awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.Colimits (goal)
Goal:
J : Type u
inst : Category.{v, u} J
F : CategoryTheory.Functor.{v, w, u, w + 1} J AddCommGrpCat
c : Cocone F
inst_1 : DecidableEq J
inst_2 : Small.{w, max u w} (Quot F)
⊢ IsColimit (colimitCocone F)
have refine : Function.Bijective (α := Quot F) (β := (↑(colimitCocone F).pt : Type w))
  (⇑(Quot.desc F (colimitCocone F)) : (a : Quot F) → (↑(colimitCocone F).pt : Type w))
Tactic: refine isColimit_of_bijective_desc F _ ?_
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.Colimits (goal)
Goal:
J : Type u
inst : Category.{v, u} J
F : CategoryTheory.Functor.{v, w, u, w + 1} J AddCommGrpCat
c : Cocone F
inst_1 : DecidableEq J
inst_2 : Small.{w, max u w} (Quot F)
⊢ Function.Bijective (α := Quot F) (β := (↑(colimitCocone F).pt : Type w))
    (⇑(Quot.desc F (colimitCocone F)) : (a : Quot F) → (↑(colimitCocone F).pt : Type w))
have rw : Function.Bijective (α := Quot F) (β := (↑(colimitCocone F).pt : Type w))
  (⇑(AddEquiv.toAddMonoidHom (AddEquiv.symm Shrink.addEquiv)) : (a : Quot F) → (↑(colimitCocone F).pt : Type w))
Tactic: rw [Quot.desc_colimitCocone]
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.Colimits (goal)
Goal:
G H : AddCommGrpCat
f : G ⟶ H
⊢ f ≫ ofHom (mk' (Hom.hom f).range) = 0
have hf : ∀ (x : (↑G : Type u)),
  Eq (α := (↑(of ((↑H : Type u) ⧸ (Hom.hom f).range)) : Type u))
    ((Hom.hom (f ≫ ofHom (mk' (Hom.hom f).range)) :
          (a : (↑G : Type u)) → (↑(of ((↑H : Type u) ⧸ (Hom.hom f).range)) : Type u))
        x :
      (↑(of ((↑H : Type u) ⧸ (Hom.hom f).range)) : Type u))
    ((Hom.hom 0 : (a : (↑G : Type u)) → (↑(of ((↑H : Type u) ⧸ (Hom.hom f).range)) : Type u)) x :
      (↑(of ((↑H : Type u) ⧸ (Hom.hom f).range)) : Type u))
Tactic: ext x
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.Colimits (goal)
Goal:
G H : AddCommGrpCat
f : G ⟶ H
⊢ cokernel.desc f (ofHom (mk' (Hom.hom f).range)) ⋯ ≫ ofHom (lift (Hom.hom f).range (Hom.hom (cokernel.π f)) ⋯) =
    𝟙 (cokernel f)
have refine : CategoryStruct.comp (Y := coequalizer f 0) (coequalizer.π f 0)
    (cokernel.desc f (ofHom (mk' (Hom.hom f).range)) sorry ≫ ofHom (lift (Hom.hom f).range (Hom.hom (cokernel.π f)) sorry)) =
  coequalizer.π f 0 ≫ 𝟙 (cokernel f)
Tactic: refine coequalizer.hom_ext ?_
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.Colimits (goal)
Goal:
G H : AddCommGrpCat
f : G ⟶ H
⊢ CategoryStruct.comp (Y := coequalizer f 0) (coequalizer.π f 0)
      (cokernel.desc f (ofHom (mk' (Hom.hom f).range)) ⋯ ≫ ofHom (lift (Hom.hom f).range (Hom.hom (cokernel.π f)) ⋯)) =
    coequalizer.π f 0 ≫ 𝟙 (cokernel f)
have simp : ofHom (mk' (Hom.hom f).range) ≫ ofHom (lift (Hom.hom f).range (Hom.hom (cokernel.π f)) sorry) = cokernel.π f
Tactic: simp only [coequalizer_as_cokernel, cokernel.π_desc_assoc, Category.comp_id]
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.Colimits (goal)
Goal:
G H : AddCommGrpCat
f : G ⟶ H
⊢ ofHom (lift (Hom.hom f).range (Hom.hom (cokernel.π f)) ⋯) ≫ cokernel.desc f (ofHom (mk' (Hom.hom f).range)) ⋯ =
    𝟙 (of ((↑H : Type u) ⧸ (Hom.hom f).range))
have hf : ∀ (x : (↑H : Type u)),
