mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
X : C
f : X ⟶ T.obj₂
hf : f ≫ T.mor₂ = 0
a : (contractibleTriangle X).obj₁ ⟶ T.obj₁
ha₁ : (contractibleTriangle X).mor₁ ≫ f = a ≫ T.mor₁
right✝ : (contractibleTriangle X).mor₃ ≫ (CategoryTheory.shiftFunctor C 1).map a = 0 ≫ T.mor₃
⊢ ∃ g, f = g ≫ T.mor₁

Tactic:
exact ⟨a, by simpa using ha₁⟩

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Preadditive C
inst✝² : HasBinaryBiproducts C
inst✝¹ : Preadditive D
inst✝ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
K₁ L₁ K₂ L₂ K₃ L₃ : CochainComplex C ℤ
φ₁ : K₁ ⟶ L₁
φ₂ : K₂ ⟶ L₂
φ₃ : K₃ ⟶ L₃
a : K₁ ⟶ K₂
b : L₁ ⟶ L₂
comm : φ₁ ≫ b = a ≫ φ₂
a' : K₂ ⟶ K₃
b' : L₂ ⟶ L₃
comm' : φ₂ ≫ b' = a' ≫ φ₃
⊢ φ₁ ≫ b ≫ b' = (a ≫ a') ≫ φ₃

Tactic:
by rw [reassoc_of% comm, comm', assoc]

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j ↦ (T j).obj₁) j) (Pi.π (fun j ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
a : A ⟶ (productTriangle T).obj₃
a' : A ⟶ Z
ha' : a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃
j : J
⊢ a ≫ Pi.π (fun j ↦ (T j).obj₃) j ≫ (T j).mor₃ - a' ≫ φ'.hom₃ ≫ Pi.π (fun j ↦ (T j).obj₃) j ≫ (T j).mor₃ = 0

Tactic:
erw [← (productTriangle.π T j).comm₃]

Goal state after:
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j ↦ (T j).obj₁) j) (Pi.π (fun j ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
a : A ⟶ (productTriangle T).obj₃
a' : A ⟶ Z
ha' : a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃
j : J
⊢ a ≫ (productTriangle T).mor₃ ≫ (shiftFunctor C 1).map (productTriangle.π T j).hom₁ -
      a' ≫ φ'.hom₃ ≫ (productTriangle T).mor₃ ≫ (shiftFunctor C 1).map (productTriangle.π T j).hom₁ =
    0
Module: Mathlib.CategoryTheory.Triangulated.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝²⁰ : Category.{v_1, u_1} C
inst✝¹⁹ : Category.{v_2, u_2} D
inst✝¹⁸ : Category.{v_3, u_3} E
inst✝¹⁷ : HasShift C ℤ
inst✝¹⁶ : HasShift D ℤ
inst✝¹⁵ : HasShift E ℤ
F : C ⥤ D
inst✝¹⁴ : F.CommShift ℤ
G : D ⥤ E
inst✝¹³ : G.CommShift ℤ
inst✝¹² : HasZeroObject C
inst✝¹¹ : HasZeroObject D
inst✝¹⁰ : HasZeroObject E
inst✝⁹ : Preadditive C
inst✝⁸ : Preadditive D
inst✝⁷ : Preadditive E
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor E n).Additive
inst✝³ : Pretriangulated C
inst✝² : Pretriangulated D
inst✝¹ : Pretriangulated E
inst✝ : F.IsTriangulated
⊢ PreservesLimitsOfShape (Discrete WalkingPair) F

Tactic:
suffices ∀ (X₁ X₃ : C), IsIso (prodComparison F X₁ X₃) by
    have := fun (X₁ X₃ : C) ↦ PreservesLimitPair.of_iso_prod_comparison F X₁ X₃
    exact ⟨fun {K} ↦ preservesLimit_of_iso_diagram F (diagramIsoPair K).symm⟩

