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

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasZeroMorphisms C
inst✝⁴ : HasFiniteCoproducts C
inst✝³ : HasCokernels C
inst✝² : IsNormalEpiCategory C
X Y Z : C
a : X ⟶ Y
b : X ⟶ Z
inst✝¹ : Epi a
inst✝ : Epi b
P : C
f : P ⟶ X
hfa : f ≫ a = 0
i : IsColimit (CokernelCofork.ofπ a hfa)
Q : C
g : Q ⟶ X
hgb : g ≫ b = 0
i' : IsColimit (CokernelCofork.ofπ b hgb)
a' : (CokernelCofork.ofπ a hfa).pt ⟶ cokernel (coprod.desc f g)
ha' : Cofork.π (CokernelCofork.ofπ a hfa) ≫ a' = cokernel.π (coprod.desc f g)
b' : (CokernelCofork.ofπ b hgb).pt ⟶ cokernel (coprod.desc f g)
hb' : Cofork.π (CokernelCofork.ofπ b hgb) ≫ b' = cokernel.π (coprod.desc f g)
s : PushoutCocone a b
⊢ 0 ≫ s.inr = coprod.inr ≫ 0

Tactic:
by rw [comp_zero, zero_comp]

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 : X ⟶ Y
⊢ a + b = b + a

Tactic:
rw [add_def]

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

Tactic:
rw [neg_def, neg_def, neg_def, sub_sub_sub]

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b : X ⟶ Y
⊢ 0 - 0 - (0 - a - b) = b + a
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 : X ⟶ Y
⊢ a + -a = 0

Tactic:
add_neg,

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a : X ⟶ Y
⊢ a - a = 0
Module: Mathlib.CategoryTheory.Abelian.NonPreadditive

Goal state before:
pp: case a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b c : X ⟶ Y
| a + b

pp: case a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b c : X ⟶ Y
| c

Tactic:
rw [add_def]

Goal state after:
case a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b c : X ⟶ Y
| a - -b

case a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
a b c : X ⟶ Y
| c
Module: Mathlib.CategoryTheory.Abelian.Basic

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasKernels C
inst✝³ : HasCokernels C
inst✝² : HasZeroObject C
X Y : C
f : X ⟶ Y
inst✝¹ : Mono f
inst✝ : IsIso (coimageImageComparison f)
⊢ IsIso (imageMonoFactorisation f).e

Tactic:
rw [imageMonoFactorisation_e']

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasKernels C
inst✝³ : HasCokernels C
inst✝² : HasZeroObject C
X Y : C
f : X ⟶ Y
inst✝¹ : Mono f
inst✝ : IsIso (coimageImageComparison f)
⊢ IsIso (cokernel.π (kernel.ι f) ≫ coimageImageComparison f)
Module: Mathlib.CategoryTheory.Abelian.Basic

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
⊢ (imageIsoImage f).hom ≫ Limits.image.ι f = kernel.ι (cokernel.π f)

Tactic:
by
  simp only [IsImage.isoExt_hom, IsImage.lift_ι, imageStrongEpiMonoFactorisation_m]

Goal state after:
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
X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : Epi g
s : PullbackCone f g
hs : IsLimit s
⊢ Epi s.fst

Tactic:
by
  have : Epi (NatTrans.app (limit.cone (cospan f g)).π WalkingCospan.left) :=
    Abelian.epi_pullback_of_epi_g f g
  apply epi_of_epi_fac (IsLimit.conePointUniqueUpToIso_hom_comp (limit.isLimit _) hs _)

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasPushouts C
W X Y Z : C
f : X ⟶ Y
g : X ⟶ Z
inst✝ : Mono f
R : C
e : R ⟶ Z
h : e ≫ pushout.inr f g = 0
u : R ⟶ Y ⊞ Z := biprod.lift 0 e
hu : u ≫ BiproductToPushoutIsCokernel.biproductToPushout f g = 0
this : IsLimit (KernelFork.ofι (biprod.lift f (-g)) ⋯)
d : R ⟶ X
hd : d ≫ biprod.lift f (-g) = biprod.lift 0 e
⊢ d ≫ biprod.lift f (-g) ≫ biprod.fst = u ≫ biprod.fst

