mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Limits.Constructions.Equalizers

Goal state before:
pp: case refine_2
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u'
inst✝⁴ : Category.{v', u'} D
G : C ⥤ D
inst✝³ : HasBinaryProducts C
inst✝² : HasPullbacks C
inst✝¹ : PreservesLimitsOfShape (Discrete WalkingPair) G
inst✝ : PreservesLimitsOfShape WalkingCospan G
K : WalkingParallelPair ⥤ C
c : Cone (K ⋙ G)
⊢ c.pt ⟶ G.obj (K.obj WalkingParallelPair.zero)

pp: case refine_3
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u'
inst✝⁴ : Category.{v', u'} D
G : C ⥤ D
inst✝³ : HasBinaryProducts C
inst✝² : HasPullbacks C
inst✝¹ : PreservesLimitsOfShape (Discrete WalkingPair) G
inst✝ : PreservesLimitsOfShape WalkingCospan G
K : WalkingParallelPair ⥤ C
c : Cone (K ⋙ G)
⊢ c.π.app WalkingParallelPair.zero ≫
      G.map (prod.lift (𝟙 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left)) =
    ?refine_2 ≫ G.map (prod.lift (𝟙 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))

Tactic:
· exact c.π.app WalkingParallelPair.zero

Goal state after:
case refine_3
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u'
inst✝⁴ : Category.{v', u'} D
G : C ⥤ D
inst✝³ : HasBinaryProducts C
inst✝² : HasPullbacks C
inst✝¹ : PreservesLimitsOfShape (Discrete WalkingPair) G
inst✝ : PreservesLimitsOfShape WalkingCospan G
K : WalkingParallelPair ⥤ C
c : Cone (K ⋙ G)
⊢ c.π.app WalkingParallelPair.zero ≫
      G.map (prod.lift (𝟙 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left)) =
    c.π.app WalkingParallelPair.zero ≫
      G.map (prod.lift (𝟙 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))
Module: Mathlib.CategoryTheory.Limits.Constructions.Equalizers

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u'
inst✝⁴ : Category.{v', u'} D
G : C ⥤ D
inst✝³ : HasBinaryCoproducts C
inst✝² : HasPushouts C
inst✝¹ : PreservesColimitsOfShape (Discrete WalkingPair) G
inst✝ : PreservesColimitsOfShape WalkingSpan G
K : WalkingParallelPair ⥤ C
s : Cocone (K ⋙ G)
m : (G.mapCocone (coequalizerCocone K)).pt ⟶ s.pt
h : ∀ (j : WalkingParallelPair), (G.mapCocone (coequalizerCocone K)).ι.app j ≫ m = s.ι.app j
⊢ m =
    (PreservesPushout.iso G (coprod.desc (𝟙 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left))
          (coprod.desc (𝟙 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right))).inv ≫
      pushout.desc (s.ι.app WalkingParallelPair.one) (s.ι.app WalkingParallelPair.one) ⋯

Tactic:
Iso.eq_inv_comp

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u'
inst✝⁴ : Category.{v', u'} D
G : C ⥤ D
inst✝³ : HasBinaryCoproducts C
inst✝² : HasPushouts C
inst✝¹ : PreservesColimitsOfShape (Discrete WalkingPair) G
inst✝ : PreservesColimitsOfShape WalkingSpan G
K : WalkingParallelPair ⥤ C
s : Cocone (K ⋙ G)
m : (G.mapCocone (coequalizerCocone K)).pt ⟶ s.pt
h : ∀ (j : WalkingParallelPair), (G.mapCocone (coequalizerCocone K)).ι.app j ≫ m = s.ι.app j
⊢ (PreservesPushout.iso G (coprod.desc (𝟙 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left))
          (coprod.desc (𝟙 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right))).hom ≫
      m =
    pushout.desc (s.ι.app WalkingParallelPair.one) (s.ι.app WalkingParallelPair.one) ⋯
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsReflexive R M
s : Set (M →ₗ[R] R)
inst✝ : Projective R ((M →ₗ[R] R) ⧸ span R s)
h : ∀ (z : M), z ≠ 0 → ∃ f ∈ s, f z ≠ 0
⊢ span R s = ⊤

