awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : IsConnected.{v, u} C
s : Cocone.{v, w, u, w + 1} (constPUnitFunctor C)
j : C
⊢ TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (CategoryStruct.comp (Y :=
          Functor.obj.{v, w, u, w + 1} (Functor.obj (Functor.const C) (Cocone.pt.{v, w, u, w + 1} (pUnitCocone C))) j)
          (NatTrans.app (Cocone.ι (pUnitCocone C)) j) (s.ι.app Classical.ofNonempty)))
      PUnit.unit =
    (ConcreteCategory.hom (FC := TypeCat.Fun) (s.ι.app j)).toFun PUnit.unit
have h : ∀ (j₁ j₂ : C) (x : j₁ ⟶ j₂),
  Eq.{w + 1} (α := s.pt)
    ((fun (k : C) ↦
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (s.ι.app k) :
              (a : (fun (X : Type w) ↦ X) ((constPUnitFunctor C).obj k)) →
                (fun (X : Type w) ↦ X) (((Functor.const C).obj s.pt).obj k))
            PUnit.unit :
          (fun (X : Type w) ↦ X) (Functor.obj.{v, w, u, w + 1} (Functor.obj (Functor.const C) s.pt) k)))
      j₁)
    ((fun (k : C) ↦
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (s.ι.app k) :
              (a : (fun (X : Type w) ↦ X) ((constPUnitFunctor C).obj k)) →
                (fun (X : Type w) ↦ X) (((Functor.const C).obj s.pt).obj k))
            PUnit.unit :
          (fun (X : Type w) ↦ X) (Functor.obj.{v, w, u, w + 1} (Functor.obj (Functor.const C) s.pt) k)))
      j₂)
Tactic: refine constant_of_preserves_morphisms (α := s.pt)
      (fun (k : C) ↦ s.ι.app k PUnit.unit) ?_ Classical.ofNonempty j
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : IsConnected.{v, u} C
s : Cocone.{v, w, u, w + 1} (constPUnitFunctor C)
m : Cocone.pt.{v, w, u, w + 1} (pUnitCocone C) ⟶ s.pt
h : ∀ (j : C), NatTrans.app (Cocone.ι (pUnitCocone C)) j ≫ m = s.ι.app j
⊢ m = s.ι.app Classical.ofNonempty
have h₁ : (ConcreteCategory.hom (FC := TypeCat.Fun) m).toFun PUnit.unit =
  TypeCat.Fun.toFun (ConcreteCategory.hom (FC := TypeCat.Fun) (s.ι.app Classical.ofNonempty)) PUnit.unit
Tactic: ext ⟨⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : IsPreconnected.{v, u} C
inst_2 : HasColimit.{v, u, w, w + 1} (constPUnitFunctor C)
a b : colimit.{v, u, w, w + 1} (constPUnitFunctor C)
⊢ a = b
have obtain : ∀ (c : C),
  (ConcreteCategory.hom (FC := TypeCat.Fun) (colimit.ι (constPUnitFunctor C) c) :
        (a : (fun (X : Type w) ↦ X) ((constPUnitFunctor C).obj c)) →
          (fun (X : Type w) ↦ X) (colimit (constPUnitFunctor C)))
      PUnit.unit =
    b
Tactic: obtain ⟨c, ⟨⟩, rfl⟩ := jointly_surjective' a
Cleared hypotheses: a
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : IsPreconnected.{v, u} C
inst_2 : HasColimit.{v, u, w, w + 1} (constPUnitFunctor C)
b : colimit.{v, u, w, w + 1} (constPUnitFunctor C)
c : C
⊢ (ConcreteCategory.hom (FC := TypeCat.Fun) (colimit.ι (constPUnitFunctor C) c) :