Tactic:
by rw [← Category.assoc, hd]

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

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Category.{v_3, u_3} E
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroMorphisms D
inst✝ : HasZeroMorphisms E
S S₁ S₂ S₃ : ShortComplex C
e₁ : S₁.X₁ ≅ S₂.X₁
e₂ : S₁.X₂ ≅ S₂.X₂
e₃ : S₁.X₃ ≅ S₂.X₃
comm₁₂ : e₁.hom ≫ S₂.f = S₁.f ≫ e₂.hom
comm₂₃ : e₂.hom ≫ S₂.g = S₁.g ≫ e₃.hom
⊢ e₂.inv ≫ S₁.g = S₂.g ≫ e₃.inv

Tactic:
by rw [← cancel_mono e₃.hom, assoc, assoc, e₃.inv_hom_id, comp_id,
          ← comm₂₃, e₂.inv_hom_id_assoc]

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

Goal state before:
pp: A B : GrpCat
f : A ⟶ B
b : ↑B
hb : h b = g b
r : b ∉ ↑(Hom.hom f).range
eq1 : (h b) (fromCoset ⟨↑(Hom.hom f).range, ⋯⟩) = fromCoset ⟨↑(Hom.hom f).range, ⋯⟩
eq2 : (g b) (fromCoset ⟨↑(Hom.hom f).range, ⋯⟩) = fromCoset ⟨b • ↑(Hom.hom f).range, ⋯⟩
⊢ fromCoset ⟨↑(Hom.hom f).range, ⋯⟩ = fromCoset ⟨b • ↑(Hom.hom f).range, ⋯⟩

Tactic:
rw [← eq1, ← eq2, DFunLike.congr_fun hb]

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasPushouts C
W X Y Z : C
f : X ⟶ Y
g : X ⟶ Z
inst✝ : Mono g
R : C
e : R ⟶ Y
h : e ≫ pushout.inl f g = 0
u : R ⟶ Y ⊞ Z := biprod.lift e 0
hu : u ≫ BiproductToPushoutIsCokernel.biproductToPushout f g = 0
this : IsLimit (KernelFork.ofι (biprod.lift f (-g)) ⋯)
d : R ⟶ X
hd : d ≫ biprod.lift f (-g) = biprod.lift e 0
⊢ d ≫ biprod.lift f (-g) ≫ biprod.snd = biprod.lift e 0 ≫ biprod.snd

Tactic:
by rw [← Category.assoc, hd]

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasPushouts C
W X Y Z : C
f : X ⟶ Y
g : X ⟶ Z
inst✝ : Mono g
R : C
e : R ⟶ Y
h : e ≫ pushout.inl f g = 0
⊢ e = 0

Tactic:
let u := biprod.lift e (0 : R ⟶ Z)
    have hu : u ≫ BiproductToPushoutIsCokernel.biproductToPushout f g = 0 := by simpa [u]
    have :=
      monoIsKernelOfCokernel _
        (BiproductToPushoutIsCokernel.isColimitBiproductToPushout f g)
    obtain ⟨d, hd⟩ := KernelFork.IsLimit.lift' this u hu
    dsimp at d
    dsimp [u] at hd
    have : d ≫ (-g) = 0 := calc
      d ≫ (-g) = d ≫ biprod.lift f (-g) ≫ biprod.snd := by rw [biprod.lift_snd]
      _ = biprod.lift e (0 : R ⟶ Z) ≫ biprod.snd := by rw [← Category.assoc, hd]
      _ = 0 := biprod.lift_snd _ _
    have : d = 0 := (cancel_mono (-g)).1 (by simpa)
    calc
      e = biprod.lift e (0 : R ⟶ Z) ≫ biprod.fst := by rw [biprod.lift_fst]
      _ = (d ≫ biprod.lift f (-g)) ≫ biprod.fst := by rw [← hd]
      _ = (0 ≫ biprod.lift f (-g)) ≫ biprod.fst := by rw [this]
      _ = 0 ≫ biprod.lift f (-g) ≫ biprod.fst := by rw [Category.assoc]
      _ = 0 := zero_comp