Tactic:
by_contra! hn

Goal state after:
R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsReflexive R M
s : Set (M →ₗ[R] R)
inst✝ : Projective R ((M →ₗ[R] R) ⧸ span R s)
h : ∀ (z : M), z ≠ 0 → ∃ f ∈ s, f z ≠ 0
hn : span R s ≠ ⊤
⊢ False
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: K : Type u_1
V₁ : Type u_2
V₂ : Type u_3
inst✝⁴ : Field K
inst✝³ : AddCommGroup V₁
inst✝² : Module K V₁
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
f : V₁ →ₗ[K] V₂
⊢ Function.Injective ⇑f.dualMap ↔ Function.Surjective ⇑f

Tactic:
by
  refine ⟨Function.mtr fun not_surj inj ↦ ?_, dualMap_injective_of_surjective⟩
  rw [← range_eq_top, ← Ne, ← lt_top_iff_ne_top] at not_surj
  obtain ⟨φ, φ0, range_le_ker⟩ := (range f).exists_le_ker_of_lt_top not_surj
  exact φ0 (inj <| ext fun x ↦ range_le_ker ⟨x, rfl⟩)

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
X Y : C
inst✝¹ : HasZeroMorphisms C
inst✝ : IsNormalEpiCategory C
X✝ Y✝ : C
f : X✝ ⟶ Y✝
x✝ : Epi f
⊢ IsRegularEpi f

Tactic:
have := normalEpiOfEpi f

Goal state after:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
X Y : C
inst✝¹ : HasZeroMorphisms C
inst✝ : IsNormalEpiCategory C
X✝ Y✝ : C
f : X✝ ⟶ Y✝
x✝ : Epi f
this : NormalEpi f
⊢ IsRegularEpi f
Module: Mathlib.Algebra.Category.Grp.Biproducts

Goal state before:
pp: J : Type w
f : J → AddCommGrpCat
s : Cone (Discrete.functor f)
m :
  s.pt ⟶
    { pt := of ((j : J) → ↑(f j)), π := Discrete.natTrans fun j ↦ ofHom (Pi.evalAddMonoidHom (fun j ↦ ↑(f j)) j.as) }.pt
w :
  ∀ (j : Discrete J),
    m ≫
        { pt := of ((j : J) → ↑(f j)),
                π := Discrete.natTrans fun j ↦ ofHom (Pi.evalAddMonoidHom (fun j ↦ ↑(f j)) j.as) }.π.app
          j =
      s.π.app j
⊢ m = lift f s

Tactic:
ext x j

Goal state after:
J : Type w
f : J → AddCommGrpCat
s : Cone (Discrete.functor f)
m :
  s.pt ⟶
    { pt := of ((j : J) → ↑(f j)), π := Discrete.natTrans fun j ↦ ofHom (Pi.evalAddMonoidHom (fun j ↦ ↑(f j)) j.as) }.pt
w :
  ∀ (j : Discrete J),
    m ≫
        { pt := of ((j : J) → ↑(f j)),
                π := Discrete.natTrans fun j ↦ ofHom (Pi.evalAddMonoidHom (fun j ↦ ↑(f j)) j.as) }.π.app
          j =
      s.π.app j
x : ↑s.pt
j : J
⊢ (Hom.hom m) x j = (Hom.hom (lift f s)) x j
Module: Mathlib.CategoryTheory.Abelian.NonPreadditive

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b c d : X ⟶ Y
⊢ a - c - (b - d) = a - b - (c - d)

Tactic:
by
  rw [sub_def, ← lift_sub_lift, sub_def, Category.assoc, σ_comp, prod.lift_map_assoc]; rfl

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.NonPreadditive

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b c : X ⟶ Y
⊢ a - b + c = a - (b - c)

Tactic:
by
  rw [add_def, neg_def, sub_sub_sub, sub_zero]

Goal state after:
Module: Mathlib.Algebra.Category.Grp.EpiMono

Goal state before:
pp: A : Type u
B : Type v
inst✝¹ : CommGroup A
inst✝ : CommGroup B
f : A →* B
h : ∀ (u v : B →* B ⧸ f.range), u.comp f = v.comp f → u = v
x : A
⊢ 1 = ↑(f x)

