awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.EpiMono (goal)
Goal:
R : Type u
inst : Ring R
X Y : ModuleCat.{v, u} R
f : X ⟶ Y
⊢ Epi f ↔
    Function.Surjective (α := (↑X : Type v)) (β := (↑Y : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat.{v, u} R) ↦ (↑x1 : Type v) →ₗ[R] (↑x2 : Type v)) (CC :=
            carrier) f) :
        (a : (↑X : Type v)) → (↑Y : Type v))
have rw : (Hom.hom f).range = ⊤ ↔
  Function.Surjective (α := (↑X : Type v)) (β := (↑Y : Type v))
    (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat.{v, u} R) ↦ (↑x1 : Type v) →ₗ[R] (↑x2 : Type v)) (CC :=
          carrier) f) :
      (a : (↑X : Type v)) → (↑Y : Type v))
Tactic: rw [epi_iff_range_eq_top, LinearMap.range_eq_top]
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.EpiMono (goal)
Goal:
R : Type u
inst : Ring R
X Y : ModuleCat.{v, u} R
f : X ⟶ Y
rw :
  (Hom.hom f).range = ⊤ ↔
    Function.Surjective (α := (↑X : Type v)) (β := (↑Y : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat.{v, u} R) ↦ (↑x1 : Type v) →ₗ[R] (↑x2 : Type v)) (CC :=
            carrier) f) :
        (a : (↑X : Type v)) → (↑Y : Type v))
⊢ Epi f ↔
    Function.Surjective (α := (↑X : Type v)) (β := (↑Y : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat.{v, u} R) ↦ (↑x1 : Type v) →ₗ[R] (↑x2 : Type v)) (CC :=
            carrier) f) :
        (a : (↑X : Type v)) → (↑Y : Type v))
have rw₁ : Function.Surjective (α := (↑X : Type v)) (β := (↑Y : Type v)) (⇑(Hom.hom f) : (a : (↑X : Type v)) → (↑Y : Type v)) ↔
  Function.Surjective (α := (↑X : Type v)) (β := (↑Y : Type v))
    (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat.{v, u} R) ↦ (↑x1 : Type v) →ₗ[R] (↑x2 : Type v)) (CC :=
          carrier) f) :
      (a : (↑X : Type v)) → (↑Y : Type v))
Tactic: rw [epi_iff_range_eq_top, LinearMap.range_eq_top]
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Ext.Basic (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : Small.{v, u} R
M N : ModuleCat.{v, u} R
r : R
mem_ann : Membership.mem (γ := Ideal R) (Module.annihilator.{u, v} R (↑N : Type v)) r
n : ℕ
⊢ Eq (α := AddCommGrpCat.of (Ext N M n) ⟶ AddCommGrpCat.of (Ext N M n))
    (AddCommGrpCat.ofHom (postcomp (mk₀ (HSMul.hSMul (β := M ⟶ M) r (𝟙 M))) N ⋯)) 0
have hf : ∀ (h : (↑(AddCommGrpCat.of (Ext N M n)) : Type v)),
  Eq (α := (↑(AddCommGrpCat.of (Ext N M n)) : Type v))
    ((AddCommGrpCat.Hom.hom (AddCommGrpCat.ofHom (postcomp (mk₀ (HSMul.hSMul (β := M ⟶ M) r (𝟙 M))) N sorry)) :
          (a : (↑(AddCommGrpCat.of (Ext N M n)) : Type v)) → (↑(AddCommGrpCat.of (Ext N M n)) : Type v))
        h :
      (↑(AddCommGrpCat.of (Ext N M n)) : Type v))
    ((AddCommGrpCat.Hom.hom 0 :
          (a : (↑(AddCommGrpCat.of (Ext N M n)) : Type v)) → (↑(AddCommGrpCat.of (Ext N M n)) : Type v))
        h :
      (↑(AddCommGrpCat.of (Ext N M n)) : Type v))
Tactic: ext h
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Ext.Basic (hypothesis)
Goal:
R : Type u
inst : CommRing R
inst_1 : Small.{v, u} R
M N : ModuleCat.{v, u} R
r : R
mem_ann : Membership.mem (γ := Ideal R) (Module.annihilator.{u, v} R (↑N : Type v)) r
n : ℕ
h : (↑(AddCommGrpCat.of (Ext N M n)) : Type v)
⊢ Eq (α := (↑(AddCommGrpCat.of (Ext N M n)) : Type v))
    ((AddCommGrpCat.Hom.hom (AddCommGrpCat.ofHom (postcomp (mk₀ (HSMul.hSMul (β := M ⟶ M) r (𝟙 M))) N ⋯)) :