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ QuasiIso φ ↔ IsIso (rightHomologyMap' φ h₁ h₂)

Tactic:
have γ : RightHomologyMapData φ h₁ h₂ := default

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
⊢ QuasiIso φ ↔ IsIso (rightHomologyMap' φ h₁ h₂)
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ (h₁.i ≫ φ.τ₂) ≫ S₂.g = 0

Tactic:
rw [assoc, φ.comm₂₃, h₁.wi_assoc, zero_comp]

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
S : ShortComplex C
inst✝¹ : S.HasLeftHomology
inst✝ : S.HasRightHomology
h : IsIso S.leftRightHomologyComparison
this : IsIso (leftRightHomologyComparison' S.leftHomologyData S.rightHomologyData)
⊢ S.HasHomology

Tactic:
exact hasHomology_of_isIso_leftRightHomologyComparison' S.leftHomologyData S.rightHomologyData

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ S₃ : ShortComplex C
φ₁ : S₁ ⟶ S₂
φ₂ : S₂ ⟶ S₃
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
h₃ : S₃.RightHomologyData
⊢ rightHomologyMap' (φ₁ ≫ φ₂) h₁ h₃ = rightHomologyMap' φ₁ h₁ h₂ ≫ rightHomologyMap' φ₂ h₂ h₃

Tactic:
by
  let γ₁ := rightHomologyMapData φ₁ h₁ h₂
  let γ₂ := rightHomologyMapData φ₂ h₂ h₃
  rw [γ₁.rightHomologyMap'_eq, γ₂.rightHomologyMap'_eq, (γ₁.comp γ₂).rightHomologyMap'_eq,
    RightHomologyMapData.comp_φH]

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
m : s.pt ⟶ c.pt
hm : ∀ (j : J), m ≫ c.π.app j = s.π.app j
⊢ m =
    { τ₁ := h₁.lift (π₁.mapCone s), τ₂ := h₂.lift (π₂.mapCone s), τ₃ := h₃.lift (π₃.mapCone s), comm₁₂ := ⋯,
      comm₂₃ := ⋯ }

Tactic:
ext
    · exact h₁.uniq (π₁.mapCone s) _ (fun j => π₁.congr_map (hm j))
    · exact h₂.uniq (π₂.mapCone s) _ (fun j => π₂.congr_map (hm j))
    · exact h₃.uniq (π₃.mapCone s) _ (fun j => π₃.congr_map (hm j))

Goal state after:
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 : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ rightHomologyMap' (-φ) h₁ h₂ = -rightHomologyMap' φ h₁ h₂

Tactic:
have γ : RightHomologyMapData φ h₁ h₂ := default

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
⊢ rightHomologyMap' (-φ) h₁ h₂ = -rightHomologyMap' φ h₁ h₂
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 : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ opcyclesMap' (-φ) h₁ h₂ = -opcyclesMap' φ h₁ h₂

Tactic:
have γ : RightHomologyMapData φ h₁ h₂ := default

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
⊢ opcyclesMap' (-φ) h₁ h₂ = -opcyclesMap' φ h₁ h₂
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

Goal state after:
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
γ : RightHomologyMapData φ h₁ h₂
⊢ rightHomologyMap' (φ + φ') h₁ h₂ = rightHomologyMap' φ h₁ h₂ + rightHomologyMap' φ' h₁ h₂
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
⊢ opcyclesMap' (φ + φ') h₁ h₂ = opcyclesMap' φ h₁ h₂ + opcyclesMap' φ' h₁ h₂