Tactic:
show (1 : B ⧸ f.range) = (1 : B) from QuotientGroup.mk_one _,

Goal state after:
A : Type u
B : Type v
inst✝¹ : CommGroup A
inst✝ : CommGroup B
f : A →* B
h : ∀ (u v : B →* B ⧸ f.range), u.comp f = v.comp f → u = v
x : A
⊢ ↑1 = ↑(f x)
Module: Mathlib.CategoryTheory.Abelian.Basic

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasPullbacks C
W X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : Epi g
R : C
e : X ⟶ R
h : pullback.fst f g ≫ e = 0
u : X ⊞ Y ⟶ R := biprod.desc e 0
hu : PullbackToBiproductIsKernel.pullbackToBiproduct f g ≫ u = 0
this✝¹ : IsColimit (CokernelCofork.ofπ (biprod.desc f (-g)) ⋯)
d : Z ⟶ R
hd : biprod.desc f (-g) ≫ d = biprod.desc e 0
this✝ : (-g) ≫ d = 0
this : d = 0
⊢ e = 0

Tactic:
calc
      e = biprod.inl ≫ biprod.desc e (0 : Y ⟶ R) := by rw [biprod.inl_desc]
      _ = biprod.inl ≫ biprod.desc f (-g) ≫ d := by rw [← hd]
      _ = biprod.inl ≫ biprod.desc f (-g) ≫ 0 := by rw [this]
      _ = (biprod.inl ≫ biprod.desc f (-g)) ≫ 0 := by rw [← Category.assoc]
      _ = 0 := HasZeroMorphisms.comp_zero _ _

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
inst✝ : S.HasLeftHomology
A : C
f₁ f₂ : A ⟶ S.cycles
⊢ f₁ = f₂ ↔ f₁ ≫ S.iCycles = f₂ ≫ S.iCycles

Tactic:
by
  rw [cancel_mono]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
inst✝ : S.HasRightHomology
A : C
f₁ f₂ : A ⟶ S.rightHomology
⊢ f₁ = f₂ ↔ f₁ ≫ S.rightHomologyι = f₂ ≫ S.rightHomologyι

Tactic:
by
  rw [cancel_mono]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.Limits

Goal state before:
pp: J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
inst✝ : HasZeroMorphisms C
F : J ⥤ ShortComplex C
c : Cone F
h₁ : IsLimit (π₁.mapCone c)
h₂ : IsLimit (π₂.mapCone c)
h₃ : IsLimit (π₃.mapCone c)
s : Cone F
j : J
⊢ (h₁.lift (π₁.mapCone s) ≫ c.pt.f) ≫ (π₂.mapCone c).π.app j =
    (s.pt.f ≫ h₂.lift (π₂.mapCone s)) ≫ (π₂.mapCone c).π.app j

Tactic:
have eq₁ := h₁.fac (π₁.mapCone s)

Goal state after:
J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
inst✝ : HasZeroMorphisms C
F : J ⥤ ShortComplex C
c : Cone F
h₁ : IsLimit (π₁.mapCone c)
h₂ : IsLimit (π₂.mapCone c)
h₃ : IsLimit (π₃.mapCone c)
s : Cone F
j : J
eq₁ : ∀ (j : J), h₁.lift (π₁.mapCone s) ≫ (π₁.mapCone c).π.app j = (π₁.mapCone s).π.app j
⊢ (h₁.lift (π₁.mapCone s) ≫ c.pt.f) ≫ (π₂.mapCone c).π.app j =
    (s.pt.f ≫ h₂.lift (π₂.mapCone s)) ≫ (π₂.mapCone c).π.app j
Module: Mathlib.Algebra.Homology.ShortComplex.Preadditive

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ : ShortComplex C
φ φ' : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ rightHomologyMap' (φ + φ') h₁ h₂ = rightHomologyMap' φ h₁ h₂ + rightHomologyMap' φ' h₁ h₂

Tactic:
have γ : RightHomologyMapData φ h₁ h₂ := default
  have γ' : RightHomologyMapData φ' h₁ h₂ := default
  simp only [γ.rightHomologyMap'_eq, γ'.rightHomologyMap'_eq,
    (γ.add γ').rightHomologyMap'_eq, RightHomologyMapData.add_φH]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.Abelian

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
S : ShortComplex C
kf : KernelFork S.g
hkf : IsLimit kf
⊢ hkf.lift (KernelFork.ofι S.f ⋯) = S.toCycles ≫ (S.isoCyclesOfIsLimit hkf).inv

