arrows.cocone
aux : {W : C} →
  (e : Y ⟶ W) →
    (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) →
      Cocone (Sieve.generateSingleton f).arrows.diagram :=
  fun {W : C} (e : Y ⟶ W) (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) ↦
    { pt := W,
      ι :=
        {
          app := fun (x : (Sieve.generateSingleton f).arrows.category) ↦
            match (motive :=
              (x : (Sieve.generateSingleton f).arrows.category) →
                (Sieve.generateSingleton f).arrows.diagram.obj x ⟶
                  Functor.obj (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) x)
              x with
            | { obj := obj, property := hT } => Exists.choose (p := fun (e : obj.left ⟶ Y) ↦ e ≫ f = obj.hom) hT ≫ e,
          naturality := ⋯ :
          NatTrans (Sieve.generateSingleton f).arrows.diagram
            (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) } }
W : C
e : Y ⟶ W
h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e
⊢ Exists.choose (p := fun (e : Y ⟶ Y) ↦ e ≫ f = f) ⋯ ≫ e = e
have nth_rewrite : Exists.choose (p := fun (e : Y ⟶ Y) ↦ e ≫ f = f) sorry ≫ e = 𝟙 Y ≫ e
Tactic: nth_rewrite 2 [← Category.id_comp e]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.EffectiveEpimorphic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : C
f : Y ⟶ X
Hf : IsColimit (Sieve.generateSingleton f).arrows.cocone
aux : {W : C} →
  (e : Y ⟶ W) →
    (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) →
      Cocone (Sieve.generateSingleton f).arrows.diagram :=
  fun {W : C} (e : Y ⟶ W) (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) ↦
    { pt := W,
      ι :=
        {
          app := fun (x : (Sieve.generateSingleton f).arrows.category) ↦
            match (motive :=
              (x : (Sieve.generateSingleton f).arrows.category) →
                (Sieve.generateSingleton f).arrows.diagram.obj x ⟶
                  Functor.obj (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) x)
              x with
            | { obj := obj, property := hT } => Exists.choose (p := fun (e : obj.left ⟶ Y) ↦ e ≫ f = obj.hom) hT ≫ e,
          naturality := ⋯ :
          NatTrans (Sieve.generateSingleton f).arrows.diagram
            (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) } }
W : C
e : Y ⟶ W
h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e
⊢ Exists.choose (p := fun (e : Y ⟶ Y) ↦ e ≫ f = f) ⋯ ≫ e = 𝟙 Y ≫ e
have a : Exists.choose (p := fun (e : Y ⟶ Y) ↦ e ≫ f = f) sorry ≫ f = 𝟙 Y ≫ f
Tactic: apply h
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.EffectiveEpimorphic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : C
f : Y ⟶ X
Hf : IsColimit (Sieve.generateSingleton f).arrows.cocone
aux : {W : C} →
  (e : Y ⟶ W) →
    (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) →
      Cocone (Sieve.generateSingleton f).arrows.diagram :=
  fun {W : C} (e : Y ⟶ W) (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) ↦
    { pt := W,
      ι :=
        {
          app := fun (x : (Sieve.generateSingleton f).arrows.category) ↦
            match (motive :=
              (x : (Sieve.generateSingleton f).arrows.category) →
                (Sieve.generateSingleton f).arrows.diagram.obj x ⟶
                  Functor.obj (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) x)
              x with
            | { obj := obj, property := hT } => Exists.choose (p := fun (e : obj.left ⟶ Y) ↦ e ≫ f = obj.hom) hT ≫ e,
          naturality := ⋯ :
          NatTrans (Sieve.generateSingleton f).arrows.diagram
            (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) } }
W : C
e : Y ⟶ W
h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e
⊢ Exists.choose (p := fun (e : Y ⟶ Y) ↦ e ≫ f = f) ⋯ ≫ f = 𝟙 Y ≫ f
have a : ∀ (hh : ∃ (e : Y ⟶ Y), e ≫ f = f), hh.choose (p := fun (e : Y ⟶ Y) ↦ e ≫ f = f) ≫ f = 𝟙 Y ≫ f
Tactic: generalize_proofs hh
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.EffectiveEpimorphic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : C
f : Y ⟶ X
Hf : IsColimit (Sieve.generateSingleton f).arrows.cocone
aux : {W : C} →
  (e : Y ⟶ W) →
    (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) →
      Cocone (Sieve.generateSingleton f).arrows.diagram :=
  fun {W : C} (e : Y ⟶ W) (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) ↦
    { pt := W,