  Eq (α := (↑(of ((↑H : Type u) ⧸ (Hom.hom f).range)) : Type u))
    (((Hom.hom
                (ofHom (lift (Hom.hom f).range (Hom.hom (cokernel.π f)) sorry) ≫
                  cokernel.desc f (ofHom (mk' (Hom.hom f).range)) sorry)).comp
            (mk' (Hom.hom f).range) :
          (a : (↑H : Type u)) → (↑(of ((↑H : Type u) ⧸ (Hom.hom f).range)) : Type u))
        x :
      (↑(of ((↑H : Type u) ⧸ (Hom.hom f).range)) : Type u))
    (((Hom.hom (𝟙 (of ((↑H : Type u) ⧸ (Hom.hom f).range)))).comp (mk' (Hom.hom f).range) :
          (a : (↑H : Type u)) → (↑(of ((↑H : Type u) ⧸ (Hom.hom f).range)) : Type u))
        x :
      (↑(of ((↑H : Type u) ⧸ (Hom.hom f).range)) : Type u))
Tactic: ext x
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A : Type u
B : Type v
inst : CommGroup A
inst_1 : CommGroup B
f : A →* B
h : ∀ (u v : B →* B ⧸ f.range), u.comp f = v.comp f → u = v
⊢ f.range = ⊤
have specialize : Eq (α := MonoidHom.{u, v} A (B ⧸ f.range)) (comp 1 f) (comp (QuotientGroup.mk' f.range) f)
Tactic: specialize h 1 (QuotientGroup.mk' _) _
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A : Type u
B : Type v
inst : CommGroup A
inst_1 : CommGroup B
f : A →* B
h : ∀ (u v : B →* B ⧸ f.range), u.comp f = v.comp f → u = v
specialize : Eq (α := MonoidHom.{u, v} A (B ⧸ f.range)) (comp 1 f) (comp (QuotientGroup.mk' f.range) f)
⊢ f.range = ⊤
have specialize₁ : ∀ (h : Eq (α := B →* B ⧸ f.range) 1 (QuotientGroup.mk' f.range)), f.range = ⊤
Tactic: specialize h 1 (QuotientGroup.mk' _) _
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A : Type u
B : Type v
inst : CommGroup A
inst_1 : CommGroup B
f : A →* B
h : ∀ (u v : B →* B ⧸ f.range), u.comp f = v.comp f → u = v
⊢ Eq (α := MonoidHom.{u, v} A (B ⧸ f.range)) (comp 1 f) (comp (QuotientGroup.mk' f.range) f)
have h₁ : ∀ (x : A),
  Eq.{v + 1} (α := B ⧸ f.range)
    (((comp 1 f : MonoidHom.{u, v} A (B ⧸ f.range)) : (a : A) → B ⧸ f.range) x : B ⧸ f.range)
    (((comp (QuotientGroup.mk' f.range) f : MonoidHom.{u, v} A (B ⧸ f.range)) : (a : A) → B ⧸ f.range) x : B ⧸ f.range)
Tactic: ext1 x
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A : Type u
B : Type v
inst : CommGroup A
inst_1 : CommGroup B
f : A →* B
h : ∀ (u v : B →* B ⧸ f.range), u.comp f = v.comp f → u = v
x : A
⊢ Eq.{v + 1} (α := B ⧸ f.range) (1 : B ⧸ f.range) ((↑((f : (a : A) → B) x) : B ⧸ f.range) : B ⧸ f.range)
have h₁ : Membership.mem (α := B) f.range ((f : (a : A) → B) x : B)
Tactic: rw [show (1 : B ⧸ f.range) = (1 : B) from QuotientGroup.mk_one _, QuotientGroup.eq, inv_one,
      one_mul]
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A : Type u
B : Type v
inst : CommGroup A
inst_1 : CommGroup B
f : A →* B
h : Eq (α := B →* B ⧸ f.range) 1 (QuotientGroup.mk' f.range)
⊢ ker (QuotientGroup.mk' f.range) = ker (M := B ⧸ f.range) 1
have rw : ker (QuotientGroup.mk' f.range) = ker (QuotientGroup.mk' f.range)
Tactic: rw [h]
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (hypothesis)
Goal:
A : Type u
B : Type v
inst : CommGroup A
inst_1 : CommGroup B
f : A →* B
h : Eq (α := B →* B ⧸ f.range) 1 (QuotientGroup.mk' f.range)
⊢ f.range = ⊤
have h₁ : ker (QuotientGroup.mk' f.range) = ker (M := B ⧸ f.range) 1