          (a : (↑(AddCommGrpCat.of (Ext N M n)) : Type v)) → (↑(AddCommGrpCat.of (Ext N M n)) : Type v))
        h :
      (↑(AddCommGrpCat.of (Ext N M n)) : Type v))
    ((AddCommGrpCat.Hom.hom 0 :
          (a : (↑(AddCommGrpCat.of (Ext N M n)) : Type v)) → (↑(AddCommGrpCat.of (Ext N M n)) : Type v))
        h :
      (↑(AddCommGrpCat.of (Ext N M n)) : Type v))
have this : HSMul.hSMul (β := N ⟶ N) r (𝟙 N) = 0
Tactic: have : r • 𝟙 N = 0 := by
    simp [← ModuleCat.lsmul_eq_smul_id, Module.mem_annihilator_iff_lsmul_eq_zero.mp mem_ann]
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Ext.Basic (hypothesis)
Goal:
R : Type u
inst : CommRing R
inst_1 : Small.{v, u} R
M N : ModuleCat.{v, u} R
r : R
mem_ann : Membership.mem (γ := Ideal R) (Module.annihilator.{u, v} R (↑N : Type v)) r
n : ℕ
h : (↑(AddCommGrpCat.of (Ext N M n)) : Type v)
this : HSMul.hSMul (β := N ⟶ N) r (𝟙 N) = 0
⊢ Eq (α := (↑(AddCommGrpCat.of (Ext N M n)) : Type v))
    ((AddCommGrpCat.Hom.hom (AddCommGrpCat.ofHom (postcomp (mk₀ (HSMul.hSMul (β := M ⟶ M) r (𝟙 M))) N ⋯)) :
          (a : (↑(AddCommGrpCat.of (Ext N M n)) : Type v)) → (↑(AddCommGrpCat.of (Ext N M n)) : Type v))
        h :
      (↑(AddCommGrpCat.of (Ext N M n)) : Type v))
    ((AddCommGrpCat.Hom.hom 0 :
          (a : (↑(AddCommGrpCat.of (Ext N M n)) : Type v)) → (↑(AddCommGrpCat.of (Ext N M n)) : Type v))
        h :
      (↑(AddCommGrpCat.of (Ext N M n)) : Type v))
have smul_eq : r • h = comp (mk₀ (HSMul.hSMul (β := N ⟶ N) r (𝟙 N))) h sorry
Tactic: have smul_eq : r • h = (Ext.mk₀ (r • 𝟙 N)).comp h (zero_add n) := by simp [Ext.mk₀_smul]
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Ext.Basic (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : Small.{v, u} R
M N : ModuleCat.{v, u} R
r : R
i : ℕ
⊢ Mono (X := AddCommGrpCat.of (Ext N M i))
      (AddCommGrpCat.ofHom (postcomp (mk₀ (HSMul.hSMul (β := M ⟶ M) r (𝟙 M))) N ⋯)) ↔
    IsSMulRegular.{u, v} (Ext N M i) r
have simp : Function.Injective (α := Ext N M i) (β := Ext N M i)
    (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CC :=
          AddCommGrpCat.carrier) (X := AddCommGrpCat.of (Ext N M i))
          (AddCommGrpCat.ofHom (postcomp (mk₀ (HSMul.hSMul (β := M ⟶ M) r (𝟙 M))) N sorry))) :
      (a : Ext N M i) → Ext N M i) ↔
  Function.Injective (β := Ext N M i) fun (x : Ext N M i) ↦ r • x
Tactic: simp only [IsSMulRegular, AddCommGrpCat.mono_iff_injective]
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Ext.Basic (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : Small.{v, u} R
M N : ModuleCat.{v, u} R
r : R
i : ℕ
⊢ Function.Injective (α := Ext N M i) (β := Ext N M i)
      (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CC :=
            AddCommGrpCat.carrier) (X := AddCommGrpCat.of (Ext N M i))
            (AddCommGrpCat.ofHom (postcomp (mk₀ (HSMul.hSMul (β := M ⟶ M) r (𝟙 M))) N ⋯))) :
        (a : Ext N M i) → Ext N M i) ↔
    Function.Injective (β := Ext N M i) fun (x : Ext N M i) ↦ r • x
have a : ∀ (x : Ext N M i),
  Eq (α := (a : Ext N M i) → Ext N M i)
    (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CC :=
          AddCommGrpCat.carrier) (X := AddCommGrpCat.of (Ext N M i))
          (AddCommGrpCat.ofHom (postcomp (mk₀ (HSMul.hSMul (β := M ⟶ M) r (𝟙 M))) N sorry))) :
      (a : Ext N M i) → Ext N M i)
    (HSMul.hSMul r)
Tactic: congr!