Tactic:
have γ : RightHomologyMapData φ h₁ h₂ := default

Goal state after:
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
γ : RightHomologyMapData φ h₁ h₂
⊢ opcyclesMap' (φ + φ') h₁ h₂ = opcyclesMap' φ h₁ h₂ + opcyclesMap' φ' h₁ h₂
Module: Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasZeroMorphisms D
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hl₂ : S₂.LeftHomologyData
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : hl₁.IsPreservedBy F
inst✝ : hl₂.IsPreservedBy F
γ : LeftHomologyMapData φ hl₁ hl₂
⊢ F.map γ.φH = leftHomologyMap' (F.mapShortComplex.map φ) (hl₁.map F) (hl₂.map F)

Tactic:
(γ.map F).leftHomologyMap'_eq,

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasZeroMorphisms D
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hl₂ : S₂.LeftHomologyData
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : hl₁.IsPreservedBy F
inst✝ : hl₂.IsPreservedBy F
γ : LeftHomologyMapData φ hl₁ hl₂
⊢ F.map γ.φH = (γ.map F).φH
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 S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄
⊢ (φ₁ - φ₃).τ₂ = S₁.g ≫ (h.h₂ - h'.h₂) + (h.h₁ - h'.h₁) ≫ S₂.f + (φ₂ - φ₄).τ₂

Tactic:
rw [sub_τ₂, sub_τ₂, h.comm₂, h'.comm₂, comp_sub, sub_comp]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄
⊢ S₁.g ≫ h.h₂ + h.h₁ ≫ S₂.f + φ₂.τ₂ - (S₁.g ≫ h'.h₂ + h'.h₁ ≫ S₂.f + φ₄.τ₂) =
    S₁.g ≫ h.h₂ - S₁.g ≫ h'.h₂ + (h.h₁ ≫ S₂.f - h'.h₁ ≫ S₂.f) + (φ₂.τ₂ - φ₄.τ₂)
Module: Mathlib.CategoryTheory.Preadditive.Injective.Basic

Goal state before:
pp: case pos
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X : Type u₁
inst✝ : Nonempty X
X✝ Y✝ : Type u₁
g : X✝ ⟶ X
f : X✝ ⟶ Y✝
mono : Mono f
y : X✝
h✝ : (ConcreteCategory.hom f) y ∈ Set.range ⇑(ConcreteCategory.hom f)
⊢ (ConcreteCategory.hom g) (Classical.choose h✝) = (ConcreteCategory.hom g).toFun y

Tactic:
rename_i h

Goal state after:
case pos
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X : Type u₁
inst✝ : Nonempty X
X✝ Y✝ : Type u₁
g : X✝ ⟶ X
f : X✝ ⟶ Y✝
mono : Mono f
y : X✝
h : (ConcreteCategory.hom f) y ∈ Set.range ⇑(ConcreteCategory.hom f)
⊢ (ConcreteCategory.hom g) (Classical.choose h) = (ConcreteCategory.hom g).toFun y
Module: Mathlib.CategoryTheory.Preadditive.Injective.Basic

Goal state before:
pp: case neg
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X : Type u₁
inst✝ : Nonempty X
X✝ Y✝ : Type u₁
g : X✝ ⟶ X
f : X✝ ⟶ Y✝
mono : Mono f
y : X✝
h✝ : (ConcreteCategory.hom f) y ∉ Set.range ⇑(ConcreteCategory.hom f)
⊢ ⋯.some = (ConcreteCategory.hom g).toFun y