      ι :=
        {
          app := fun (x : (Sieve.generateSingleton f).arrows.category) ↦
            match (motive :=
              (x : (Sieve.generateSingleton f).arrows.category) →
                (Sieve.generateSingleton f).arrows.diagram.obj x ⟶
                  Functor.obj (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) x)
              x with
            | { obj := obj, property := hT } => Exists.choose (p := fun (e : obj.left ⟶ Y) ↦ e ≫ f = obj.hom) hT ≫ e,
          naturality := ⋯ :
          NatTrans (Sieve.generateSingleton f).arrows.diagram
            (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) } }
W : C
e : Y ⟶ W
h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e
hh : ∃ (e : Y ⟶ Y), e ≫ f = f
⊢ hh.choose (p := fun (e : Y ⟶ Y) ↦ e ≫ f = f) ≫ f = 𝟙 Y ≫ f
have a : f = 𝟙 Y ≫ f
Tactic: rw [hh.choose_spec, Category.id_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.EffectiveEpimorphic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : C
f : Y ⟶ X
Hf : IsColimit (Sieve.generateSingleton f).arrows.cocone
aux : {W : C} →
  (e : Y ⟶ W) →
    (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) →
      Cocone (Sieve.generateSingleton f).arrows.diagram :=
  fun {W : C} (e : Y ⟶ W) (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) ↦
    { pt := W,
      ι :=
        {
          app := fun (x : (Sieve.generateSingleton f).arrows.category) ↦
            match (motive :=
              (x : (Sieve.generateSingleton f).arrows.category) →
                (Sieve.generateSingleton f).arrows.diagram.obj x ⟶
                  Functor.obj (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) x)
              x with
            | { obj := obj, property := hT } => Exists.choose (p := fun (e : obj.left ⟶ Y) ↦ e ≫ f = obj.hom) hT ≫ e,
          naturality := ⋯ :
          NatTrans (Sieve.generateSingleton f).arrows.diagram
            (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) } }
W : C
e : Y ⟶ W
h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e
hh : ∃ (e : Y ⟶ Y), e ≫ f = f
a : f = 𝟙 Y ≫ f
⊢ hh.choose (p := fun (e : Y ⟶ Y) ↦ e ≫ f = f) ≫ f = 𝟙 Y ≫ f
have a₁ : f = f
Tactic: rw [hh.choose_spec, Category.id_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.EffectiveEpimorphic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : C
f : Y ⟶ X
Hf : IsColimit (Sieve.generateSingleton f).arrows.cocone
aux : {W : C} →
  (e : Y ⟶ W) →
    (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) →
      Cocone (Sieve.generateSingleton f).arrows.diagram :=
  fun {W : C} (e : Y ⟶ W) (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) ↦
    { pt := W,
      ι :=
        {
          app := fun (x : (Sieve.generateSingleton f).arrows.category) ↦
            match (motive :=
              (x : (Sieve.generateSingleton f).arrows.category) →
                (Sieve.generateSingleton f).arrows.diagram.obj x ⟶
                  Functor.obj (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) x)
              x with
            | { obj := obj, property := hT } => Exists.choose (p := fun (e : obj.left ⟶ Y) ↦ e ≫ f = obj.hom) hT ≫ e,
          naturality := ⋯ :
          NatTrans (Sieve.generateSingleton f).arrows.diagram
            (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) } }
W : C
e : Y ⟶ W
h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e
m : X ⟶ W
hm : f ≫ m = e
⊢ m = Hf.desc (aux e ⋯)
have x : ∀ (j : (Sieve.generateSingleton f).arrows.category),
  (Sieve.generateSingleton f).arrows.cocone.ι.app j ≫ m = (aux e sorry).ι.app j
Tactic: apply Hf.uniq (aux e h)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.EffectiveEpimorphic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : C
f : Y ⟶ X
Hf : IsColimit (Sieve.generateSingleton f).arrows.cocone
aux : {W : C} →
  (e : Y ⟶ W) →
    (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) →
      Cocone (Sieve.generateSingleton f).arrows.diagram :=
  fun {W : C} (e : Y ⟶ W) (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) ↦
    { pt := W,
      ι :=
        {
          app := fun (x : (Sieve.generateSingleton f).arrows.category) ↦
            match (motive :=
              (x : (Sieve.generateSingleton f).arrows.category) →
                (Sieve.generateSingleton f).arrows.diagram.obj x ⟶