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Ext.Basic (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : Small.{v, u} R
M N : ModuleCat.{v, u} R
r : R
i : ℕ
x : Ext N M i
⊢ Eq (α := (a : Ext N M i) → Ext N M i)
    (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CC :=
          AddCommGrpCat.carrier) (X := AddCommGrpCat.of (Ext N M i))
          (AddCommGrpCat.ofHom (postcomp (mk₀ (HSMul.hSMul (β := M ⟶ M) r (𝟙 M))) N ⋯))) :
      (a : Ext N M i) → Ext N M i)
    (HSMul.hSMul r)
have a : ∀ (x : Ext N M i),
  (ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CC :=
          AddCommGrpCat.carrier) (X := AddCommGrpCat.of (Ext N M i))
          (AddCommGrpCat.ofHom (postcomp (mk₀ (HSMul.hSMul (β := M ⟶ M) r (𝟙 M))) N sorry)) :
        (a : Ext N M i) → Ext N M i)
      x =
    r • x
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Ext.DimensionShifting (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : Small.{v, u} R
M : ModuleCat.{v, u} R
⊢ M.projectiveShortComplex.ShortExact
have h : Function.Surjective.{v + 1, v + 1} (α := (↑M : Type v) →₀ Shrink.{v, u} R) (β := (↑M : Type v))
  (⇑((Module.Basis.constr { repr := Finsupp.mapRange.linearEquiv (Shrink.linearEquiv R R) } ℕ :
            (a : (↑M : Type v) → (↑M : Type v)) → ((↑M : Type v) →₀ Shrink.{v, u} R) →ₗ[R] (↑M : Type v))
          id :
        ((↑M : Type v) →₀ Shrink.{v, u} R) →ₗ[R] (↑M : Type v)) :
    (a : (↑M : Type v) →₀ Shrink.{v, u} R) → (↑M : Type v))
Tactic: apply LinearMap.shortExact_shortComplexKer
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Ext.DimensionShifting (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : Small.{v, u} R
M : ModuleCat.{v, u} R
⊢ Function.Surjective.{v + 1, v + 1} (α := (↑M : Type v) →₀ Shrink.{v, u} R) (β := (↑M : Type v))
    (⇑((Module.Basis.constr { repr := Finsupp.mapRange.linearEquiv (Shrink.linearEquiv R R) } ℕ :
              (a : (↑M : Type v) → (↑M : Type v)) → ((↑M : Type v) →₀ Shrink.{v, u} R) →ₗ[R] (↑M : Type v))
            id :
          ((↑M : Type v) →₀ Shrink.{v, u} R) →ₗ[R] (↑M : Type v)) :
      (a : (↑M : Type v) →₀ Shrink.{v, u} R) → (↑M : Type v))
have h : ∀ (m : (↑M : Type v)),
  (((Module.Basis.constr { repr := Finsupp.mapRange.linearEquiv (Shrink.linearEquiv R R) } ℕ :
              (a : (↑M : Type v) → (↑M : Type v)) → ((↑M : Type v) →₀ Shrink.{v, u} R) →ₗ[R] (↑M : Type v))
            id :
          ((↑M : Type v) →₀ Shrink.{v, u} R) →ₗ[R] (↑M : Type v)) :
        (a : (↑M : Type v) →₀ Shrink.{v, u} R) → (↑M : Type v))
      (Finsupp.single m 1) =
    m
Tactic: refine fun m ↦ ⟨Finsupp.single m 1, ?_⟩
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Ext.Finite (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : Small.{v, u} R
inst_2 : IsNoetherianRing R
M : ModuleCat.{v, u} R
inst_3 : Module.Finite.{u, v} R (↑M : Type v)
n : ℕ
ih : ∀ (N : ModuleCat.{v, u} R) [Module.Finite.{u, v} R (↑N : Type v)], Module.Finite.{u, v} R (Ext N M n)
N : ModuleCat.{v, u} R
inst_4 : Module.Finite.{u, v} R (↑N : Type v)
⊢ Module.Finite.{u, v} R (Ext N M (n + 1))
have succ : ∀ (N_1 : Type v) (w : AddCommGroup N_1) (w_1 : Module R N_1),
  Module.Free R N_1 →
    Module.Finite R N_1 →
      ∀ (f : N_1 →ₗ[R] (↑N : Type v))
        (surjf : Function.Surjective (α := N_1) (β := (↑N : Type v)) (⇑f : (a : N_1) → (↑N : Type v))),
        Module.Finite.{u, v} R (Ext N M (n + 1))