Goal state after:
C : Type u_1
D : Type u_2
E : Type u_3
inst✝²⁰ : Category.{v_1, u_1} C
inst✝¹⁹ : Category.{v_2, u_2} D
inst✝¹⁸ : Category.{v_3, u_3} E
inst✝¹⁷ : HasShift C ℤ
inst✝¹⁶ : HasShift D ℤ
inst✝¹⁵ : HasShift E ℤ
F : C ⥤ D
inst✝¹⁴ : F.CommShift ℤ
G : D ⥤ E
inst✝¹³ : G.CommShift ℤ
inst✝¹² : HasZeroObject C
inst✝¹¹ : HasZeroObject D
inst✝¹⁰ : HasZeroObject E
inst✝⁹ : Preadditive C
inst✝⁸ : Preadditive D
inst✝⁷ : Preadditive E
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor E n).Additive
inst✝³ : Pretriangulated C
inst✝² : Pretriangulated D
inst✝¹ : Pretriangulated E
inst✝ : F.IsTriangulated
⊢ ∀ (X₁ X₃ : C), IsIso (prodComparison F X₁ X₃)
Module: Mathlib.Algebra.Homology.HomologySequence

Goal state before:
pp: case refine_2
C : Type u_1
ι : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.Exact
inst✝³ : Epi S.g
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
this : Epi (S.map (eval C c i)).g
hj : IsColimit (CokernelCofork.ofπ (S.map (eval C c i)).g ⋯)
A : C
k : S.X₂.opcycles i ⟶ A
hk : opcyclesMap S.f i ≫ k = 0
H : { l // S.g.f i ≫ l = S.X₂.pOpcycles i ≫ k }
⊢ S.X₂.pOpcycles i ≫ S.X₂.descOpcycles (S.g.f i ≫ ↑H) (c.prev i) ⋯ ⋯ = S.X₂.pOpcycles i ≫ k

Tactic:
p_descOpcycles,

Goal state after:
case refine_2
C : Type u_1
ι : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.Exact
inst✝³ : Epi S.g
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
this : Epi (S.map (eval C c i)).g
hj : IsColimit (CokernelCofork.ofπ (S.map (eval C c i)).g ⋯)
A : C
k : S.X₂.opcycles i ⟶ A
hk : opcyclesMap S.f i ≫ k = 0
H : { l // S.g.f i ≫ l = S.X₂.pOpcycles i ≫ k }
⊢ S.g.f i ≫ ↑H = S.X₂.pOpcycles i ≫ k
Module: Mathlib.CategoryTheory.PathCategory.Basic

Goal state before:
pp: case h_map
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{v_1, u_1} C
F G : Paths V ⥤ C
h_obj : F.obj = G.obj
h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯
⊢ autoParam (∀ (X Y : Paths V) (f : X ⟶ Y), F.map f = eqToHom ⋯ ≫ G.map f ≫ eqToHom ⋯) Functor.ext._auto_1

Tactic:
intro X Y f

Goal state after:
case h_map
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{v_1, u_1} C
F G : Paths V ⥤ C
h_obj : F.obj = G.obj
h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯
X Y : Paths V
f : X ⟶ Y
⊢ F.map f = eqToHom ⋯ ≫ G.map f ≫ eqToHom ⋯
Module: Mathlib.CategoryTheory.Localization.Construction

Goal state before:
pp: C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G : C ⥤ D
hG : W.IsInvertedBy G
⊢ ∀ (X Y : C) (f : X ⟶ Y), (W.Q ⋙ lift G hG).map f = eqToHom ⋯ ≫ G.map f ≫ eqToHom ⋯

Tactic:
intro X Y f

Goal state after:
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G : C ⥤ D
hG : W.IsInvertedBy G
X Y : C
f : X ⟶ Y
⊢ (W.Q ⋙ lift G hG).map f = eqToHom ⋯ ≫ G.map f ≫ eqToHom ⋯
Module: Mathlib.CategoryTheory.Localization.Predicate