Tactic:
by
  have := Fork.IsLimit.mono hkf
  simp [← cancel_mono kf.ι]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.Abelian

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Abelian C
S : ShortComplex C
D : Type u'
inst✝³ : Category.{v', u'} D
inst✝² : HasZeroMorphisms D
kf : KernelFork S.g
cc : CokernelCofork S.f
hkf : IsLimit kf
hcc : IsColimit cc
H : C
π : kf.pt ⟶ H
ι : H ⟶ cc.pt
fac : Fork.ι kf ≫ Cofork.π cc = π ≫ ι
inst✝¹ : Epi π
inst✝ : Mono ι
⊢ (hkf.lift (KernelFork.ofι S.f ⋯) ≫ 𝟙 kf.pt) ≫ π = 0

Tactic:
rw [← cancel_mono (isoHomology S hkf hcc fac).hom, assoc, assoc, id_comp,
      π_comp_isoHomology_hom, zero_comp, f'_eq,
      assoc, Iso.inv_hom_id_assoc, toCycles_comp_homologyπ]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

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
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
S : ShortComplex C
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hr₁ : S₁.RightHomologyData
hl₂ : S₂.LeftHomologyData
hr₂ : S₂.RightHomologyData
ψl : LeftHomologyMapData φ hl₁ hl₂
ψr : RightHomologyMapData φ hr₁ hr₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesRightHomologyOf S₁
inst✝¹ : F.PreservesRightHomologyOf S₂
inst✝ : QuasiIso φ
⊢ QuasiIso (F.mapShortComplex.map φ)

Tactic:
have γ : RightHomologyMapData φ S₁.rightHomologyData S₂.rightHomologyData := default

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
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
S : ShortComplex C
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hr₁ : S₁.RightHomologyData
hl₂ : S₂.LeftHomologyData
hr₂ : S₂.RightHomologyData
ψl : LeftHomologyMapData φ hl₁ hl₂
ψr : RightHomologyMapData φ hr₁ hr₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesRightHomologyOf S₁
inst✝¹ : F.PreservesRightHomologyOf S₂
inst✝ : QuasiIso φ
γ : RightHomologyMapData φ S₁.rightHomologyData S₂.rightHomologyData
⊢ QuasiIso (F.mapShortComplex.map φ)
Module: Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

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
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesRightHomologyOf S₁
inst✝¹ : F.PreservesRightHomologyOf S₂
inst✝ : F.ReflectsIsomorphisms
⊢ QuasiIso (F.mapShortComplex.map φ) ↔ QuasiIso φ

Tactic:
have γ : RightHomologyMapData φ S₁.rightHomologyData S₂.rightHomologyData := default

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
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesRightHomologyOf S₁
inst✝¹ : F.PreservesRightHomologyOf S₂
inst✝ : F.ReflectsIsomorphisms
γ : RightHomologyMapData φ S₁.rightHomologyData S₂.rightHomologyData
⊢ QuasiIso (F.mapShortComplex.map φ) ↔ QuasiIso φ
Module: Mathlib.CategoryTheory.Adjunction.Reflective

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
i : D ⥤ C
inst✝ : Reflective i
A : C
h : i.essImage A
⊢ IsIso ((reflectorAdjunction i).unit.app A)

Tactic:
isIso_unit_app_iff_mem_essImage

Goal state after:
C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
i : D ⥤ C
inst✝ : Reflective i
A : C
h : i.essImage A
⊢ i.essImage A
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

Goal state before:
pp: case mp
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S : ShortComplex C
hS : S.Exact
a✝ : Mono S.g
⊢ S.f = 0

Tactic:
← cancel_mono S.g,

Goal state after:
case mp
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S : ShortComplex C
hS : S.Exact
a✝ : Mono S.g
⊢ S.f ≫ S.g = 0 ≫ S.g
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S : ShortComplex C
hS : S.Exact
hf : S.f = 0
hg : S.g = 0
this : Mono S.g
⊢ 𝟙 S.X₂ = 0

Tactic:
← cancel_mono S.g,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S : ShortComplex C
hS : S.Exact
hf : S.f = 0
hg : S.g = 0
this : Mono S.g
⊢ 𝟙 S.X₂ ≫ S.g = 0 ≫ S.g
Module: Mathlib.Algebra.Homology.ShortComplex.ExactFunctor