        (a : (fun (X : Type w) ↦ X) ((constPUnitFunctor C).obj c)) →
          (fun (X : Type w) ↦ X) (colimit (constPUnitFunctor C)))
      PUnit.unit =
    b
have obtain : ∀ (d : C),
  Eq (α := colimit.{v, u, w, w + 1} (constPUnitFunctor C))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (colimit.ι (constPUnitFunctor C) c) :
          (a : (fun (X : Type w) ↦ X) ((constPUnitFunctor C).obj c)) →
            (fun (X : Type w) ↦ X) (colimit (constPUnitFunctor C)))
        PUnit.unit :
      (fun (X : Type w) ↦ X) (colimit.{v, u, w, w + 1} (constPUnitFunctor C)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (colimit.ι (constPUnitFunctor C) d) :
          (a : (fun (X : Type w) ↦ X) ((constPUnitFunctor C).obj d)) →
            (fun (X : Type w) ↦ X) (colimit (constPUnitFunctor C)))
        PUnit.unit :
      (fun (X : Type w) ↦ X) (colimit.{v, u, w, w + 1} (constPUnitFunctor C)))
Tactic: obtain ⟨d, ⟨⟩, rfl⟩ := jointly_surjective' b
Cleared hypotheses: b
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : IsPreconnected.{v, u} C
inst_2 : HasColimit.{v, u, w, w + 1} (constPUnitFunctor C)
c d : C
⊢ Eq (α := colimit.{v, u, w, w + 1} (constPUnitFunctor C))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (colimit.ι (constPUnitFunctor C) c) :
          (a : (fun (X : Type w) ↦ X) ((constPUnitFunctor C).obj c)) →
            (fun (X : Type w) ↦ X) (colimit (constPUnitFunctor C)))
        PUnit.unit :
      (fun (X : Type w) ↦ X) (colimit.{v, u, w, w + 1} (constPUnitFunctor C)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (colimit.ι (constPUnitFunctor C) d) :
          (a : (fun (X : Type w) ↦ X) ((constPUnitFunctor C).obj d)) →
            (fun (X : Type w) ↦ X) (colimit (constPUnitFunctor C)))
        PUnit.unit :
      (fun (X : Type w) ↦ X) (colimit.{v, u, w, w + 1} (constPUnitFunctor C)))
have h : ∀ (j₁ j₂ : C) (x : j₁ ⟶ j₂),
  Eq (α := (fun (X : Type w) ↦ X) (colimit.{v, u, w, w + 1} (constPUnitFunctor C)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (colimit.ι (constPUnitFunctor C) j₁) :
          (a : (fun (X : Type w) ↦ X) ((constPUnitFunctor C).obj j₁)) →
            (fun (X : Type w) ↦ X) (colimit (constPUnitFunctor C)))
        PUnit.unit :
      (fun (X : Type w) ↦ X) (colimit.{v, u, w, w + 1} (constPUnitFunctor C)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (colimit.ι (constPUnitFunctor C) j₂) :
          (a : (fun (X : Type w) ↦ X) ((constPUnitFunctor C).obj j₂)) →
            (fun (X : Type w) ↦ X) (colimit (constPUnitFunctor C)))
        PUnit.unit :
      (fun (X : Type w) ↦ X) (colimit.{v, u, w, w + 1} (constPUnitFunctor C)))
Tactic: apply constant_of_preserves_morphisms (colimit.ι (constPUnitFunctor C) · PUnit.unit)