Tactic: obtain ⟨N, _, _, _, _, f, surjf⟩ := Module.exists_finite_presentation R N
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Ext.Finite (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : Small.{v, u} R
inst_2 : IsNoetherianRing R
N M : ModuleCat.{v, u} R
inst_3 : Module.Finite.{u, v} R (↑N : Type v)
inst_4 : Module.Finite.{u, v} R (↑M : Type v)
i : ℕ
⊢ Module.Finite.{u, v} R (Ext N M i)
have zero : ∀ (N : ModuleCat.{v, u} R) [Module.Finite.{u, v} R (↑N : Type v)], Module.Finite.{u, v} R (Ext N M 0)
Tactic: induction i generalizing N with
  | zero => exact Module.Finite.equiv (Ext.linearEquiv₀.trans ModuleCat.homLinearEquiv).symm
  | succ n ih =>
    obtain ⟨N, _, _, _, _, f, surjf⟩ := Module.exists_finite_presentation R N
    let exac := LinearMap.shortExact_shortComplexKer surjf
    exact Module.Finite.of_surjective (exac.extClass.precompOfLinear R M (add_comm 1 n))
      (precomp_extClass_surjective_of_projective_X₂ M exac n)
Cleared hypotheses: N, inst_3, i
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Ext.Finite (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : Small.{v, u} R
inst_2 : IsNoetherianRing R
N M : ModuleCat.{v, u} R
inst_3 : Module.Finite.{u, v} R (↑N : Type v)
inst_4 : Module.Finite.{u, v} R (↑M : Type v)
i : ℕ
zero : ∀ (N : ModuleCat.{v, u} R) [Module.Finite.{u, v} R (↑N : Type v)], Module.Finite.{u, v} R (Ext N M 0)
⊢ Module.Finite.{u, v} R (Ext N M i)
have succ : ∀ (n : ℕ) (ih : ∀ (N : ModuleCat.{v, u} R) [Module.Finite.{u, v} R (↑N : Type v)], Module.Finite.{u, v} R (Ext N M n))
  (N : ModuleCat.{v, u} R) [Module.Finite.{u, v} R (↑N : Type v)], Module.Finite.{u, v} R (Ext N M (n + 1))
Tactic: induction i generalizing N with
  | zero => exact Module.Finite.equiv (Ext.linearEquiv₀.trans ModuleCat.homLinearEquiv).symm
  | succ n ih =>
    obtain ⟨N, _, _, _, _, f, surjf⟩ := Module.exists_finite_presentation R N
    let exac := LinearMap.shortExact_shortComplexKer surjf
    exact Module.Finite.of_surjective (exac.extClass.precompOfLinear R M (add_comm 1 n))
      (precomp_extClass_surjective_of_projective_X₂ M exac n)
Cleared hypotheses: N, inst_3, i
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.ExteriorPower (goal)
Goal:
R : Type u
inst : CommRing R
M : ModuleCat.{v, u} R
N : ModuleCat R
n : ℕ
f g : M.exteriorPower n ⟶ N
h : AlternatingMap.postcomp mk f = AlternatingMap.postcomp mk g
⊢ f = g
have hf : Hom.hom f = Hom.hom g
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.FilteredColimits (goal)
Goal:
R : Type u
inst : Ring R
J : Type v
inst_1 : SmallCategory J
inst_2 : IsFiltered J
F : J ⥤ ModuleCat R
r : R
x y : (j : J) × (↑(F.obj j) : Type (max u v))
h :
  Types.FilteredColimit.Rel
    (F ⋙
      forget (ModuleCat R) (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type (max u v)) →ₗ[R] (↑x2 : Type (max u v)))
        (CC := carrier))
    x y
⊢ Eq (α := (↑(M F) : Type (max u v))) (colimitSMulAux F r x) (colimitSMulAux F r y)
have h₁ : ∃ (k : J) (f :
  Sigma.fst (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v)))
      (Sigma.mk (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v)))
        (x.fst (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v))))
        (r • x.snd (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v))))) ⟶
    k)
  (g :
  Sigma.fst (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v)))
      (Sigma.mk (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v)))