Tactic: replace h : (QuotientGroup.mk' f.range).ker = (1 : B →* B ⧸ f.range).ker := by rw [h]
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A : Type u
B : Type v
inst : CommGroup A
inst_1 : CommGroup B
f : A →* B
h : ker (QuotientGroup.mk' f.range) = ker (M := B ⧸ f.range) 1
⊢ f.range = ⊤
have rwa : ∀ (h : ker (QuotientGroup.mk' f.range) = ⊤), f.range = ⊤
Tactic: rwa [ker_one, QuotientGroup.ker_mk'] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A : Type u
B : Type v
inst : CommGroup A
inst_1 : CommGroup B
f : A →* B
h : ker (QuotientGroup.mk' f.range) = ker (M := B ⧸ f.range) 1
rwa : ∀ (h : ker (QuotientGroup.mk' f.range) = ⊤), f.range = ⊤
⊢ f.range = ⊤
have rwa₁ : ∀ (h : f.range = ⊤), f.range = ⊤
Tactic: rwa [ker_one, QuotientGroup.ker_mk'] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A B : GrpCat
f : A ⟶ B
b : (↑B : Type u)
x : X'
y : (↑X : Type u)
⊢ Membership.mem (γ := Set (Set (↑B : Type u))) X (b • (↑y : Set (↑B : Type u)))
have rw : Membership.mem (γ := Set (Set (↑B : Type u))) X
  ((b *
      Exists.choose (p := fun (y_1 : (↑B : Type u)) ↦
        (fun (x : (↑B : Type u)) ↦ x • (↑(Hom.hom f).range : Set (↑B : Type u))) y_1 = (↑y : Set (↑B : Type u))) sorry) •
    (↑(Hom.hom f).range : Set (↑B : Type u)))
Tactic: rw [← y.2.choose_spec, leftCoset_assoc]
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A B : GrpCat
f : A ⟶ B
b : (↑B : Type u)
x : X'
y : (↑X : Type u)
b' : (↑B : Type u) :=
  Exists.choose (p := fun (y_1 : (↑B : Type u)) ↦
    (fun (x : (↑B : Type u)) ↦ x • (↑(Hom.hom f).range : Set (↑B : Type u))) y_1 = (↑y : Set (↑B : Type u))) ⋯
w : (↑B : Type u)
⊢ Membership.mem (γ := Set (Set (↑B : Type u))) X
    ((b *
        Exists.choose (p := fun (y_1 : (↑B : Type u)) ↦
          (fun (x : (↑B : Type u)) ↦ x • (↑(Hom.hom f).range : Set (↑B : Type u))) y_1 = (↑y : Set (↑B : Type u))) ⋯) •
      (↑(Hom.hom f).range : Set (↑B : Type u)))
have h : (fun (x : (↑B : Type u)) ↦ x • (↑(Hom.hom f).range : Set (↑B : Type u))) (b * b') =
  (b *
      Exists.choose (p := fun (y_1 : (↑B : Type u)) ↦
        (fun (x : (↑B : Type u)) ↦ x • (↑(Hom.hom f).range : Set (↑B : Type u))) y_1 = (↑y : Set (↑B : Type u))) sorry) •
    (↑(Hom.hom f).range : Set (↑B : Type u))
Tactic: use b * b'
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A B : GrpCat
f : A ⟶ B
b : (↑B : Type u)
hb : b ∈ (Hom.hom f).range
⊢ Eq (α := X')
    (fromCoset
      (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
        (b • (↑(Hom.hom f).range : Set (↑B : Type u))) ⋯))
    (fromCoset
      (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
        (↑(Hom.hom f).range : Set (↑B : Type u)) ⋯))
have e_a : b • (↑(Hom.hom f).range : Set (↑B : Type u)) = (↑(Hom.hom f).range : Set (↑B : Type u))
Tactic: congr
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A B : GrpCat
f : A ⟶ B
b : (↑B : Type u)
hb : b ∈ (Hom.hom f).range
⊢ b • (↑(Hom.hom f).range : Set (↑B : Type u)) = (↑(Hom.hom f).range : Set (↑B : Type u))
have e_a : b • (↑(Hom.hom f).range : Set (↑B : Type u)) =
  HSMul.hSMul (α := (↑B : Type u)) 1 (↑(Hom.hom f).range : Set (↑B : Type u))