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 : Abelian C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
tfae_1_to_2 :
  (∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Mono (F.map S.f)) →
    ∀ (S : ShortComplex C), S.Exact ∧ Mono S.f → (S.map F).Exact ∧ Mono (F.map S.f)
hF : ∀ (S : ShortComplex C), S.Exact ∧ Mono S.f → (S.map F).Exact ∧ Mono (F.map S.f)
X Y : C
f : X ⟶ Y
⊢ IsLimit (F.mapCone (Fork.ofι (kernel.ι f) ⋯))

Tactic:
apply (KernelFork.isLimitMapConeEquiv _ F).2

Goal state after:
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 : Abelian C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
tfae_1_to_2 :
  (∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Mono (F.map S.f)) →
    ∀ (S : ShortComplex C), S.Exact ∧ Mono S.f → (S.map F).Exact ∧ Mono (F.map S.f)
hF : ∀ (S : ShortComplex C), S.Exact ∧ Mono S.f → (S.map F).Exact ∧ Mono (F.map S.f)
X Y : C
f : X ⟶ Y
⊢ IsLimit (KernelFork.map (Fork.ofι (kernel.ι f) ⋯) F)
Module: Mathlib.CategoryTheory.Abelian.Exact

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Abelian C
S : ShortComplex C
cg : KernelFork S.g
hg : IsLimit cg
cf : CokernelCofork S.f
hf : IsColimit cf
e₁ : (Fork.ofι (kernel.ι S.g) ⋯).pt ≅ cg.pt := (kernelIsKernel S.g).conePointUniqueUpToIso hg
e₂ : (Cofork.ofπ (cokernel.π S.f) ⋯).pt ≅ cf.pt := (cokernelIsCokernel S.f).coconePointUniqueUpToIso hf
⊢ kernel.ι S.g ≫ cokernel.π S.f = 0 ↔ Fork.ι cg ≫ Cofork.π cf = 0

Tactic:
have : cg.ι ≫ cf.π = e₁.inv ≫ kernel.ι S.g ≫ cokernel.π S.f ≫ e₂.hom := by
    have eq₁ := IsLimit.conePointUniqueUpToIso_inv_comp (kernelIsKernel S.g) hg (.zero)
    have eq₂ := IsColimit.comp_coconePointUniqueUpToIso_hom (cokernelIsCokernel S.f) hf (.one)
    dsimp at eq₁ eq₂
    rw [← eq₁, ← eq₂, Category.assoc]

Goal state after:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Abelian C
S : ShortComplex C
cg : KernelFork S.g
hg : IsLimit cg
cf : CokernelCofork S.f
hf : IsColimit cf
e₁ : (Fork.ofι (kernel.ι S.g) ⋯).pt ≅ cg.pt := (kernelIsKernel S.g).conePointUniqueUpToIso hg
e₂ : (Cofork.ofπ (cokernel.π S.f) ⋯).pt ≅ cf.pt := (cokernelIsCokernel S.f).coconePointUniqueUpToIso hf
this : Fork.ι cg ≫ Cofork.π cf = e₁.inv ≫ kernel.ι S.g ≫ cokernel.π S.f ≫ e₂.hom
⊢ kernel.ι S.g ≫ cokernel.π S.f = 0 ↔ Fork.ι cg ≫ Cofork.π cf = 0
Module: Mathlib.CategoryTheory.Abelian.Exact

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Abelian C
S : ShortComplex C
cg : KernelFork S.g
hg : IsLimit cg
cf : CokernelCofork S.f
hf : IsColimit cf
e₁ : (Fork.ofι (kernel.ι S.g) ⋯).pt ≅ cg.pt := (kernelIsKernel S.g).conePointUniqueUpToIso hg
e₂ : (Cofork.ofπ (cokernel.π S.f) ⋯).pt ≅ cf.pt := (cokernelIsCokernel S.f).coconePointUniqueUpToIso hf
this : Fork.ι cg ≫ Cofork.π cf = e₁.inv ≫ kernel.ι S.g ≫ cokernel.π S.f ≫ e₂.hom
⊢ kernel.ι S.g ≫ cokernel.π S.f = 0 ↔ Fork.ι cg ≫ Cofork.π cf = 0