                  Functor.obj (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) x)
              x with
            | { obj := obj, property := hT } => Exists.choose (p := fun (e : obj.left ⟶ Y) ↦ e ≫ f = obj.hom) hT ≫ e,
          naturality := ⋯ :
          NatTrans (Sieve.generateSingleton f).arrows.diagram
            (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) } }
W : C
e : Y ⟶ W
h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e
m : X ⟶ W
hm : f ≫ m = e
A : Over.{v, u} X
g : A.left ⟶ Y
hA : g ≫ f = A.hom
⊢ Eq (α :=
    (Sieve.generateSingleton f).arrows.diagram.obj
        (ObjectProperty.FullSubcategory.mk (P := fun (f_1 : Over.{v, u} X) ↦ (Sieve.generateSingleton f).arrows f_1.hom)
          A ⋯) ⟶
      (aux e ⋯).pt)
    ((Sieve.generateSingleton f).arrows.cocone.ι.app
        (ObjectProperty.FullSubcategory.mk (P := fun (f_1 : Over.{v, u} X) ↦ (Sieve.generateSingleton f).arrows f_1.hom)
          A ⋯) ≫
      m)
    ((aux e ⋯).ι.app
      (ObjectProperty.FullSubcategory.mk (P := fun (f_1 : Over.{v, u} X) ↦ (Sieve.generateSingleton f).arrows f_1.hom) A
        ⋯))
have x : A.hom ≫ m =
  (aux e sorry).ι.app
    (ObjectProperty.FullSubcategory.mk (P := fun (f_1 : Over.{v, u} X) ↦ (Sieve.generateSingleton f).arrows f_1.hom) A
      sorry)
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.EffectiveEpimorphic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : C
f : Y ⟶ X
Hf : IsColimit (Sieve.generateSingleton f).arrows.cocone
aux : {W : C} →
  (e : Y ⟶ W) →
    (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) →
      Cocone (Sieve.generateSingleton f).arrows.diagram :=
  fun {W : C} (e : Y ⟶ W) (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) ↦
    { pt := W,
      ι :=
        {
          app := fun (x : (Sieve.generateSingleton f).arrows.category) ↦
            match (motive :=
              (x : (Sieve.generateSingleton f).arrows.category) →
                (Sieve.generateSingleton f).arrows.diagram.obj x ⟶
                  Functor.obj (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) x)
              x with
            | { obj := obj, property := hT } => Exists.choose (p := fun (e : obj.left ⟶ Y) ↦ e ≫ f = obj.hom) hT ≫ e,
          naturality := ⋯ :
          NatTrans (Sieve.generateSingleton f).arrows.diagram
            (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) } }
W : C
e : Y ⟶ W
h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e
m : X ⟶ W
hm : f ≫ m = e
A : Over.{v, u} X
g : A.left ⟶ Y
hA : g ≫ f = A.hom
⊢ A.hom ≫ m =
    (aux e ⋯).ι.app
      (ObjectProperty.FullSubcategory.mk (P := fun (f_1 : Over.{v, u} X) ↦ (Sieve.generateSingleton f).arrows f_1.hom) A
        ⋯)
have x : g ≫ e =
  (aux e sorry).ι.app
    (ObjectProperty.FullSubcategory.mk (P := fun (f_1 : Over.{v, u} X) ↦ (Sieve.generateSingleton f).arrows f_1.hom) A
      sorry)
Tactic: simp only [← hA, Category.assoc, hm]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.EffectiveEpimorphic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : C
f : Y ⟶ X
Hf : IsColimit (Sieve.generateSingleton f).arrows.cocone
aux : {W : C} →
  (e : Y ⟶ W) →
    (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) →
      Cocone (Sieve.generateSingleton f).arrows.diagram :=
  fun {W : C} (e : Y ⟶ W) (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) ↦
    { pt := W,
      ι :=
        {
          app := fun (x : (Sieve.generateSingleton f).arrows.category) ↦
            match (motive :=
              (x : (Sieve.generateSingleton f).arrows.category) →
                (Sieve.generateSingleton f).arrows.diagram.obj x ⟶
                  Functor.obj (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) x)
              x with
            | { obj := obj, property := hT } => Exists.choose (p := fun (e : obj.left ⟶ Y) ↦ e ≫ f = obj.hom) hT ≫ e,
          naturality := ⋯ :
          NatTrans (Sieve.generateSingleton f).arrows.diagram
            (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) } }
W : C
e : Y ⟶ W
h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e
m : X ⟶ W
hm : f ≫ m = e
A : Over.{v, u} X
g : A.left ⟶ Y
hA : g ≫ f = A.hom
⊢ g ≫ e =
    (aux e ⋯).ι.app
      (ObjectProperty.FullSubcategory.mk (P := fun (f_1 : Over.{v, u} X) ↦ (Sieve.generateSingleton f).arrows f_1.hom) A
        ⋯)