Tactic: nth_rw 2 [show (f.hom.range : Set B) = (1 : B) • f.hom.range from (one_leftCoset _).symm]
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A B : GrpCat
f : A ⟶ B
b : (↑B : Type u)
hb : b ∈ (Hom.hom f).range
⊢ b • (↑(Hom.hom f).range : Set (↑B : Type u)) =
    HSMul.hSMul (α := (↑B : Type u)) 1 (↑(Hom.hom f).range : Set (↑B : Type u))
have e_a : b⁻¹ ∈ (Hom.hom f).range
Tactic: rw [leftCoset_eq_iff, mul_one]
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (hypothesis)
Goal:
A B : GrpCat
f : A ⟶ B
b : (↑B : Type u)
hb : b ∉ (Hom.hom f).range
r :
  Eq (α := X')
    (fromCoset
      (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
        (b • (↑(Hom.hom f).range : Set (↑B : Type u))) ⋯))
    (fromCoset
      (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
        (↑(Hom.hom f).range : Set (↑B : Type u)) ⋯))
⊢ False
have r₁ : b • (↑(Hom.hom f).range : Set (↑B : Type u)) = (↑(Hom.hom f).range : Set (↑B : Type u))
Tactic: simp only [fromCoset.injEq, Subtype.mk.injEq] at r
Cleared hypotheses: r
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (hypothesis)
Goal:
A B : GrpCat
f : A ⟶ B
b : (↑B : Type u)
hb : b ∉ (Hom.hom f).range
r : b • (↑(Hom.hom f).range : Set (↑B : Type u)) = (↑(Hom.hom f).range : Set (↑B : Type u))
⊢ False
have r₁ : b • (↑(Hom.hom f).range : Set (↑B : Type u)) =
  HSMul.hSMul (α := (↑B : Type u)) 1 (↑(Hom.hom f).range : Set (↑B : Type u))
Tactic: nth_rw 2 [show (f.hom.range : Set B) = (1 : B) • f.hom.range from (one_leftCoset _).symm] at r
Cleared hypotheses: r
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (hypothesis)
Goal:
A B : GrpCat
f : A ⟶ B
b : (↑B : Type u)
hb : b ∉ (Hom.hom f).range
r :
  b • (↑(Hom.hom f).range : Set (↑B : Type u)) =
    HSMul.hSMul (α := (↑B : Type u)) 1 (↑(Hom.hom f).range : Set (↑B : Type u))
⊢ False
have r₁ : b⁻¹ ∈ (Hom.hom f).range
Tactic: rw [leftCoset_eq_iff, mul_one] at r
Cleared hypotheses: r
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A B : GrpCat
f : A ⟶ B
⊢ Eq (α := X')
    ((Equiv.symm τ : (a : X') → X')
        (fromCoset
          (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
            (↑(Hom.hom f).range : Set (↑B : Type u)) ⋯)) :
      X')
    (∞ : X')
have rw : Eq (α := X')
  ((Equiv.symm
          (Equiv.swap
            (fromCoset
              (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
                (↑(Hom.hom f).range : Set (↑B : Type u)) sorry))
            ∞) :
        (a : X') → X')
      (fromCoset
        (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
          (↑(Hom.hom f).range : Set (↑B : Type u)) sorry)) :
    X')
  (∞ : X')
Tactic: rw [tau, Equiv.symm_swap, Equiv.swap_apply_left]
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A B : GrpCat
f : A ⟶ B
rw :
  Eq (α := X')
    ((Equiv.symm
            (Equiv.swap
              (fromCoset
                (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
                  (↑(Hom.hom f).range : Set (↑B : Type u)) ⋯))
              ∞) :
          (a : X') → X')
        (fromCoset
          (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
            (↑(Hom.hom f).range : Set (↑B : Type u)) ⋯)) :