        (y.fst (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v))))
        (r • y.snd (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v))))) ⟶
    k),
  Eq (α := (↑(F.obj k) : Type (max u v)))
    ((hom (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type (max u v)) →ₗ[R] (↑x2 : Type (max u v))) (CC := carrier)
            (F.map f) :
          (a : (↑(F.obj ⟨x.fst, r • x.snd⟩.fst) : Type (max u v))) → (↑(F.obj k) : Type (max u v)))
        (Sigma.snd (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v)))
          (Sigma.mk (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v)))
            (x.fst (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v))))
            (r • x.snd (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v)))))) :
      (↑(F.obj k) : Type (max u v)))
    ((hom (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type (max u v)) →ₗ[R] (↑x2 : Type (max u v))) (CC := carrier)
            (F.map g) :
          (a : (↑(F.obj ⟨y.fst, r • y.snd⟩.fst) : Type (max u v))) → (↑(F.obj k) : Type (max u v)))
        (Sigma.snd (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v)))
          (Sigma.mk (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v)))
            (y.fst (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v))))
            (r • y.snd (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v)))))) :
      (↑(F.obj k) : Type (max u v)))
Tactic: apply M.mk_eq
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.FilteredColimits (goal)
Goal:
R : Type u
inst : Ring R
J : Type v
inst_1 : SmallCategory J
inst_2 : IsFiltered J
F : J ⥤ ModuleCat R
r : R
x : (↑(M F) : Type (max u v))
⊢ (↑(M F) : Type (max u v))
have refine : ∀ (a b : (j : J) × (↑(F.obj j) : Type (max u v)))
  (a_1 :
    ((((F ⋙
                forget₂ (ModuleCat R) (FC := fun (x1 x2 : ModuleCat R) ↦
                  (↑x1 : Type (max u v)) →ₗ[R] (↑x2 : Type (max u v))) (CC := carrier) AddCommGrpCat (FD :=
                  fun (x1 x2 : AddCommGrpCat) ↦ (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CD :=
                  AddCommGrpCat.carrier)) ⋙
              forget₂ AddCommGrpCat (FC := fun (x1 x2 : AddCommGrpCat) ↦
                (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CC := AddCommGrpCat.carrier) AddGrpCat (FD :=
                fun (x1 x2 : AddGrpCat) ↦ (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CD := AddGrpCat.carrier)) ⋙
            forget₂ AddGrpCat (FC := fun (x1 x2 : AddGrpCat) ↦ (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CC :=
              AddGrpCat.carrier) AddMonCat (FD := fun (x1 x2 : AddMonCat) ↦
              (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CD := AddMonCat.carrier)) ⋙
          forget AddMonCat (FC := fun (x1 x2 : AddMonCat) ↦ (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CC :=
            AddMonCat.carrier)).ColimitTypeRel
      a b),
  Eq (α := (↑(M F) : Type (max u v))) (colimitSMulAux F r a) (colimitSMulAux F r b)
Tactic: refine Quot.lift (colimitSMulAux F r) ?_ x
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.FilteredColimits (goal)
Goal:
R : Type u
inst : Ring R
J : Type v
inst_1 : SmallCategory J
inst_2 : IsFiltered J
F : J ⥤ ModuleCat R
r : R
x_1 : (↑(M F) : Type (max u v))
x y : (j : J) × (↑(F.obj j) : Type (max u v))
h :
  ((((F ⋙
              forget₂ (ModuleCat R) (FC := fun (x1 x2 : ModuleCat R) ↦