Tactic:
rename_i h

Goal state after:
case neg
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X : Type u₁
inst✝ : Nonempty X
X✝ Y✝ : Type u₁
g : X✝ ⟶ X
f : X✝ ⟶ Y✝
mono : Mono f
y : X✝
h : (ConcreteCategory.hom f) y ∉ Set.range ⇑(ConcreteCategory.hom f)
⊢ ⋯.some = (ConcreteCategory.hom g).toFun y
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 S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h₀ : S₁.X₁ ⟶ S₂.X₁
h₀_f : h₀ ≫ S₂.f = 0
h₁ : S₁.X₂ ⟶ S₂.X₁
h₂ : S₁.X₃ ⟶ S₂.X₂
h₃ : S₁.X₃ ⟶ S₂.X₃
g_h₃ : S₁.g ≫ h₃ = 0
⊢ (S₁.nullHomotopic S₂ h₀ h₀_f h₁ h₂ h₃ g_h₃).τ₃ = h₃ + h₂ ≫ S₂.g + Hom.τ₃ 0

Tactic:
by rw [nullHomotopic_τ₃, zero_τ₃, add_zero]; abel

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasZeroMorphisms D
S : ShortComplex C
h : S.Exact
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : F.PreservesLeftHomologyOf S
inst✝ : (S.map F).HasHomology
this : S.HasHomology
⊢ (S.map F).Exact

Tactic:
S.leftHomologyData.exact_iff,

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasZeroMorphisms D
S : ShortComplex C
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : F.PreservesLeftHomologyOf S
inst✝ : (S.map F).HasHomology
this : S.HasHomology
h : IsZero S.leftHomologyData.H
⊢ (S.map F).Exact
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

Goal state before:
pp: case mp
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : HasZeroMorphisms C
inst✝⁵ : HasZeroMorphisms D
S : ShortComplex C
inst✝⁴ : S.HasHomology
F : C ⥤ D
inst✝³ : F.PreservesZeroMorphisms
inst✝² : F.PreservesLeftHomologyOf S
inst✝¹ : F.PreservesRightHomologyOf S
inst✝ : F.Faithful
h : (S.map F).Exact
⊢ S.Exact

Tactic:
S.leftHomologyData.exact_iff,

Goal state after:
case mp
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : HasZeroMorphisms C
inst✝⁵ : HasZeroMorphisms D
S : ShortComplex C
inst✝⁴ : S.HasHomology
F : C ⥤ D
inst✝³ : F.PreservesZeroMorphisms
inst✝² : F.PreservesLeftHomologyOf S
inst✝¹ : F.PreservesRightHomologyOf S
inst✝ : F.Faithful
h : (S.map F).Exact
⊢ IsZero S.leftHomologyData.H
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
inst✝¹ : Preadditive C
inst✝ : Balanced C
S : ShortComplex C
hS : S.Exact
s : S.X₃ ⟶ S.X₂
s_g : s ≫ S.g = 𝟙 S.X₃
hf : Mono S.f
⊢ S.f - S.f ≫ S.g ≫ s = 𝟙 S.X₁ ≫ S.f

Tactic:
zero_assoc,

Goal state after:
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
inst✝¹ : Preadditive C
inst✝ : Balanced C
S : ShortComplex C
hS : S.Exact
s : S.X₃ ⟶ S.X₂
s_g : s ≫ S.g = 𝟙 S.X₃
hf : Mono S.f
⊢ S.f - 0 ≫ s = 𝟙 S.X₁ ≫ S.f
Module: Mathlib.CategoryTheory.Subobject.Basic

Goal state before:
pp: case f
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : HasPullbacks C
X Y Z W : C
f : X ⟶ Y
g : X ⟶ Z
h : Y ⟶ W
k : Z ⟶ W
inst✝¹ : Mono h
inst✝ : Mono g
comm : f ≫ h = g ≫ k
t : IsLimit (PullbackCone.mk f g comm)
a : MonoOver Y
⊢ (MonoOver.map g).obj ((MonoOver.pullback f).obj a) ⟶ (MonoOver.pullback k).obj ((MonoOver.map h).obj a)

Tactic:
refine MonoOver.homMk (pullback.lift (pullback.fst _ _) _ ?_) (pullback.lift_snd _ _ _)
    simp [← comm, pullback.condition_assoc]