      X')
    (∞ : X')
⊢ Eq (α := X')
    ((Equiv.symm τ : (a : X') → X')
        (fromCoset
          (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
            (↑(Hom.hom f).range : Set (↑B : Type u)) ⋯)) :
      X')
    (∞ : X')
have rw₁ : Eq (α := X')
  ((Equiv.swap
          (fromCoset
            (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
              (↑(Hom.hom f).range : Set (↑B : Type u)) sorry))
          ∞ :
        (a : X') → X')
      (fromCoset
        (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
          (↑(Hom.hom f).range : Set (↑B : Type u)) sorry)) :
    X')
  (∞ : X')
Tactic: rw [tau, Equiv.symm_swap, Equiv.swap_apply_left]
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A B : GrpCat
f : A ⟶ B
rw :
  Eq (α := X')
    ((Equiv.symm
            (Equiv.swap
              (fromCoset
                (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
                  (↑(Hom.hom f).range : Set (↑B : Type u)) ⋯))
              ∞) :
          (a : X') → X')
        (fromCoset
          (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
            (↑(Hom.hom f).range : Set (↑B : Type u)) ⋯)) :
      X')
    (∞ : X')
rw₁ :
  Eq (α := X')
    ((Equiv.swap
            (fromCoset
              (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
                (↑(Hom.hom f).range : Set (↑B : Type u)) ⋯))
            ∞ :
          (a : X') → X')
        (fromCoset
          (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
            (↑(Hom.hom f).range : Set (↑B : Type u)) ⋯)) :
      X')
    (∞ : X')
⊢ Eq (α := X')
    ((Equiv.symm τ : (a : X') → X')
        (fromCoset
          (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
            (↑(Hom.hom f).range : Set (↑B : Type u)) ⋯)) :
      X')
    (∞ : X')
have rw₂ : Eq.{u + 1} (α := X') ∞ ∞
Tactic: rw [tau, Equiv.symm_swap, Equiv.swap_apply_left]
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A B : GrpCat
f : A ⟶ B
⊢ Eq (α := X') ((Equiv.symm τ : (a : X') → X') ∞ : X')
    (fromCoset
        (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
          (↑(Hom.hom f).range : Set (↑B : Type u)) ⋯) :
      X')
have rw : Eq (α := X')
  ((Equiv.symm
          (Equiv.swap
            (fromCoset
              (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
                (↑(Hom.hom f).range : Set (↑B : Type u)) sorry))
            ∞) :
        (a : X') → X')
      ∞ :
    X')
  (fromCoset
      (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
        (↑(Hom.hom f).range : Set (↑B : Type u)) sorry) :
    X')
Tactic: rw [tau, Equiv.symm_swap, Equiv.swap_apply_right]
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A B : GrpCat
f : A ⟶ B
rw :
  Eq (α := X')
    ((Equiv.symm
            (Equiv.swap
              (fromCoset
                (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
                  (↑(Hom.hom f).range : Set (↑B : Type u)) ⋯))
              ∞) :
          (a : X') → X')
        ∞ :
      X')
    (fromCoset
        (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
          (↑(Hom.hom f).range : Set (↑B : Type u)) ⋯) :