                (↑x1 : Type (max u v)) →ₗ[R] (↑x2 : Type (max u v))) (CC := carrier) AddCommGrpCat (FD :=
                fun (x1 x2 : AddCommGrpCat) ↦ (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CD :=
                AddCommGrpCat.carrier)) ⋙
            forget₂ AddCommGrpCat (FC := fun (x1 x2 : AddCommGrpCat) ↦ (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v)))
              (CC := AddCommGrpCat.carrier) AddGrpCat (FD := fun (x1 x2 : AddGrpCat) ↦
              (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CD := AddGrpCat.carrier)) ⋙
          forget₂ AddGrpCat (FC := fun (x1 x2 : AddGrpCat) ↦ (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CC :=
            AddGrpCat.carrier) AddMonCat (FD := fun (x1 x2 : AddMonCat) ↦
            (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CD := AddMonCat.carrier)) ⋙
        forget AddMonCat (FC := fun (x1 x2 : AddMonCat) ↦ (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CC :=
          AddMonCat.carrier)).ColimitTypeRel
    x y
⊢ Eq (α := (↑(M F) : Type (max u v))) (colimitSMulAux F r x) (colimitSMulAux F r y)
have h₁ : Types.FilteredColimit.Rel
  (F ⋙
    forget (ModuleCat R) (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type (max u v)) →ₗ[R] (↑x2 : Type (max u v))) (CC :=
      carrier))
  x y
Tactic: apply colimitSMulAux_eq_of_rel
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.FilteredColimits (goal)
Goal:
R : Type u
inst : Ring R
J : Type v
inst_1 : SmallCategory J
inst_2 : IsFiltered J
F : J ⥤ ModuleCat R
r : R
x_1 : (↑(M F) : Type (max u v))
x y : (j : J) × (↑(F.obj j) : Type (max u v))
h :
  ((((F ⋙
              forget₂ (ModuleCat R) (FC := fun (x1 x2 : ModuleCat R) ↦
                (↑x1 : Type (max u v)) →ₗ[R] (↑x2 : Type (max u v))) (CC := carrier) AddCommGrpCat (FD :=
                fun (x1 x2 : AddCommGrpCat) ↦ (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CD :=
                AddCommGrpCat.carrier)) ⋙
            forget₂ AddCommGrpCat (FC := fun (x1 x2 : AddCommGrpCat) ↦ (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v)))
              (CC := AddCommGrpCat.carrier) AddGrpCat (FD := fun (x1 x2 : AddGrpCat) ↦
              (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CD := AddGrpCat.carrier)) ⋙
          forget₂ AddGrpCat (FC := fun (x1 x2 : AddGrpCat) ↦ (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CC :=
            AddGrpCat.carrier) AddMonCat (FD := fun (x1 x2 : AddMonCat) ↦
            (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CD := AddMonCat.carrier)) ⋙
        forget AddMonCat (FC := fun (x1 x2 : AddMonCat) ↦ (↑x1 : Type (max u v)) →+ (↑x2 : Type (max u v))) (CC :=
          AddMonCat.carrier)).ColimitTypeRel
    x y
⊢ Types.FilteredColimit.Rel
    (F ⋙
      forget (ModuleCat R) (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type (max u v)) →ₗ[R] (↑x2 : Type (max u v)))
        (CC := carrier))
    x y
have h₁ : (F ⋙
      forget (ModuleCat R) (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type (max u v)) →ₗ[R] (↑x2 : Type (max u v)))
        (CC := carrier)).ColimitTypeRel
  x y
Tactic: apply Types.FilteredColimit.rel_of_colimitTypeRel
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.FilteredColimits (goal)
Goal:
R : Type u
inst : Ring R
J : Type v
inst_1 : SmallCategory J
inst_2 : IsFiltered J
F : J ⥤ ModuleCat R
r s : R