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
h₁ : Localization.StrictUniversalPropertyFixedTarget L W D
h₂ : Localization.StrictUniversalPropertyFixedTarget L W W.Localization
⊢ W.Q ⋙ 𝟭 W.Localization = W.Q ⋙ Localization.Construction.lift L ⋯ ⋙ h₂.lift W.Q ⋯

Tactic:
by
        simp only [← Functor.assoc, Localization.Construction.fac, h₂.fac, Functor.comp_id]

Goal state after:
Module: Mathlib.CategoryTheory.Localization.Construction

Goal state before:
pp: C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G : C ⥤ D
hG : W.IsInvertedBy G
F₁ F₂ : W.Localization ⥤ D
τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂
this : MorphismProperty.naturalityProperty (NatTransExtension.app τ) = ⊤
⊢ ∀ ⦃X Y : W.Localization⦄ (f : X ⟶ Y), F₁.map f ≫ NatTransExtension.app τ Y = NatTransExtension.app τ X ≫ F₂.map f

Tactic:
intro X Y f

Goal state after:
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G : C ⥤ D
hG : W.IsInvertedBy G
F₁ F₂ : W.Localization ⥤ D
τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂
this : MorphismProperty.naturalityProperty (NatTransExtension.app τ) = ⊤
X Y : W.Localization
f : X ⟶ Y
⊢ F₁.map f ≫ NatTransExtension.app τ Y = NatTransExtension.app τ X ≫ F₂.map f
Module: Mathlib.CategoryTheory.Localization.Construction

Goal state before:
pp: C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G : C ⥤ D
hG : W.IsInvertedBy G
F₁ F₂ : W.Localization ⥤ D
τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂
⊢ ∀ ⦃X Y : C⦄ (f : X ⟶ Y), MorphismProperty.naturalityProperty (NatTransExtension.app τ) (W.Q.map f)

Tactic:
intro X Y f

Goal state after:
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G : C ⥤ D
hG : W.IsInvertedBy G
F₁ F₂ : W.Localization ⥤ D
τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂
X Y : C
f : X ⟶ Y
⊢ MorphismProperty.naturalityProperty (NatTransExtension.app τ) (W.Q.map f)
Module: Mathlib.CategoryTheory.Localization.Construction

Goal state before:
pp: case refine_2
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G : C ⥤ D
hG : W.IsInvertedBy G
⊢ ∀ (X Y : W.FunctorsInverting D) (f : X ⟶ Y),
    (inverse W D ⋙ functor W D).map f = eqToHom ⋯ ≫ (𝟭 (W.FunctorsInverting D)).map f ≫ eqToHom ⋯

Tactic:
rintro ⟨G₁, hG₁⟩ ⟨G₂, hG₂⟩ f

Goal state after:
case refine_2
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G : C ⥤ D
hG : W.IsInvertedBy G
G₁ : C ⥤ D
hG₁ : W.IsInvertedBy G₁
G₂ : C ⥤ D
hG₂ : W.IsInvertedBy G₂
f : { obj := G₁, property := hG₁ } ⟶ { obj := G₂, property := hG₂ }
⊢ (inverse W D ⋙ functor W D).map f = eqToHom ⋯ ≫ (𝟭 (W.FunctorsInverting D)).map f ≫ eqToHom ⋯
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f : L.obj X ⟶ L.obj Y
α : W.LeftFraction X Y
hα : f = α.map L ⋯
⊢ L.map 0 = { Y' := α.Y', f := α.f, f' := 0, s := α.s, hs := ⋯ }.snd.map L ⋯

Tactic:
rw [← cancel_mono (L.map α.s), ← L.map_comp, Limits.zero_comp,
      LeftFraction.map_comp_map_s]

Goal state after:
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions

Goal state before:
pp: case mp
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
φ ψ : W.LeftFraction₂ X Y
⊢ LeftFractionRel φ.fst ψ.fst ∧ LeftFractionRel φ.snd ψ.snd → LeftFraction₂Rel φ ψ

pp: case mpr
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
φ ψ : W.LeftFraction₂ X Y
⊢ LeftFraction₂Rel φ ψ → LeftFractionRel φ.fst ψ.fst ∧ LeftFractionRel φ.snd ψ.snd

Tactic:
· intro ⟨h, h'⟩
    obtain ⟨Z, t₁, t₂, hst, hft, ht⟩ := h
    obtain ⟨Z', t₁', t₂', hst', hft', ht'⟩ := h'
    dsimp at t₁ t₂ t₁' t₂' hst hft hst' hft' ht ht'
    have ⟨α, hα⟩ := (RightFraction.mk _ ht (φ.s ≫ t₁')).exists_leftFraction
    simp only [Category.assoc] at hα
    obtain ⟨Z'', u, hu, fac⟩ := HasLeftCalculusOfFractions.ext _ _ _ φ.hs hα
    have hα' : ψ.s ≫ t₂ ≫ α.f ≫ u = ψ.s ≫ t₂' ≫ α.s ≫ u := by
      rw [← reassoc_of% hst, ← reassoc_of% hα, ← reassoc_of% hst']
    obtain ⟨Z''', u', hu', fac'⟩ := HasLeftCalculusOfFractions.ext _ _ _ ψ.hs hα'
    simp only [Category.assoc] at fac fac'
    refine ⟨Z''', t₁' ≫ α.s ≫ u ≫ u', t₂' ≫ α.s ≫ u ≫ u', ?_, ?_, ?_, ?_⟩
    · rw [reassoc_of% hst']
    · rw [reassoc_of% fac, reassoc_of% hft, fac']
    · rw [reassoc_of% hft']
    · rw [← Category.assoc]
      exact W.comp_mem _ _ ht' (W.comp_mem _ _ α.hs (W.comp_mem _ _ hu hu'))

Goal state after:
case mpr
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
φ ψ : W.LeftFraction₂ X Y
⊢ LeftFraction₂Rel φ ψ → LeftFractionRel φ.fst ψ.fst ∧ LeftFractionRel φ.snd ψ.snd
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
W : MorphismProperty C
X Y : C
φ : W.RightFraction₂ X Y
inst✝ : W.HasLeftCalculusOfFractions
⊢ ∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f ∧ φ.f' ≫ ψ.s = φ.s ≫ ψ.f'

Tactic:
by
  obtain ⟨ψ₁, hψ₁⟩ := φ.fst.exists_leftFraction
  obtain ⟨ψ₂, hψ₂⟩ := φ.snd.exists_leftFraction
  obtain ⟨α, hα⟩ := (RightFraction.mk _ ψ₁.hs ψ₂.s).exists_leftFraction
  dsimp at hψ₁ hψ₂ hα
  refine ⟨LeftFraction₂.mk (ψ₁.f ≫ α.f) (ψ₂.f ≫ α.s) (ψ₂.s ≫ α.s)
      (W.comp_mem _ _ ψ₂.hs α.hs), ?_, ?_⟩
  · dsimp
    rw [hα, reassoc_of% hψ₁]
  · rw [reassoc_of% hψ₂]

Goal state after:
Module: Mathlib.SetTheory.Cardinal.HasCardinalLT

Goal state before:
pp: X : Type u
κ : Cardinal.{v}
h : HasCardinalLT X κ
Y : Type u'
f : Y → X
hf : Function.Injective f
⊢ HasCardinalLT Y κ

Tactic:
dsimp [HasCardinalLT] at h ⊢

Goal state after:
X : Type u
κ : Cardinal.{v}
h : Cardinal.lift.{v, u} (Cardinal.mk X) < Cardinal.lift.{u, v} κ
Y : Type u'
f : Y → X
hf : Function.Injective f
⊢ Cardinal.lift.{v, u'} (Cardinal.mk Y) < Cardinal.lift.{u', v} κ
Module: Mathlib.SetTheory.Cardinal.HasCardinalLT

Goal state before:
pp: X : Type u
κ : Cardinal.{v}
h : HasCardinalLT X κ
Y : Type u'
f : X → Y
hf : Function.Surjective f
⊢ HasCardinalLT Y κ

Tactic:
dsimp [HasCardinalLT] at h ⊢

Goal state after:
X : Type u
κ : Cardinal.{v}
h : Cardinal.lift.{v, u} (Cardinal.mk X) < Cardinal.lift.{u, v} κ
Y : Type u'
f : X → Y
hf : Function.Surjective f
⊢ Cardinal.lift.{v, u'} (Cardinal.mk Y) < Cardinal.lift.{u', v} κ
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: case refine_2.refine_2
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y Z : C
z₁✝ : Hom W X Y
z₂✝ : Hom W Y Z
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z
U : C
t₁ : a₁.Y' ⟶ U
t₂ : a₂.Y' ⟶ U
hst : a₁.s ≫ t₁ = a₂.s ≫ t₂
hft : a₁.f ≫ t₁ = a₂.f ≫ t₂
ht : W (a₁.s ≫ t₁)
z₁ : W.LeftFraction a₁.Y' b.Y'
fac₁ : b.f ≫ z₁.s = a₁.s ≫ z₁.f
z₂ : W.LeftFraction a₂.Y' b.Y'
fac₂ : b.f ≫ z₂.s = a₂.s ≫ z₂.f
w₁ : W.LeftFraction U z₁.Y'
w₂ : W.LeftFraction U z₂.Y'
p₁ : W.LeftFraction X Z := { Y' := w₁.Y', f := a₁.f ≫ t₁ ≫ w₁.f, s := b.s ≫ z₁.s ≫ w₁.s, hs := ⋯ }
p₂ : W.LeftFraction X Z := { Y' := w₂.Y', f := a₂.f ≫ t₂ ≫ w₂.f, s := b.s ≫ z₂.s ≫ w₂.s, hs := ⋯ }
fac₁' : b.f ≫ z₁.s ≫ w₁.s = a₁.s ≫ t₁ ≫ w₁.f
fac₂' : b.f ≫ z₂.s ≫ w₂.s = a₂.s ≫ t₂ ≫ w₂.f
q : W.LeftFraction w₁.Y' w₂.Y'
fac₃ : z₂.s ≫ w₂.s ≫ q.s = z₁.s ≫ w₁.s ≫ q.f
eq : a₁.s ≫ t₁ ≫ w₁.f ≫ q.f = a₁.s ≫ t₁ ≫ w₂.f ≫ q.s
⊢ LeftFractionRel p₁ p₂

Tactic:
obtain ⟨Z, u, hu, fac₄⟩ := HasLeftCalculusOfFractions.ext _ _ _ a₁.hs eq

Goal state after:
case refine_2.refine_2
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y Z✝ : C
z₁✝ : Hom W X Y
z₂✝ : Hom W Y Z✝
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z✝
U : C
t₁ : a₁.Y' ⟶ U
t₂ : a₂.Y' ⟶ U
hst : a₁.s ≫ t₁ = a₂.s ≫ t₂
hft : a₁.f ≫ t₁ = a₂.f ≫ t₂
ht : W (a₁.s ≫ t₁)
z₁ : W.LeftFraction a₁.Y' b.Y'
fac₁ : b.f ≫ z₁.s = a₁.s ≫ z₁.f
z₂ : W.LeftFraction a₂.Y' b.Y'
fac₂ : b.f ≫ z₂.s = a₂.s ≫ z₂.f
w₁ : W.LeftFraction U z₁.Y'
w₂ : W.LeftFraction U z₂.Y'
p₁ : W.LeftFraction X Z✝ := { Y' := w₁.Y', f := a₁.f ≫ t₁ ≫ w₁.f, s := b.s ≫ z₁.s ≫ w₁.s, hs := ⋯ }
p₂ : W.LeftFraction X Z✝ := { Y' := w₂.Y', f := a₂.f ≫ t₂ ≫ w₂.f, s := b.s ≫ z₂.s ≫ w₂.s, hs := ⋯ }
fac₁' : b.f ≫ z₁.s ≫ w₁.s = a₁.s ≫ t₁ ≫ w₁.f
fac₂' : b.f ≫ z₂.s ≫ w₂.s = a₂.s ≫ t₂ ≫ w₂.f
q : W.LeftFraction w₁.Y' w₂.Y'
fac₃ : z₂.s ≫ w₂.s ≫ q.s = z₁.s ≫ w₁.s ≫ q.f
eq : a₁.s ≫ t₁ ≫ w₁.f ≫ q.f = a₁.s ≫ t₁ ≫ w₂.f ≫ q.s
Z : C
u : q.Y' ⟶ Z
hu : W u
fac₄ : (t₁ ≫ w₁.f ≫ q.f) ≫ u = (t₁ ≫ w₂.f ≫ q.s) ≫ u
⊢ LeftFractionRel p₁ p₂
Module: Mathlib.SetTheory.Cardinal.HasCardinalLT

Goal state before:
pp: X : Type u
κ : Cardinal.{w}
hκ : Cardinal.aleph0 ≤ κ
⊢ HasCardinalLT (Option X) κ ↔ HasCardinalLT X κ

Tactic:
hasCardinalLT_iff_of_equiv (Equiv.optionEquivSumPUnit.{0} X),

Goal state after:
X : Type u
κ : Cardinal.{w}
hκ : Cardinal.aleph0 ≤ κ
⊢ HasCardinalLT (X ⊕ PUnit.{1}) κ ↔ HasCardinalLT X κ
Module: Mathlib.SetTheory.Cardinal.HasCardinalLT

Goal state before:
pp: X : Type u
κ : Cardinal.{w}
hκ : Cardinal.aleph0 ≤ κ
⊢ HasCardinalLT X κ → HasCardinalLT PUnit.{1} κ

Tactic:
refine fun _ ↦ HasCardinalLT.of_le ?_ hκ

Goal state after:
X : Type u
κ : Cardinal.{w}
hκ : Cardinal.aleph0 ≤ κ
x✝ : HasCardinalLT X κ
⊢ HasCardinalLT PUnit.{1} Cardinal.aleph0
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsClosure

Goal state before:
pp: C : Type u
inst✝¹¹ : Category.{v, u} C
P : ObjectProperty C
α : Type t
J : α → Type u'
inst✝¹⁰ : (a : α) → Category.{v', u'} (J a)
β : Type w'
inst✝⁹ : LinearOrder β
inst✝⁸ : OrderBot β
inst✝⁷ : SuccOrder β
inst✝⁶ : WellFoundedLT β
inst✝⁵ : ObjectProperty.Small.{w, v, u} P
inst✝⁴ : LocallySmall.{w, v, u} C
inst✝³ : Small.{w, t} α
inst✝² : ∀ (a : α), Small.{w, u'} (J a)
inst✝¹ : ∀ (a : α), LocallySmall.{w, v', u'} (J a)
b✝ : β
hb₀ : Small.{w, w'} ↑(Set.Iio b✝)
b c : β
hbc : b ≤ c
inst✝ : Small.{w, w'} ↑(Set.Iio c)
x✝² x✝¹ : ↑(Set.Iio b)
x✝ : (fun x ↦ ⟨↑x, ⋯⟩) x✝² = (fun x ↦ ⟨↑x, ⋯⟩) x✝¹
⊢ x✝² = x✝¹

Tactic:
by aesop

Goal state after:
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: case refine_2.refine_2.refine_2
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y Z✝ : C
z₁✝ : Hom W X Y
z₂✝ : Hom W Y Z✝
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z✝
U : C
t₁ : a₁.Y' ⟶ U
t₂ : a₂.Y' ⟶ U
hst : a₁.s ≫ t₁ = a₂.s ≫ t₂
hft : a₁.f ≫ t₁ = a₂.f ≫ t₂
ht : W (a₁.s ≫ t₁)
z₁ : W.LeftFraction a₁.Y' b.Y'
fac₁ : b.f ≫ z₁.s = a₁.s ≫ z₁.f
z₂ : W.LeftFraction a₂.Y' b.Y'
fac₂ : b.f ≫ z₂.s = a₂.s ≫ z₂.f
w₁ : W.LeftFraction U z₁.Y'
w₂ : W.LeftFraction U z₂.Y'
p₁ : W.LeftFraction X Z✝ := { Y' := w₁.Y', f := a₁.f ≫ t₁ ≫ w₁.f, s := b.s ≫ z₁.s ≫ w₁.s, hs := ⋯ }
p₂ : W.LeftFraction X Z✝ := { Y' := w₂.Y', f := a₂.f ≫ t₂ ≫ w₂.f, s := b.s ≫ z₂.s ≫ w₂.s, hs := ⋯ }
fac₁' : b.f ≫ z₁.s ≫ w₁.s = a₁.s ≫ t₁ ≫ w₁.f
fac₂' : b.f ≫ z₂.s ≫ w₂.s = a₂.s ≫ t₂ ≫ w₂.f
q : W.LeftFraction w₁.Y' w₂.Y'
fac₃ : z₂.s ≫ w₂.s ≫ q.s = z₁.s ≫ w₁.s ≫ q.f
eq : a₁.s ≫ t₁ ≫ w₁.f ≫ q.f = a₁.s ≫ t₁ ≫ w₂.f ≫ q.s
Z : C
u : q.Y' ⟶ Z
hu : W u
fac₄ : t₁ ≫ w₁.f ≫ q.f ≫ u = t₁ ≫ w₂.f ≫ q.s ≫ u
⊢ a₁.f ≫ t₁ ≫ w₁.f ≫ q.f ≫ u = a₂.f ≫ t₂ ≫ w₂.f ≫ q.s ≫ u

Tactic:
fac₄,

Goal state after:
case refine_2.refine_2.refine_2
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y Z✝ : C
z₁✝ : Hom W X Y
z₂✝ : Hom W Y Z✝
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z✝
U : C
t₁ : a₁.Y' ⟶ U
t₂ : a₂.Y' ⟶ U
hst : a₁.s ≫ t₁ = a₂.s ≫ t₂
hft : a₁.f ≫ t₁ = a₂.f ≫ t₂
ht : W (a₁.s ≫ t₁)
z₁ : W.LeftFraction a₁.Y' b.Y'
fac₁ : b.f ≫ z₁.s = a₁.s ≫ z₁.f
z₂ : W.LeftFraction a₂.Y' b.Y'
fac₂ : b.f ≫ z₂.s = a₂.s ≫ z₂.f
w₁ : W.LeftFraction U z₁.Y'
w₂ : W.LeftFraction U z₂.Y'
p₁ : W.LeftFraction X Z✝ := { Y' := w₁.Y', f := a₁.f ≫ t₁ ≫ w₁.f, s := b.s ≫ z₁.s ≫ w₁.s, hs := ⋯ }
p₂ : W.LeftFraction X Z✝ := { Y' := w₂.Y', f := a₂.f ≫ t₂ ≫ w₂.f, s := b.s ≫ z₂.s ≫ w₂.s, hs := ⋯ }
fac₁' : b.f ≫ z₁.s ≫ w₁.s = a₁.s ≫ t₁ ≫ w₁.f
fac₂' : b.f ≫ z₂.s ≫ w₂.s = a₂.s ≫ t₂ ≫ w₂.f
q : W.LeftFraction w₁.Y' w₂.Y'
fac₃ : z₂.s ≫ w₂.s ≫ q.s = z₁.s ≫ w₁.s ≫ q.f
eq : a₁.s ≫ t₁ ≫ w₁.f ≫ q.f = a₁.s ≫ t₁ ≫ w₂.f ≫ q.s
Z : C
u : q.Y' ⟶ Z
hu : W u
fac₄ : t₁ ≫ w₁.f ≫ q.f ≫ u = t₁ ≫ w₂.f ≫ q.s ≫ u
⊢ a₁.f ≫ t₁ ≫ w₂.f ≫ q.s ≫ u = a₂.f ≫ t₂ ≫ w₂.f ≫ q.s ≫ u
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝ : Category.{v_3, u_3} E
F₁ F₂ : Localization W ⥤ E
h : Q W ⋙ F₁ = Q W ⋙ F₂
X Y : C
f : W.LeftFraction X Y
⊢ F₁.map (Hom.mk f) = eqToHom ⋯ ≫ F₂.map (Hom.mk f) ≫ eqToHom ⋯