have x : g ≫ f = Exists.choose (p := fun (e : A.left ⟶ Y) ↦ e ≫ f = A.hom) sorry ≫ f
Tactic: apply h
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.EffectiveEpimorphic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : C
f : Y ⟶ X
Hf : IsColimit (Sieve.generateSingleton f).arrows.cocone
aux : {W : C} →
  (e : Y ⟶ W) →
    (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) →
      Cocone (Sieve.generateSingleton f).arrows.diagram :=
  fun {W : C} (e : Y ⟶ W) (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) ↦
    { pt := W,
      ι :=
        {
          app := fun (x : (Sieve.generateSingleton f).arrows.category) ↦
            match (motive :=
              (x : (Sieve.generateSingleton f).arrows.category) →
                (Sieve.generateSingleton f).arrows.diagram.obj x ⟶
                  Functor.obj (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) x)
              x with
            | { obj := obj, property := hT } => Exists.choose (p := fun (e : obj.left ⟶ Y) ↦ e ≫ f = obj.hom) hT ≫ e,
          naturality := ⋯ :
          NatTrans (Sieve.generateSingleton f).arrows.diagram
            (Functor.obj (Functor.const (Sieve.generateSingleton f).arrows.category) W) } }
W : C
e : Y ⟶ W
h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e
m : X ⟶ W
hm : f ≫ m = e
A : Over.{v, u} X
g : A.left ⟶ Y
hA : g ≫ f = A.hom
⊢ g ≫ f = Exists.choose (p := fun (e : A.left ⟶ Y) ↦ e ≫ f = A.hom) ⋯ ≫ f
have x : ∀ (hh : ∃ (e : A.left ⟶ Y), e ≫ f = A.hom), g ≫ f = hh.choose (p := fun (e : A.left ⟶ Y) ↦ e ≫ f = A.hom) ≫ f
Tactic: generalize_proofs hh
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.EffectiveEpimorphic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : C
f : Y ⟶ X
⊢ (Presieve.singleton f).EffectiveEpimorphic ↔ EffectiveEpi f
have mp : (Presieve.singleton f).EffectiveEpimorphic → EffectiveEpi f
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.EffectiveEpimorphic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : C
f : Y ⟶ X
mp : (Presieve.singleton f).EffectiveEpimorphic → EffectiveEpi f
⊢ (Presieve.singleton f).EffectiveEpimorphic ↔ EffectiveEpi f
have mpr : EffectiveEpi f → (Presieve.singleton f).EffectiveEpimorphic
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.EffectiveEpimorphic (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : C
f : Y ⟶ X
h : Nonempty (IsColimit (generate (Presieve.singleton f)).arrows.cocone)
⊢ EffectiveEpi f
have h₁ : Nonempty (IsColimit (generateSingleton f).arrows.cocone)
Tactic: rw [Sieve.generateSingleton_eq] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.EffectiveEpimorphic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : C
f : Y ⟶ X
h : Nonempty (IsColimit (generateSingleton f).arrows.cocone)
⊢ EffectiveEpi f
have mp : Nonempty (EffectiveEpiStruct f)
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.EffectiveEpimorphic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : C
f : Y ⟶ X
h : Nonempty (EffectiveEpiStruct f)
⊢ Nonempty (IsColimit (generate (Presieve.singleton f)).arrows.cocone)
have mpr : Nonempty (IsColimit (generateSingleton f).arrows.cocone)
Tactic: rw [Sieve.generateSingleton_eq]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.EffectiveEpimorphic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
B : C
α : Type u_1
X : α → C
π : (a : α) → X a ⟶ B
⊢ generate (Presieve.ofArrows X π) = generateFamily X π
have h : ∀ ⦃Y : C⦄ (g : Y ⟶ B), (generate (Presieve.ofArrows X π)).arrows g ↔ (generateFamily X π).arrows g
Tactic: ext Y g
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.EffectiveEpimorphic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
B : C
α : Type u_1
X : α → C
π : (a : α) → X a ⟶ B
Y : C
g : Y ⟶ B
⊢ (generate (Presieve.ofArrows X π)).arrows g ↔ (generateFamily X π).arrows g
have h : (generate (Presieve.ofArrows X π)).arrows g → (generateFamily X π).arrows g
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.EffectiveEpimorphic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
B : C
α : Type u_1
X : α → C
π : (a : α) → X a ⟶ B
Y : C
g : Y ⟶ B
h : (generate (Presieve.ofArrows X π)).arrows g → (generateFamily X π).arrows g
⊢ (generate (Presieve.ofArrows X π)).arrows g ↔ (generateFamily X π).arrows g
have h₁ : (generateFamily X π).arrows g → (generate (Presieve.ofArrows X π)).arrows g
Tactic: constructor