      X')
⊢ Eq (α := X') ((Equiv.symm τ : (a : X') → X') ∞ : X')
    (fromCoset
        (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
          (↑(Hom.hom f).range : Set (↑B : Type u)) ⋯) :
      X')
have rw₁ : Eq (α := X')
  ((Equiv.swap
          (fromCoset
            (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
              (↑(Hom.hom f).range : Set (↑B : Type u)) sorry))
          ∞ :
        (a : X') → X')
      ∞ :
    X')
  (fromCoset
      (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
        (↑(Hom.hom f).range : Set (↑B : Type u)) sorry) :
    X')
Tactic: rw [tau, Equiv.symm_swap, Equiv.swap_apply_right]
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A B : GrpCat
f : A ⟶ B
rw :
  Eq (α := X')
    ((Equiv.symm
            (Equiv.swap
              (fromCoset
                (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
                  (↑(Hom.hom f).range : Set (↑B : Type u)) ⋯))
              ∞) :
          (a : X') → X')
        ∞ :
      X')
    (fromCoset
        (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
          (↑(Hom.hom f).range : Set (↑B : Type u)) ⋯) :
      X')
rw₁ :
  Eq (α := X')
    ((Equiv.swap
            (fromCoset
              (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
                (↑(Hom.hom f).range : Set (↑B : Type u)) ⋯))
            ∞ :
          (a : X') → X')
        ∞ :
      X')
    (fromCoset
        (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
          (↑(Hom.hom f).range : Set (↑B : Type u)) ⋯) :
      X')
⊢ Eq (α := X') ((Equiv.symm τ : (a : X') → X') ∞ : X')
    (fromCoset
        (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
          (↑(Hom.hom f).range : Set (↑B : Type u)) ⋯) :
      X')
have rw₂ : Eq (α := X')
  (fromCoset
    (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
      (↑(Hom.hom f).range : Set (↑B : Type u)) sorry))
  (fromCoset
    (Subtype.mk (p := fun (x : Set (↑B : Type u)) ↦ Membership.mem (γ := Set (Set (↑B : Type u))) X x)
      (↑(Hom.hom f).range : Set (↑B : Type u)) sorry))
Tactic: rw [tau, Equiv.symm_swap, Equiv.swap_apply_right]
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A B : GrpCat
f : A ⟶ B
β : (↑B : Type u)
x : X'
⊢ β⁻¹ • β • x = x
have rw : (β⁻¹ * β) • x = x
Tactic: rw [← mul_smul, inv_mul_cancel, one_smul]
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A B : GrpCat
f : A ⟶ B
β : (↑B : Type u)
x : X'
rw : (β⁻¹ * β) • x = x
⊢ β⁻¹ • β • x = x
have rw₁ : HSMul.hSMul (α := (↑B : Type u)) 1 x = x
Tactic: rw [← mul_smul, inv_mul_cancel, one_smul]
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A B : GrpCat
f : A ⟶ B
β : (↑B : Type u)
x : X'
rw : (β⁻¹ * β) • x = x
rw₁ : HSMul.hSMul (α := (↑B : Type u)) 1 x = x
⊢ β⁻¹ • β • x = x
have rw₂ : x = x
Tactic: rw [← mul_smul, inv_mul_cancel, one_smul]
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A B : GrpCat
f : A ⟶ B
β : (↑B : Type u)
x : X'
⊢ β • β⁻¹ • x = x
have rw : (β * β⁻¹) • x = x
Tactic: rw [← mul_smul, mul_inv_cancel, one_smul]
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A B : GrpCat
f : A ⟶ B
β : (↑B : Type u)