Tactic:
show Hom.mk f = homMk (mk f.f f.s f.hs) by rfl,

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝ : Category.{v_3, u_3} E
F₁ F₂ : Localization W ⥤ E
h : Q W ⋙ F₁ = Q W ⋙ F₂
X Y : C
f : W.LeftFraction X Y
⊢ F₁.map (homMk { Y' := f.Y', f := f.f, s := f.s, hs := ⋯ }) =
    eqToHom ⋯ ≫ F₂.map (homMk { Y' := f.Y', f := f.f, s := f.s, hs := ⋯ }) ≫ eqToHom ⋯
Module: Mathlib.CategoryTheory.Localization.Triangulated

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
W : MorphismProperty C
inst✝² : L.IsLocalization W
inst✝¹ : W.HasLeftCalculusOfFractions
inst✝ : W.IsCompatibleWithTriangulation
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
a : L.obj T₁.obj₁ ⟶ L.obj T₂.obj₁
b : L.obj T₁.obj₂ ⟶ L.obj T₂.obj₂
fac : L.map T₁.mor₁ ≫ b = a ≫ L.map T₂.mor₁
α : W.LeftFraction T₁.obj₁ T₂.obj₁
hα : a = α.map L ⋯
β : W.LeftFraction α.Y' T₂.obj₂
hβ : T₂.mor₁ ≫ β.s = α.s ≫ β.f
γ : W.LeftFraction T₁.obj₂ β.Y'
hγ : b ≫ L.map β.s = γ.map L ⋯
⊢ L.map (α.f ≫ β.f ≫ γ.s) = L.map (T₁.mor₁ ≫ γ.f)

Tactic:
rw [← cancel_mono (L.map β.s), assoc, assoc, hγ, ← cancel_mono (L.map γ.s),
        assoc, assoc, assoc, hα, MorphismProperty.LeftFraction.map_comp_map_s,
        ← Functor.map_comp] at fac

Goal state after:
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
W : MorphismProperty C
inst✝² : L.IsLocalization W
inst✝¹ : W.HasLeftCalculusOfFractions
inst✝ : W.IsCompatibleWithTriangulation
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
a : L.obj T₁.obj₁ ⟶ L.obj T₂.obj₁
b : L.obj T₁.obj₂ ⟶ L.obj T₂.obj₂
α : W.LeftFraction T₁.obj₁ T₂.obj₁
hα : a = α.map L ⋯
β : W.LeftFraction α.Y' T₂.obj₂
hβ : T₂.mor₁ ≫ β.s = α.s ≫ β.f
γ : W.LeftFraction T₁.obj₂ β.Y'
fac : L.map (T₁.mor₁ ≫ γ.f) = α.map L ⋯ ≫ L.map T₂.mor₁ ≫ L.map β.s ≫ L.map γ.s
hγ : b ≫ L.map β.s = γ.map L ⋯
⊢ L.map (α.f ≫ β.f ≫ γ.s) = L.map (T₁.mor₁ ≫ γ.f)