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : Functor.{v, w, u, w + 1} C (Type w)
c d : (j : C) × F.obj j
h : Relation.EqvGen F.ColimitTypeRel c d
⊢ Zigzag.{v, u} (c.fst (β := fun (j : C) ↦ F.obj j)) (d.fst (β := fun (j : C) ↦ F.obj j))
have rel : ∀ (x y : (j : C) × F.obj j) (h : F.ColimitTypeRel x y),
  Zigzag.{v, u} (x.fst (β := fun (j : C) ↦ F.obj j)) (y.fst (β := fun (j : C) ↦ F.obj j))
Tactic: induction h with
  | rel _ _ h => exact Zigzag.of_hom <| Exists.choose h
  | refl _ => exact Zigzag.refl _
  | symm _ _ _ ih => exact zigzag_symmetric ih
  | trans _ _ _ _ _ ih₁ ih₂ => exact ih₁.trans ih₂
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : Functor.{v, w, u, w + 1} C (Type w)
c d : (j : C) × F.obj j
h : Relation.EqvGen F.ColimitTypeRel c d
rel :
  ∀ (x y : (j : C) × F.obj j) (h : F.ColimitTypeRel x y),
    Zigzag.{v, u} (x.fst (β := fun (j : C) ↦ F.obj j)) (y.fst (β := fun (j : C) ↦ F.obj j))
⊢ Zigzag.{v, u} (c.fst (β := fun (j : C) ↦ F.obj j)) (d.fst (β := fun (j : C) ↦ F.obj j))
have refl : ∀ (x : (j : C) × F.obj j), Zigzag.{v, u} (x.fst (β := fun (j : C) ↦ F.obj j)) (x.fst (β := fun (j : C) ↦ F.obj j))
Tactic: induction h with
  | rel _ _ h => exact Zigzag.of_hom <| Exists.choose h
  | refl _ => exact Zigzag.refl _
  | symm _ _ _ ih => exact zigzag_symmetric ih
  | trans _ _ _ _ _ ih₁ ih₂ => exact ih₁.trans ih₂
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : Functor.{v, w, u, w + 1} C (Type w)
c d : (j : C) × F.obj j
h : Relation.EqvGen F.ColimitTypeRel c d
rel :
  ∀ (x y : (j : C) × F.obj j) (h : F.ColimitTypeRel x y),
    Zigzag.{v, u} (x.fst (β := fun (j : C) ↦ F.obj j)) (y.fst (β := fun (j : C) ↦ F.obj j))
refl :
  ∀ (x : (j : C) × F.obj j), Zigzag.{v, u} (x.fst (β := fun (j : C) ↦ F.obj j)) (x.fst (β := fun (j : C) ↦ F.obj j))
⊢ Zigzag.{v, u} (c.fst (β := fun (j : C) ↦ F.obj j)) (d.fst (β := fun (j : C) ↦ F.obj j))
have symm : ∀ (x y : (j : C) × F.obj j),
  Relation.EqvGen F.ColimitTypeRel x y →
    ∀ (ih : Zigzag.{v, u} (x.fst (β := fun (j : C) ↦ F.obj j)) (y.fst (β := fun (j : C) ↦ F.obj j))),
      Zigzag.{v, u} (y.fst (β := fun (j : C) ↦ F.obj j)) (x.fst (β := fun (j : C) ↦ F.obj j))
Tactic: induction h with
  | rel _ _ h => exact Zigzag.of_hom <| Exists.choose h
  | refl _ => exact Zigzag.refl _
  | symm _ _ _ ih => exact zigzag_symmetric ih
  | trans _ _ _ _ _ ih₁ ih₂ => exact ih₁.trans ih₂
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : Functor.{v, w, u, w + 1} C (Type w)
c d : (j : C) × F.obj j
h : Relation.EqvGen F.ColimitTypeRel c d
rel :
  ∀ (x y : (j : C) × F.obj j) (h : F.ColimitTypeRel x y),
    Zigzag.{v, u} (x.fst (β := fun (j : C) ↦ F.obj j)) (y.fst (β := fun (j : C) ↦ F.obj j))
refl :
  ∀ (x : (j : C) × F.obj j), Zigzag.{v, u} (x.fst (β := fun (j : C) ↦ F.obj j)) (x.fst (β := fun (j : C) ↦ F.obj j))