Tactic:
rw [this, IsIso.comp_left_eq_zero e₁.inv, ← Category.assoc,
    IsIso.comp_right_eq_zero _ e₂.hom]

Goal state after:
Module: Mathlib.CategoryTheory.Comma.StructuredArrow.Functor

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
T : C ⥤ D
X✝ Y✝ Z✝ : D
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ (map (f ≫ g)).toCatHom = (map f).toCatHom ≫ (map g).toCatHom

Tactic:
by
    ext
    exact Functor.ext (fun _ => by simp [CostructuredArrow.map, Comma.mapRight])

Goal state after:
Module: Mathlib.CategoryTheory.Comma.StructuredArrow.Small

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
S : D
T : C ⥤ D
P : ObjectProperty C
inst✝¹ : ObjectProperty.Small.{v₁, v₁, u₁} P
inst✝ : LocallySmall.{v₁, v₂, u₂} D
X : StructuredArrow S T
⊢ P.inverseImage (proj S T) X ↔ ObjectProperty.ofObj (fun f ↦ mk f.snd) X

Tactic:
simp only [ObjectProperty.prop_inverseImage_iff, proj_obj, ObjectProperty.ofObj_iff,
    Sigma.exists, Subtype.exists, exists_prop]

Goal state after:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
S : D
T : C ⥤ D
P : ObjectProperty C
inst✝¹ : ObjectProperty.Small.{v₁, v₁, u₁} P
inst✝ : LocallySmall.{v₁, v₂, u₂} D
X : StructuredArrow S T
⊢ P X.right ↔ ∃ a, P a ∧ ∃ b, mk b = X
Module: Mathlib.CategoryTheory.Comma.StructuredArrow.Small

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
S : C ⥤ D
T : D
P : ObjectProperty C
inst✝¹ : ObjectProperty.Small.{v₁, v₁, u₁} P
inst✝ : LocallySmall.{v₁, v₂, u₂} D
X : CostructuredArrow S T
⊢ P.inverseImage (proj S T) X ↔ ObjectProperty.ofObj (fun f ↦ mk f.snd) X

Tactic:
simp only [ObjectProperty.prop_inverseImage_iff, proj_obj, ObjectProperty.ofObj_iff,
    Sigma.exists, Subtype.exists, exists_prop]

Goal state after:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
S : C ⥤ D
T : D
P : ObjectProperty C
inst✝¹ : ObjectProperty.Small.{v₁, v₁, u₁} P
inst✝ : LocallySmall.{v₁, v₂, u₂} D
X : CostructuredArrow S T
⊢ P X.left ↔ ∃ a, P a ∧ ∃ b, mk b = X
Module: Mathlib.CategoryTheory.Limits.Presheaf

Goal state before:
pp: case mpr
C : Type u₁
inst✝² : Category.{v₁, u₁} C
ℰ : Type u₂
inst✝¹ : Category.{v₂, u₂} ℰ
A : C ⥤ ℰ
inst✝ : uliftYoneda.{max w v₂, v₁, u₁}.HasPointwiseLeftKanExtension A
L : (Cᵒᵖ ⥤ Type (max w v₁ v₂)) ⥤ ℰ
α : A ⟶ uliftYoneda.{max w v₂, v₁, u₁} ⋙ L
left✝ : IsIso α
right✝ :
  PreservesColimitsOfSize.{v₁, max w u₁ v₁ v₂, max (max (max u₁ v₁) v₂) w, v₂,
      max (max (max u₁ (v₁ + 1)) (v₂ + 1)) (w + 1), u₂}
    L
P : Cᵒᵖ ⥤ Type (max (max w v₂) v₁)
⊢ (Functor.LeftExtension.mk L α).IsPointwiseLeftKanExtensionAt P

Tactic:
dsimp [Functor.LeftExtension.IsPointwiseLeftKanExtensionAt]