Goal state after:
Module: Mathlib.CategoryTheory.Subobject.Limits

Goal state before:
pp: case refine_2
C : Type u
inst✝³ : Category.{v, u} C
X✝ Y Z : C
inst✝² : HasZeroMorphisms C
f✝ : X✝ ⟶ Y
inst✝¹ : HasKernel f✝
inst✝ : HasKernels C
X : C
A B : Cᵒᵖ
f : A ⟶ op X
g : B ⟶ op X
hf : Mono f
hg : Mono g
h : Subobject.mk f ≤ Subobject.mk g
⊢ kernel.lift g.unop (kernel.ι f.unop) ⋯ ≫ kernel.ι g.unop = kernel.ι f.unop

Tactic:
· exact Quiver.Hom.op_inj (by simp)

Goal state after:
Module: Mathlib.CategoryTheory.Subobject.Lattice

Goal state before:
pp: case f.refine_1
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : LocallySmall.{w, v₁, u₁} C
inst✝² : WellPowered.{w, v₁, u₁} C
inst✝¹ : HasCoproducts C
inst✝ : HasImages C
A : C
s : Set (Subobject A)
f : Subobject A
k : ∀ g ∈ s, g ≤ f
g : Shrink.{w, max u₁ v₁} (Subobject A)
m : g ∈ ⇑(equivShrink (Subobject A)) '' s
⊢ underlying.obj ((equivShrink (Subobject A)).symm ↑⟨g, m⟩) ⟶ underlying.obj f

Tactic:
refine underlying.map (homOfLE (k _ ?_))

Goal state after:
case f.refine_1
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : LocallySmall.{w, v₁, u₁} C
inst✝² : WellPowered.{w, v₁, u₁} C
inst✝¹ : HasCoproducts C
inst✝ : HasImages C
A : C
s : Set (Subobject A)
f : Subobject A
k : ∀ g ∈ s, g ≤ f
g : Shrink.{w, max u₁ v₁} (Subobject A)
m : g ∈ ⇑(equivShrink (Subobject A)) '' s
⊢ (equivShrink (Subobject A)).symm ↑⟨g, m⟩ ∈ s
Module: Mathlib.CategoryTheory.Generator.Basic

Goal state before:
pp: case refine_1
C : Type u₁
inst✝ : Category.{v₁, u₁} C
𝒢 : ObjectProperty C
h𝒢 : 𝒢.IsDetecting
X : C
P Q : Subobject X
h₁ : P ≤ Q
h₂ : ∀ (G : C), 𝒢 G → ∀ {f : G ⟶ X}, Q.Factors f → P.Factors f
G : C
hG : 𝒢 G
f : G ⟶ underlying.obj Q
this : P.Factors (f ≫ Q.arrow)
⊢ (fun h' ↦ h' ≫ P.ofLE Q h₁ = f) (P.factorThru (f ≫ Q.arrow) this)

Tactic:
simp only [← cancel_mono Q.arrow, Category.assoc, ofLE_arrow, factorThru_arrow]

Goal state after:
Module: Mathlib.CategoryTheory.Generator.Basic

Goal state before:
pp: case refine_2
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
G : C
inst✝ : ∀ (A : C), HasCoproduct fun x ↦ G
h : ∀ (A : C), Epi (Sigma.desc fun f ↦ f)
X Y : C
f g : X ⟶ Y
hh : ∀ (h : G ⟶ X), h ≫ f = h ≫ g
⊢ f = g

Tactic:
have := h X

Goal state after:
case refine_2
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
G : C
inst✝ : ∀ (A : C), HasCoproduct fun x ↦ G
h : ∀ (A : C), Epi (Sigma.desc fun f ↦ f)
X Y : C
f g : X ⟶ Y
hh : ∀ (h : G ⟶ X), h ≫ f = h ≫ g
this : Epi (Sigma.desc fun f ↦ f)
⊢ f = g