x : X'
rw : (β * β⁻¹) • x = x
⊢ β • β⁻¹ • x = x
have rw₁ : HSMul.hSMul (α := (↑B : Type u)) 1 x = x
Tactic: rw [← mul_smul, mul_inv_cancel, one_smul]
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A B : GrpCat
f : A ⟶ B
β : (↑B : Type u)
x : X'
rw : (β * β⁻¹) • x = x
rw₁ : HSMul.hSMul (α := (↑B : Type u)) 1 x = x
⊢ β • β⁻¹ • x = x
have rw₂ : x = x
Tactic: rw [← mul_smul, mul_inv_cancel, one_smul]
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A B : GrpCat
f : A ⟶ B
⊢ Eq.{u + 1} (α := SX')
    { toFun := fun (x : X') ↦ HSMul.hSMul (α := (↑B : Type u)) 1 x,
      invFun := fun (x : X') ↦ HSMul.hSMul (α := (↑B : Type u)) 1⁻¹ x, left_inv := ⋯, right_inv := ⋯ }
    1
have H : ∀ (x : X'),
  Eq (α := X')
    (({ toFun := fun (x : X') ↦ HSMul.hSMul (α := (↑B : Type u)) 1 x,
            invFun := fun (x : X') ↦ HSMul.hSMul (α := (↑B : Type u)) 1⁻¹ x, left_inv := sorry, right_inv := sorry } :
          (a : X') → X')
        x :
      X')
    ((1 : (a : X') → X') x : X')
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A B : GrpCat
f : A ⟶ B
b1 b2 : (↑B : Type u)
⊢ { toFun := fun (x : X') ↦ (b1 * b2) • x, invFun := fun (x : X') ↦ (b1 * b2)⁻¹ • x, left_inv := ⋯, right_inv := ⋯ } =
    HMul.hMul (α := SX')
      { toFun := fun (x : X') ↦ b1 • x, invFun := fun (x : X') ↦ b1⁻¹ • x, left_inv := ⋯, right_inv := ⋯ : X' ≃ X' }
      { toFun := fun (x : X') ↦ b2 • x, invFun := fun (x : X') ↦ b2⁻¹ • x, left_inv := ⋯, right_inv := ⋯ : X' ≃ X' }
have H : ∀ (x : X'),
  Eq (α := X')
    (({ toFun := fun (x : X') ↦ (b1 * b2) • x, invFun := fun (x : X') ↦ (b1 * b2)⁻¹ • x, left_inv := sorry,
            right_inv := sorry } :
          (a : X') → X')
        x :
      X')
    ((HMul.hMul (α := SX')
            { toFun := fun (x : X') ↦ b1 • x, invFun := fun (x : X') ↦ b1⁻¹ • x, left_inv := sorry, right_inv := sorry :
              X' ≃ X' }
            { toFun := fun (x : X') ↦ b2 • x, invFun := fun (x : X') ↦ b2⁻¹ • x, left_inv := sorry, right_inv := sorry :
              X' ≃ X' } :
          (a : X') → X')
        x :
      X')
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A B : GrpCat
f : A ⟶ B
⊢ Equiv.trans ((Equiv.symm τ).trans ((g : (a : (↑B : Type u)) → SX') 1)) τ = 1
have H : ∀ (x : X'),
  Eq (α := X') ((Equiv.trans ((Equiv.symm τ).trans ((g : (a : (↑B : Type u)) → SX') 1)) τ : (a : X') → X') x : X')
    ((1 : (a : X') → X') x : X')
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Category.Grp.EpiMono (goal)
Goal:
A B : GrpCat
f : A ⟶ B
b1 b2 : (↑B : Type u)
⊢ Equiv.trans ((Equiv.symm τ).trans ((g : (a : (↑B : Type u)) → SX') (b1 * b2))) τ =
    Equiv.trans ((Equiv.symm τ).trans ((g : (a : (↑B : Type u)) → SX') b1)) τ *
      Equiv.trans ((Equiv.symm τ).trans ((g : (a : (↑B : Type u)) → SX') b2)) τ
have H : ∀ (x : X'),
  Eq (α := X')
    ((Equiv.trans ((Equiv.symm τ).trans ((g : (a : (↑B : Type u)) → SX') (b1 * b2))) τ : (a : X') → X') x : X')
    ((Equiv.trans ((Equiv.symm τ).trans ((g : (a : (↑B : Type u)) → SX') b1)) τ *
            Equiv.trans ((Equiv.symm τ).trans ((g : (a : (↑B : Type u)) → SX') b2)) τ :
          (a : X') → X')
        x :
      X')
Tactic: ext