x : (↑(M F) : Type (max u v))
⊢ (r * s) • x = r • s • x
have obtain : ∀ (j : J) (x : (↑(F.obj j) : Type (max u v))),
  HSMul.hSMul (β := (↑(M F) : Type (max u v))) (r * s)
      (M.mk F (Sigma.mk (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v))) j x)) =
    r •
      HSMul.hSMul (β := (↑(M F) : Type (max u v))) s
        (M.mk F (Sigma.mk (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v))) j x))
Tactic: obtain ⟨j, x, rfl⟩ := M.mk_surjective F x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.FilteredColimits (goal)
Goal:
R : Type u
inst : Ring R
J : Type v
inst_1 : SmallCategory J
inst_2 : IsFiltered J
F : J ⥤ ModuleCat R
x : (↑(M F) : Type (max u v))
⊢ HSMul.hSMul (α := R) 1 x = x
have obtain : ∀ (j : J) (x : (↑(F.obj j) : Type (max u v))),
  HSMul.hSMul (α := R) (β := (↑(M F) : Type (max u v))) 1
      (M.mk F (Sigma.mk (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v))) j x)) =
    M.mk F (Sigma.mk (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v))) j x)
Tactic: obtain ⟨j, x, rfl⟩ := M.mk_surjective F x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.FilteredColimits (goal)
Goal:
R : Type u
inst : Ring R
J : Type v
inst_1 : SmallCategory J
inst_2 : IsFiltered J
F : J ⥤ ModuleCat R
r : R
⊢ HSMul.hSMul (β := (↑(M F) : Type (max u v))) r 0 = 0
have rw : HSMul.hSMul (β := (↑(M F) : Type (max u v))) r
    (M.mk F (Sigma.mk (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v))) (Nonempty.some sorry) 0)) =
  M.mk F (Sigma.mk (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v))) (Nonempty.some sorry) 0)
Tactic: rw [colimit_zero_eq _ (IsFiltered.nonempty.some : J), colimit_smul_mk_eq, smul_zero]
-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.FilteredColimits (goal)
Goal:
R : Type u
inst : Ring R
J : Type v
inst_1 : SmallCategory J
inst_2 : IsFiltered J
F : J ⥤ ModuleCat R
r : R
rw :
  HSMul.hSMul (β := (↑(M F) : Type (max u v))) r
      (M.mk F (Sigma.mk (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v))) (Nonempty.some ⋯) 0)) =
    M.mk F (Sigma.mk (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v))) (Nonempty.some ⋯) 0)
⊢ HSMul.hSMul (β := (↑(M F) : Type (max u v))) r 0 = 0
have rw₁ : Eq (α := (↑(M F) : Type (max u v)))
  (M.mk F
    (Sigma.mk (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v)))
      (Sigma.fst (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v)))
        (Sigma.mk (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v))) (Nonempty.some sorry) 0))
      (HSMul.hSMul (β :=
        (↑(F.obj
              (Sigma.fst (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v)))
                (Sigma.mk (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v))) (Nonempty.some sorry) 0))) :
          Type (max u v)))
        r
        (Sigma.snd (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v)))
            (Sigma.mk (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v))) (Nonempty.some sorry) 0) :
          (↑(F.obj
                (Sigma.fst (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v)))
                  (Sigma.mk (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v))) (Nonempty.some sorry) 0))) :
            Type (max u v))))))
  (M.mk F (Sigma.mk (β := fun (j : J) ↦ (↑(F.obj j) : Type (max u v))) (Nonempty.some sorry) 0))
Tactic: rw [colimit_zero_eq _ (IsFiltered.nonempty.some : J), colimit_smul_mk_eq, smul_zero]