symm :
  ∀ (x y : (j : C) × F.obj j),
    Relation.EqvGen F.ColimitTypeRel x y →
      ∀ (ih : Zigzag.{v, u} (x.fst (β := fun (j : C) ↦ F.obj j)) (y.fst (β := fun (j : C) ↦ F.obj j))),
        Zigzag.{v, u} (y.fst (β := fun (j : C) ↦ F.obj j)) (x.fst (β := fun (j : C) ↦ F.obj j))
⊢ Zigzag.{v, u} (c.fst (β := fun (j : C) ↦ F.obj j)) (d.fst (β := fun (j : C) ↦ F.obj j))
have trans : ∀ (x y z : (j : C) × F.obj j),
  Relation.EqvGen F.ColimitTypeRel x y →
    Relation.EqvGen F.ColimitTypeRel y z →
      ∀ (ih₁ : Zigzag.{v, u} (x.fst (β := fun (j : C) ↦ F.obj j)) (y.fst (β := fun (j : C) ↦ F.obj j)))
        (ih₂ : Zigzag.{v, u} (y.fst (β := fun (j : C) ↦ F.obj j)) (z.fst (β := fun (j : C) ↦ F.obj j))),
        Zigzag.{v, u} (x.fst (β := fun (j : C) ↦ F.obj j)) (z.fst (β := fun (j : C) ↦ F.obj j))
Tactic: induction h with
  | rel _ _ h => exact Zigzag.of_hom <| Exists.choose h
  | refl _ => exact Zigzag.refl _
  | symm _ _ _ ih => exact zigzag_symmetric ih
  | trans _ _ _ _ _ ih₁ ih₂ => exact ih₁.trans ih₂
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimit.{v, u, w, w + 1} (constPUnitFunctor C)
⊢ IsConnected.{v, u} C ↔ Nonempty.{w + 1} (colimit.{v, u, w, w + 1} (constPUnitFunctor C) ≅ PUnit.{w + 1})
have refine : Nonempty.{w + 1} (colimit.{v, u, w, w + 1} (constPUnitFunctor C) ≅ PUnit.{w + 1}) →
  ∀ (h : colimit.{v, u, w, w + 1} (constPUnitFunctor C) ≅ PUnit.{w + 1}), IsConnected.{v, u} C
Tactic: refine ⟨fun _ => ⟨colimitConstPUnitIsoPUnit.{w} C⟩, fun ⟨h⟩ => ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimit.{v, u, w, w + 1} (constPUnitFunctor C)
h_1 : Nonempty.{w + 1} (colimit.{v, u, w, w + 1} (constPUnitFunctor C) ≅ PUnit.{w + 1})
h : colimit.{v, u, w, w + 1} (constPUnitFunctor C) ≅ PUnit.{w + 1}
⊢ IsConnected.{v, u} C
have this : Nonempty C
Tactic: have : Nonempty C := nonempty_of_nonempty_colimit <| Nonempty.map h.inv inferInstance
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimit.{v, u, w, w + 1} (constPUnitFunctor C)
h_1 : Nonempty.{w + 1} (colimit.{v, u, w, w + 1} (constPUnitFunctor C) ≅ PUnit.{w + 1})
h : colimit.{v, u, w, w + 1} (constPUnitFunctor C) ≅ PUnit.{w + 1}
this : Nonempty C
⊢ IsConnected.{v, u} C
have refine : ∀ (c d : C), Zigzag.{v, u} c d
Tactic: refine zigzag_isConnected <| fun c d => ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimit.{v, u, w, w + 1} (constPUnitFunctor C)
h_1 : Nonempty.{w + 1} (colimit.{v, u, w, w + 1} (constPUnitFunctor C) ≅ PUnit.{w + 1})
h : colimit.{v, u, w, w + 1} (constPUnitFunctor C) ≅ PUnit.{w + 1}
this : Nonempty C
c d : C
⊢ Zigzag.{v, u} c d
have refine : Relation.EqvGen (α := (j : C) × Functor.obj.{v, w, u, w + 1} (constPUnitFunctor C) j)
  (Functor.ColimitTypeRel (constPUnitFunctor C))
  (Sigma.mk (β := fun (j : C) ↦ Functor.obj.{v, w, u, w + 1} (constPUnitFunctor C) j) c PUnit.unit)
  (Sigma.mk (β := fun (j : C) ↦ Functor.obj.{v, w, u, w + 1} (constPUnitFunctor C) j) d PUnit.unit)
Tactic: refine zigzag_of_eqvGen_colimitTypeRel _ (constPUnitFunctor C) ⟨c, PUnit.unit⟩ ⟨d, PUnit.unit⟩ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (goal)
Goal:
C : Type u
inst : Category.{v, u} C
⊢ IsConnected.{v, u} C ↔ Nonempty (IsColimit.{v, w, u, w + 1} (pUnitCocone C))
have refine : Nonempty (IsColimit.{v, w, u, w + 1} (pUnitCocone C)) →
  ∀ (h : IsColimit.{v, w, u, w + 1} (pUnitCocone C)), IsConnected.{v, u} C
Tactic: refine ⟨fun inst => ⟨isColimitPUnitCocone C⟩, fun ⟨h⟩ => ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
h_1 : Nonempty (IsColimit.{v, w, u, w + 1} (pUnitCocone C))
h : IsColimit.{v, w, u, w + 1} (pUnitCocone C)
colimitCocone : ColimitCocone.{v, u, w, w + 1} (constPUnitFunctor C) := { cocone := pUnitCocone C, isColimit := h }
⊢ IsConnected.{v, u} C
have this : HasColimit.{v, u, w, w + 1} (constPUnitFunctor C)
Tactic: have : HasColimit (constPUnitFunctor.{w} C) := ⟨⟨colimitCocone⟩⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (goal)
Goal:
C : Type u
inst : Category.{v, u} C
h_1 : Nonempty (IsColimit.{v, w, u, w + 1} (pUnitCocone C))
h : IsColimit.{v, w, u, w + 1} (pUnitCocone C)
colimitCocone : ColimitCocone.{v, u, w, w + 1} (constPUnitFunctor C) := { cocone := pUnitCocone C, isColimit := h }
this : HasColimit.{v, u, w, w + 1} (constPUnitFunctor C)
⊢ IsConnected.{v, u} C
have simp : Nonempty.{w + 1} (colimit.{v, u, w, w + 1} (constPUnitFunctor C) ≅ PUnit.{w + 1})
Tactic: simp only [isConnected_iff_colimit_constPUnitFunctor_iso_pUnit.{w} C]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v, v₂, u, u₂} C D
inst_2 : F.Final
⊢ IsConnected.{v, u} C ↔ IsConnected.{v₂, u₂} D
have rw : Nonempty (Limits.colimit (constPUnitFunctor C) ≅ PUnit.{(max v u v₂ u₂) + 1}) ↔
  Nonempty (Limits.colimit (constPUnitFunctor D) ≅ PUnit.{(max v u v₂ u₂) + 1})
Tactic: rw [isConnected_iff_colimit_constPUnitFunctor_iso_pUnit.{max v u v₂ u₂} C,
    isConnected_iff_colimit_constPUnitFunctor_iso_pUnit.{max v u v₂ u₂} D]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v, v₂, u, u₂} C D
inst_2 : F.Initial
⊢ IsConnected.{v, u} C ↔ IsConnected.{v₂, u₂} D
have rw : IsConnected.{v, u} Cᵒᵖ ↔ IsConnected.{v₂, u₂} Dᵒᵖ
Tactic: rw [← isConnected_op_iff_isConnected C, ← isConnected_op_iff_isConnected D]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
x : C
h : Limits.IsInitial.{v_1, u_1} x
this : Nonempty C := ⋯
⊢ IsConnected.{v_1, u_1} C
have h₁ : ∀ (j₁ j₂ : C), ∃ (l : List C), List.IsChain Zag.{v_1, u_1} (j₁ :: l) ∧ (j₁ :: l).getLast sorry = j₂
Tactic: apply isConnected_of_zigzag
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
x : C
h : Limits.IsInitial.{v_1, u_1} x
this : Nonempty C := ⋯
j₁ j₂ : C
⊢ ∃ (l : List C), List.IsChain Zag.{v_1, u_1} (j₁ :: l) ∧ (j₁ :: l).getLast ⋯ = j₂
have h₁ : List.IsChain Zag.{v_1, u_1} [j₁, x, j₂] ∧ [j₁, x, j₂].getLast sorry = j₂
Tactic: use [x, j₂]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
x : C
h : Limits.IsInitial.{v_1, u_1} x
this : Nonempty C := ⋯
j₁ j₂ : C
⊢ List.IsChain Zag.{v_1, u_1} [j₁, x, j₂] ∧ [j₁, x, j₂].getLast ⋯ = j₂
have h₁ : Zag.{v_1, u_1} j₁ x ∧ Zag.{v_1, u_1} x j₂
Tactic: simp only [List.isChain_cons_cons, List.isChain_singleton, and_true, ne_eq,
    reduceCtorEq, not_false_eq_true, List.getLast_cons, List.cons_ne_self, List.getLast_singleton]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
x : C
h : Limits.IsTerminal.{v_1, u_1} x
this : Nonempty C := ⋯
⊢ IsConnected.{v_1, u_1} C
have h₁ : ∀ (j₁ j₂ : C), ∃ (l : List C), List.IsChain Zag.{v_1, u_1} (j₁ :: l) ∧ (j₁ :: l).getLast sorry = j₂
Tactic: apply isConnected_of_zigzag
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
x : C
h : Limits.IsTerminal.{v_1, u_1} x
this : Nonempty C := ⋯
j₁ j₂ : C
⊢ ∃ (l : List C), List.IsChain Zag.{v_1, u_1} (j₁ :: l) ∧ (j₁ :: l).getLast ⋯ = j₂
have h₁ : List.IsChain Zag.{v_1, u_1} [j₁, x, j₂] ∧ [j₁, x, j₂].getLast sorry = j₂
Tactic: use [x, j₂]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsConnected (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
x : C
h : Limits.IsTerminal.{v_1, u_1} x
this : Nonempty C := ⋯
j₁ j₂ : C
⊢ List.IsChain Zag.{v_1, u_1} [j₁, x, j₂] ∧ [j₁, x, j₂].getLast ⋯ = j₂
have h₁ : Zag.{v_1, u_1} j₁ x ∧ Zag.{v_1, u_1} x j₂
Tactic: simp only [List.isChain_cons_cons, List.isChain_singleton, and_true, ne_eq,
    reduceCtorEq, not_false_eq_true, List.getLast_cons, List.cons_ne_self, List.getLast_singleton]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsLimit (goal)
Goal:
J : Type u₁
inst : Category.{v₁, u₁} J
K : Type u₂
inst_1 : Category.{v₂, u₂} K
C : Type u₃
inst_2 : Category.{v₃, u₃} C
F : Functor.{v₁, v₃, u₁, u₃} J C
t : Cone F
P Q : IsLimit t
⊢ P = Q
have mk : ∀ (lift : (s : Cone F) → s.pt ⟶ t.pt) (fac : ∀ (s : Cone F) (j : J), lift s ≫ t.π.app j = s.π.app j)
  (uniq : ∀ (s : Cone F) (m : s.pt ⟶ t.pt), (∀ (j : J), m ≫ t.π.app j = s.π.app j) → m = lift s),
  { lift := lift, fac := fac, uniq := uniq : IsLimit t } = Q
Tactic: cases P
Cleared hypotheses: P
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsLimit (goal)
Goal:
J : Type u₁
inst : Category.{v₁, u₁} J
K : Type u₂
inst_1 : Category.{v₂, u₂} K
C : Type u₃
inst_2 : Category.{v₃, u₃} C
F : Functor.{v₁, v₃, u₁, u₃} J C
t : Cone F
Q : IsLimit t
lift : (s : Cone F) → s.pt ⟶ t.pt
fac : ∀ (s : Cone F) (j : J), lift s ≫ t.π.app j = s.π.app j
uniq : ∀ (s : Cone F) (m : s.pt ⟶ t.pt), (∀ (j : J), m ≫ t.π.app j = s.π.app j) → m = lift s
⊢ { lift := lift, fac := fac, uniq := uniq : IsLimit t } = Q
have mk : ∀ (lift_1 : (s : Cone F) → s.pt ⟶ t.pt) (fac_1 : ∀ (s : Cone F) (j : J), lift_1 s ≫ t.π.app j = s.π.app j)
  (uniq_1 : ∀ (s : Cone F) (m : s.pt ⟶ t.pt), (∀ (j : J), m ≫ t.π.app j = s.π.app j) → m = lift_1 s),
  { lift := lift, fac := fac, uniq := uniq : IsLimit t } = { lift := lift_1, fac := fac_1, uniq := uniq_1 }
Tactic: cases Q
Cleared hypotheses: Q
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsLimit (goal)
Goal:
J : Type u₁
inst : Category.{v₁, u₁} J
K : Type u₂
inst_1 : Category.{v₂, u₂} K
C : Type u₃
inst_2 : Category.{v₃, u₃} C
F : Functor.{v₁, v₃, u₁, u₃} J C
t : Cone F
lift : (s : Cone F) → s.pt ⟶ t.pt
fac : ∀ (s : Cone F) (j : J), lift s ≫ t.π.app j = s.π.app j
uniq : ∀ (s : Cone F) (m : s.pt ⟶ t.pt), (∀ (j : J), m ≫ t.π.app j = s.π.app j) → m = lift s
lift_1 : (s : Cone F) → s.pt ⟶ t.pt
fac_1 : ∀ (s : Cone F) (j : J), lift_1 s ≫ t.π.app j = s.π.app j
uniq_1 : ∀ (s : Cone F) (m : s.pt ⟶ t.pt), (∀ (j : J), m ≫ t.π.app j = s.π.app j) → m = lift_1 s
⊢ { lift := lift, fac := fac, uniq := uniq : IsLimit t } = { lift := lift_1, fac := fac_1, uniq := uniq_1 }
have mk : lift = lift_1
Tactic: congr
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsLimit (goal)
Goal:
J : Type u₁
inst : Category.{v₁, u₁} J
C : Type u₃
inst_1 : Category.{v₃, u₃} C
F : Functor.{v₁, v₃, u₁, u₃} J C
s t : Cone F
h : IsLimit t
f f' g : s ⟶ t
⊢ g = h.liftConeMorphism s
have w : g.hom = (h.liftConeMorphism s).hom
Tactic: apply ConeMorphism.ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsLimit (goal)
Goal:
J : Type u₁
inst : Category.{v₁, u₁} J
K : Type u₂
inst_1 : Category.{v₂, u₂} K
C : Type u₃
inst_2 : Category.{v₃, u₃} C
F : Functor.{v₁, v₃, u₁, u₃} J C
t : Cone F
ht : ∀ (s : Cone F), ∃! l : s.pt ⟶ t.pt, ∀ (j : J), l ≫ t.π.app j = s.π.app j
⊢ IsLimit t
have choose : (s : (s : Cone F) → s.pt ⟶ t.pt) →
  (hs : ∀ (s_1 : Cone F), (fun (l : s_1.pt ⟶ t.pt) ↦ ∀ (j : J), l ≫ t.π.app j = s_1.π.app j) (s s_1)) →
    (hs' :
        ∀ (s_1 : Cone F) (y : s_1.pt ⟶ t.pt),
          (fun (l : s_1.pt ⟶ t.pt) ↦ ∀ (j : J), l ≫ t.π.app j = s_1.π.app j) y → y = s s_1) →
      IsLimit t
Tactic: choose s hs hs' using ht
Cleared hypotheses: ht
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsLimit (goal)
Goal:
J : Type u₁
inst : Category.{v₁, u₁} J
K : Type u₂
inst_1 : Category.{v₂, u₂} K
C : Type u₃
inst_2 : Category.{v₃, u₃} C
F : Functor.{v₁, v₃, u₁, u₃} J C
r t : Cone F
P : IsLimit r
i : r ≅ t
s : Cone F
m : s ⟶ t
⊢ m = (fun (s : Cone F) ↦ P.liftConeMorphism s ≫ i.hom) s
have rw : m ≫ i.inv = P.liftConeMorphism s
Tactic: rw [← i.comp_inv_eq]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsLimit (hypothesis)
Goal:
J : Type u₁
inst : Category.{v₁, u₁} J
K : Type u₂
inst_1 : Category.{v₂, u₂} K
C : Type u₃
inst_2 : Category.{v₃, u₃} C
F : Functor.{v₁, v₃, u₁, u₃} J C
r t : Cone F
P : IsLimit r
i : IsIso (P.lift t)
⊢ r ≅ t
have this : IsIso (P.liftConeMorphism t).hom
Tactic: haveI : IsIso (P.liftConeMorphism t).hom := i
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsLimit (hypothesis)
Goal:
J : Type u₁
inst : Category.{v₁, u₁} J
K : Type u₂
inst_1 : Category.{v₂, u₂} K
C : Type u₃
inst_2 : Category.{v₃, u₃} C
F : Functor.{v₁, v₃, u₁, u₃} J C
r t : Cone F
P : IsLimit r
i : IsIso (P.lift t)
this : IsIso (P.liftConeMorphism t).hom
⊢ r ≅ t
have this₁ : IsIso (P.liftConeMorphism t)
Tactic: haveI : IsIso (P.liftConeMorphism t) := Cone.cone_iso_of_hom_iso _
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsLimit (goal)
Goal:
J : Type u₁
inst : Category.{v₁, u₁} J
K : Type u₂
inst_1 : Category.{v₂, u₂} K
C : Type u₃
inst_2 : Category.{v₃, u₃} C
F : Functor.{v₁, v₃, u₁, u₃} J C
r t : Cone F
P : IsLimit r
i : IsIso (P.lift t)
this_1 : IsIso (P.liftConeMorphism t).hom
this : IsIso (P.liftConeMorphism t)
⊢ r ≅ t
have symm : t ≅ r
Tactic: symm
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsLimit (goal)
Goal:
J : Type u₁
inst : Category.{v₁, u₁} J
C : Type u₃
inst_1 : Category.{v₃, u₃} C
F : Functor.{v₁, v₃, u₁, u₃} J C
t : Cone F
h : IsLimit t
W : C
f f' : W ⟶ t.pt
w : ∀ (j : J), f ≫ t.π.app j = f' ≫ t.π.app j
⊢ f = f'
have rw : Eq (α := W ⟶ t.pt)
  (h.lift { pt := W, π := { app := fun (b : J) ↦ f ≫ t.π.app b, naturality := sorry : NatTrans (obj (const J) W) F } } :
    { pt := W, π := { app := fun (b : J) ↦ f ≫ t.π.app b, naturality := sorry : NatTrans (obj (const J) W) F } :
          Cone F }.pt ⟶
      t.pt)
  (h.lift { pt := W, π := { app := fun (b : J) ↦ f' ≫ t.π.app b, naturality := sorry : NatTrans (obj (const J) W) F } })
Tactic: rw [h.hom_lift f, h.hom_lift f']
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.IsLimit (goal)
Goal:
J : Type u₁
inst : Category.{v₁, u₁} J
C : Type u₃
inst_1 : Category.{v₃, u₃} C
F : Functor.{v₁, v₃, u₁, u₃} J C
t : Cone F
h : IsLimit t
W : C
f f' : W ⟶ t.pt
w : ∀ (j : J), f ≫ t.π.app j = f' ≫ t.π.app j
⊢ Eq (α := W ⟶ t.pt)
    (h.lift { pt := W, π := { app := fun (b : J) ↦ f ≫ t.π.app b, naturality := ⋯ : NatTrans (obj (const J) W) F } } :
      { pt := W, π := { app := fun (b : J) ↦ f ≫ t.π.app b, naturality := ⋯ : NatTrans (obj (const J) W) F } :
            Cone F }.pt ⟶
        t.pt)
    (h.lift { pt := W, π := { app := fun (b : J) ↦ f' ≫ t.π.app b, naturality := ⋯ : NatTrans (obj (const J) W) F } })
have h₁ : Eq (α := (X : J) → obj.{v₁, v₃, u₁, u₃} (obj (const J) W) X ⟶ F.obj X) (fun (b : J) ↦ f ≫ t.π.app b) fun (b : J) ↦
  f' ≫ t.π.app b
Tactic: congr