Goal state after:
case mpr
C : Type u₁
inst✝² : Category.{v₁, u₁} C
ℰ : Type u₂
inst✝¹ : Category.{v₂, u₂} ℰ
A : C ⥤ ℰ
inst✝ : uliftYoneda.{max w v₂, v₁, u₁}.HasPointwiseLeftKanExtension A
L : (Cᵒᵖ ⥤ Type (max w v₁ v₂)) ⥤ ℰ
α : A ⟶ uliftYoneda.{max w v₂, v₁, u₁} ⋙ L
left✝ : IsIso α
right✝ :
  PreservesColimitsOfSize.{v₁, max w u₁ v₁ v₂, max (max (max u₁ v₁) v₂) w, v₂,
      max (max (max u₁ (v₁ + 1)) (v₂ + 1)) (w + 1), u₂}
    L
P : Cᵒᵖ ⥤ Type (max (max w v₂) v₁)
⊢ IsColimit ((Functor.LeftExtension.mk L α).coconeAt P)
Module: Mathlib.CategoryTheory.Limits.Presheaf

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
G : (Cᵒᵖ ⥤ Type (max w v₁ v₂)) ⥤ Dᵒᵖ ⥤ Type (max w v₁ v₂)
φ : F ⋙ uliftYoneda.{max w v₁, v₂, u₂} ⟶ uliftYoneda.{max w v₂, v₁, u₁} ⋙ G
P Q : Cᵒᵖ ⥤ Type (max w v₁ v₂)
f : P ⟶ Q
X : C
p : uliftYoneda.{max w v₂, v₁, u₁}.obj X ⟶ P
⊢ uliftYonedaEquiv (p ≫ presheafHom φ P ≫ F.op.whiskerLeft (G.map f)) = uliftYonedaEquiv (p ≫ f ≫ presheafHom φ Q)

Tactic:
← assoc p f,

Goal state after:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
G : (Cᵒᵖ ⥤ Type (max w v₁ v₂)) ⥤ Dᵒᵖ ⥤ Type (max w v₁ v₂)
φ : F ⋙ uliftYoneda.{max w v₁, v₂, u₂} ⟶ uliftYoneda.{max w v₂, v₁, u₁} ⋙ G
P Q : Cᵒᵖ ⥤ Type (max w v₁ v₂)
f : P ⟶ Q
X : C
p : uliftYoneda.{max w v₂, v₁, u₁}.obj X ⟶ P
⊢ uliftYonedaEquiv (p ≫ presheafHom φ P ≫ F.op.whiskerLeft (G.map f)) = uliftYonedaEquiv ((p ≫ f) ≫ presheafHom φ Q)
Module: Mathlib.CategoryTheory.Limits.Types.Coproducts

Goal state before:
pp: C : Type u
F : C → Type v
c : CofanTypes F
hc : Functor.CoconeTypes.IsColimit c
i : C
⊢ Function.Injective (c.inj i)

Tactic:
intro y₁

Goal state after:
C : Type u
F : C → Type v
c : CofanTypes F
hc : Functor.CoconeTypes.IsColimit c
i : C
y₁ : F i
⊢ ∀ ⦃a₂ : F i⦄, c.inj i y₁ = c.inj i a₂ → y₁ = a₂
Module: Mathlib.Data.ZMod.IntUnitsPower

Goal state before:
pp: z₁ z₂ : ZMod 2
au : Additive ℤˣ
⊢ Additive.toMul au ^ (z₁ + z₂).val = Additive.toMul au ^ z₁.val * Additive.toMul au ^ z₂.val

Tactic:
← pow_add,

Goal state after:
z₁ z₂ : ZMod 2
au : Additive ℤˣ
⊢ Additive.toMul au ^ (z₁ + z₂).val = Additive.toMul au ^ (z₁.val + z₂.val)
Module: Mathlib.Data.ZMod.IntUnitsPower

Goal state before:
pp: R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Module R (Additive ℤˣ)
u : ℤˣ
n : ℕ
⊢ Additive.toMul (↑n • Additive.ofMul u) = u ^ n

Tactic:
rw [Nat.cast_smul_eq_nsmul, toMul_nsmul, toMul_ofMul]

Goal state after:
Module: Mathlib.Algebra.Ring.Periodic

Goal state before:
pp: α : Type u_1
β : Type u_2
f : α → β
c : α
inst✝¹ : AddGroup α
inst✝ : InvolutiveNeg β
h : Antiperiodic f c
⊢ Antiperiodic f (-c)

Tactic:
by simpa only [sub_eq_add_neg, Antiperiodic] using h.sub_eq

Goal state after: