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

Goal state before:
pp: case a
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
G : C ⥤ D
X Y : C
f g : X ⟶ Y
inst✝³ : HasCoequalizer f g
inst✝² : HasCoequalizer (G.map f) (G.map g)
inst✝¹ : PreservesColimit (parallelPair f g) G
X' Y' : D
f' g' : X' ⟶ Y'
inst✝ : HasCoequalizer f' g'
p : G.obj X ⟶ X'
q : G.obj Y ⟶ Y'
wf : G.map f ≫ q = p ≫ f'
wg : G.map g ≫ q = p ≫ g'
Z' : D
h : Y' ⟶ Z'
wh : f' ≫ h = g' ≫ h
| coequalizer.π f' g' ≫ coequalizer.desc h wh

pp: case a
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
G : C ⥤ D
X Y : C
f g : X ⟶ Y
inst✝³ : HasCoequalizer f g
inst✝² : HasCoequalizer (G.map f) (G.map g)
inst✝¹ : PreservesColimit (parallelPair f g) G
X' Y' : D
f' g' : X' ⟶ Y'
inst✝ : HasCoequalizer f' g'
p : G.obj X ⟶ X'
q : G.obj Y ⟶ Y'
wf : G.map f ≫ q = p ≫ f'
wg : G.map g ≫ q = p ≫ g'
Z' : D
h : Y' ⟶ Z'
wh : f' ≫ h = g' ≫ h
| q

Tactic:
rw [coequalizer.π_desc]

Goal state after:
case a
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
G : C ⥤ D
X Y : C
f g : X ⟶ Y
inst✝³ : HasCoequalizer f g
inst✝² : HasCoequalizer (G.map f) (G.map g)
inst✝¹ : PreservesColimit (parallelPair f g) G
X' Y' : D
f' g' : X' ⟶ Y'
inst✝ : HasCoequalizer f' g'
p : G.obj X ⟶ X'
q : G.obj Y ⟶ Y'
wf : G.map f ≫ q = p ≫ f'
wg : G.map g ≫ q = p ≫ g'
Z' : D
h : Y' ⟶ Z'
wh : f' ≫ h = g' ≫ h
| h

case a
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
G : C ⥤ D
X Y : C
f g : X ⟶ Y
inst✝³ : HasCoequalizer f g
inst✝² : HasCoequalizer (G.map f) (G.map g)
inst✝¹ : PreservesColimit (parallelPair f g) G
X' Y' : D
f' g' : X' ⟶ Y'
inst✝ : HasCoequalizer f' g'
p : G.obj X ⟶ X'
q : G.obj Y ⟶ Y'
wf : G.map f ≫ q = p ≫ f'
wg : G.map g ≫ q = p ≫ g'
Z' : D
h : Y' ⟶ Z'
wh : f' ≫ h = g' ≫ h
| q
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
M : Type u_5
inst✝⁴ : CommRing R
inst✝³ : LinearOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : LinearMap.BilinForm R M
hs : ∀ (x : M), 0 ≤ (B x) x
hB : IsSymm B
⊢ Nondegenerate B ↔ ∀ (x : M), x ≠ 0 → 0 < (B x) x

Tactic:
B.nondegenerate_iff hs hB

Goal state after:
R : Type u_1
M : Type u_5
inst✝⁴ : CommRing R
inst✝³ : LinearOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : LinearMap.BilinForm R M
hs : ∀ (x : M), 0 ≤ (B x) x
hB : IsSymm B
⊢ (∀ (x : M), (B x) x = 0 ↔ x = 0) ↔ ∀ (x : M), x ≠ 0 → 0 < (B x) x
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
M : Type u_5
inst✝⁴ : CommRing R
inst✝³ : LinearOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : LinearMap.BilinForm R M
hs : ∀ (x : M), 0 ≤ (B x) x
hB : IsSymm B
W : Submodule R M
hW : Disjoint W (ker B)
hB' : (domRestrict₁₂ B W W).IsRefl
x : M
hx : x ∈ W
h : (B x) x = 0
⊢ ⟨x, hx⟩ = 0

Tactic:
B.apply_apply_same_eq_zero_iff hs hB

Goal state after:
R : Type u_1
M : Type u_5
inst✝⁴ : CommRing R
inst✝³ : LinearOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : LinearMap.BilinForm R M
hs : ∀ (x : M), 0 ≤ (B x) x
hB : IsSymm B
W : Submodule R M
hW : Disjoint W (ker B)
hB' : (domRestrict₁₂ B W W).IsRefl
x : M
hx : x ∈ W
h : x ∈ ker B
⊢ ⟨x, hx⟩ = 0
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: K : Type u_1
V₁ : Type u_2
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V₁
inst✝¹ : Module K V₁
inst✝ : FiniteDimensional K V₁
f g : Dual K V₁
x : V₁
h : LinearMap.ker f = LinearMap.ker g
h' : f x = g x
hx : f x ≠ 0
p : Submodule K V₁ := K ∙ x
hp : p ≠ ⊥
hpf : Disjoint (LinearMap.ker f) p
hf : f ≠ 0
v : V₁
⊢ ∃ y ∈ LinearMap.ker f, ∃ z ∈ p, y + z = v

Tactic:
by
    have : v ∈ (⊤ : Submodule K V₁) := Submodule.mem_top
    rwa [← (isCompl_ker_of_disjoint_of_ne_bot hf hpf hp).sup_eq_top, Submodule.mem_sup] at this

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.Pullbacks

Goal state before:
pp: C : Type u
𝒞 : Category.{v, u} C
X Y Z : C
f : X ⟶ Y
g : X ⟶ Z
inst✝¹ : HasColimit (pair Y Z)
inst✝ : HasColimit (parallelPair (f ≫ coprod.inl) (g ≫ coprod.inr))
ι₁ : Y ⟶ Y ⨿ Z := coprod.inl
ι₂ : Z ⟶ Y ⨿ Z := coprod.inr
c : Y ⨿ Z ⟶ coequalizer (f ≫ ι₁) (g ≫ ι₂) := coequalizer.π (f ≫ ι₁) (g ≫ ι₂)
⊢ ∀ (s : PushoutCocone f g),
    (ι₁ ≫ c) ≫ coequalizer.desc (coprod.desc (s.ι.app WalkingSpan.left) (s.ι.app WalkingSpan.right)) ⋯ = s.inl

Tactic:
simp [ι₁, c]

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
inst✝² : HasFiniteProducts C
inst✝¹ : HasKernels C
inst✝ : IsNormalMonoCategory C
X Y : C
f : X ⟶ Y
Z : C
l : IsColimit (CokernelCofork.ofπ 0 ⋯)
Z✝ : C
u v : Y ⟶ Z✝
huv : f ≫ u = f ≫ v
W : C
w : Y ⟶ W
hw : equalizer.ι u v ≫ w = 0
hl : IsLimit (KernelFork.ofι (equalizer.ι u v) hw)
⊢ u = v

Tactic:
obtain ⟨m, hm⟩ := equalizer.lift' f huv

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
inst✝² : HasFiniteProducts C
inst✝¹ : HasKernels C
inst✝ : IsNormalMonoCategory C
X Y : C
f : X ⟶ Y
Z : C
l : IsColimit (CokernelCofork.ofπ 0 ⋯)
Z✝ : C
u v : Y ⟶ Z✝
huv : f ≫ u = f ≫ v
W : C
w : Y ⟶ W
hw : equalizer.ι u v ≫ w = 0
hl : IsLimit (KernelFork.ofι (equalizer.ι u v) hw)
m : X ⟶ equalizer u v
hm : m ≫ equalizer.ι u v = f
⊢ u = v
Module: Mathlib.CategoryTheory.Abelian.NonPreadditive

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X : C
⊢ (prod.lift (𝟙 X) 0 ≫ cokernel.π (diag X)) ≫ inv (r X) = 𝟙 X

Tactic:
IsIso.hom_inv_id

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X : C
⊢ 𝟙 X = 𝟙 X
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:
rw [sub_def, ← lift_sub_lift, sub_def, Category.assoc, σ_comp, prod.lift_map_assoc]; rfl

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
inst✝² : HasFiniteCoproducts C
inst✝¹ : HasCokernels C
inst✝ : IsNormalEpiCategory C
X Y : C
f : X ⟶ Y
Z : C
l : IsLimit (KernelFork.ofι 0 ⋯)
Z✝ : C
u v : Z✝ ⟶ X
huv : u ≫ f = v ≫ f
W : C
w : W ⟶ X
hw : w ≫ coequalizer.π u v = 0
hl : IsColimit (CokernelCofork.ofπ (coequalizer.π u v) hw)
m : coequalizer u v ⟶ Y
hm : coequalizer.π u v ≫ m = f
reassoced : ∀ {W_1 : C} (h : coequalizer u v ⟶ W_1), w ≫ coequalizer.π u v ≫ h = 0 ≫ h
hwf : w ≫ f = 0
n : W ⟶ (KernelFork.ofι 0 ⋯).pt
hn : 0 = w
this : IsIso (coequalizer.π u v)
⊢ u = v

Tactic:
apply (cancel_mono (coequalizer.π u v)).1

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
inst✝² : HasFiniteCoproducts C
inst✝¹ : HasCokernels C
inst✝ : IsNormalEpiCategory C
X Y : C
f : X ⟶ Y
Z : C
l : IsLimit (KernelFork.ofι 0 ⋯)
Z✝ : C
u v : Z✝ ⟶ X
huv : u ≫ f = v ≫ f
W : C
w : W ⟶ X
hw : w ≫ coequalizer.π u v = 0
hl : IsColimit (CokernelCofork.ofπ (coequalizer.π u v) hw)
m : coequalizer u v ⟶ Y
hm : coequalizer.π u v ≫ m = f
reassoced : ∀ {W_1 : C} (h : coequalizer u v ⟶ W_1), w ≫ coequalizer.π u v ≫ h = 0 ≫ h
hwf : w ≫ f = 0
n : W ⟶ (KernelFork.ofι 0 ⋯).pt
hn : 0 = w
this : IsIso (coequalizer.π u v)
⊢ u ≫ coequalizer.π u v = v ≫ coequalizer.π u v
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
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ RightHomologyMapData φ h₁ h₂

Tactic:
let φQ : h₁.Q ⟶ h₂.Q := h₁.descQ (φ.τ₂ ≫ h₂.p) (by rw [← φ.comm₁₂_assoc, h₂.wp, comp_zero])

Goal state after:
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.RightHomologyData
h₂ : S₂.RightHomologyData
φQ : h₁.Q ⟶ h₂.Q := h₁.descQ (φ.τ₂ ≫ h₂.p) ⋯
⊢ RightHomologyMapData φ h₁ h₂
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : S₂.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
⊢ S₁.LeftHomologyData

Tactic:
let i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : S₂.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
⊢ S₁.LeftHomologyData
Module: Mathlib.Algebra.Homology.ShortComplex.Homology

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
inst✝ : S.HasHomology
h : S.RightHomologyData
⊢ h.homologyIso.inv ≫ S.homologyι = h.ι ≫ h.opcyclesIso.inv

Tactic:
dsimp only [homologyι, homologyIso]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
inst✝ : S.HasHomology
h : S.RightHomologyData
⊢ (S.rightHomologyIso.symm ≪≫ h.rightHomologyIso).inv ≫ S.rightHomologyIso.inv ≫ S.rightHomologyι =
    h.ι ≫ h.opcyclesIso.inv
Module: Mathlib.Algebra.Homology.ShortComplex.Homology

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
inst✝ : S.HasHomology
h : S.RightHomologyData
⊢ h.homologyIso.hom ≫ h.ι = S.homologyι ≫ h.opcyclesIso.hom

Tactic:
dsimp only [homologyι, homologyIso]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
inst✝ : S.HasHomology
h : S.RightHomologyData
⊢ (S.rightHomologyIso.symm ≪≫ h.rightHomologyIso).hom ≫ h.ι =
    (S.rightHomologyIso.inv ≫ S.rightHomologyι) ≫ h.opcyclesIso.hom
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
S : ShortComplex C
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
hₗ : S.LeftHomologyData
hᵣ : S.RightHomologyData
inst✝¹ : hₗ.IsPreservedBy F
inst✝ : hᵣ.IsPreservedBy F
⊢ (hₗ.map F).π ≫ leftRightHomologyComparison' (hₗ.map F) (hᵣ.map F) ≫ (hᵣ.map F).ι =
    (Cofork.π (CokernelCofork.ofπ (hₗ.map F).π ⋯) ≫ leftRightHomologyComparison' (hₗ.map F) (hᵣ.map F)) ≫
      Fork.ι (KernelFork.ofι (hᵣ.map F).ι ⋯)

Tactic:
· simp

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
S : ShortComplex C
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
⊢ (isoHomology S hkf hcc fac).hom ≫ S.homologyι = ι ≫ (S.isoOpcyclesOfIsColimit hcc).hom

Tactic:
by
  rw [← cancel_mono (S.isoOpcyclesOfIsColimit hcc).inv, assoc, assoc, Iso.hom_inv_id,
    comp_id, ← isoHomology_hom_comp_ι S hkf hcc fac, Iso.hom_inv_id_assoc]

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 S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄
⊢ (φ₁ - φ₃).τ₁ = S₁.f ≫ (h.h₁ - h'.h₁) + (h.h₀ - h'.h₀) + (φ₂ - φ₄).τ₁

Tactic:
sub_τ₁,

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
⊢ φ₁.τ₁ - φ₃.τ₁ = S₁.f ≫ (h.h₁ - h'.h₁) + (h.h₀ - h'.h₀) + (φ₂ - φ₄).τ₁
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.τ₁ - φ₃.τ₁ = S₁.f ≫ (h.h₁ - h'.h₁) + (h.h₀ - h'.h₀) + (φ₂ - φ₄).τ₁

Tactic:
sub_τ₁,

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
⊢ φ₁.τ₁ - φ₃.τ₁ = S₁.f ≫ (h.h₁ - h'.h₁) + (h.h₀ - h'.h₀) + (φ₂.τ₁ - φ₄.τ₁)
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:
nullHomotopic_τ₃,

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
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
⊢ h₂ ≫ S₂.g + h₃ = h₃ + h₂ ≫ S₂.g + Hom.τ₃ 0
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
H₁ : S₁.RightHomologyData
H₂ : S₂.RightHomologyData
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.g ≫ h₂ ≫ H₂.p) ≫ H₂.g' = S₁.g ≫ (S₁.nullHomotopic S₂ h₀ h₀_f h₁ h₂ h₃ g_h₃).τ₃

Tactic:
nullHomotopic_τ₃,

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
H₁ : S₁.RightHomologyData
H₂ : S₂.RightHomologyData
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.g ≫ h₂ ≫ H₂.p) ≫ H₂.g' = S₁.g ≫ (h₂ ≫ S₂.g + h₃)
Module: Mathlib.CategoryTheory.Adjunction.Reflective

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
j : C ⥤ D
inst✝¹ : Coreflective j
A : D
inst✝ : IsSplitEpi ((coreflectorAdjunction j).counit.app A)
⊢ j.essImage A

Tactic:
by
  let ε : coreflector j ⋙ j ⟶ 𝟭 D := (coreflectorAdjunction j).counit
  have : IsIso (ε.app (j.obj ((coreflector j).obj A))) :=
    Functor.essImage.counit_isIso ((j.obj_mem_essImage _))
  have : Mono (ε.app A) := by
    refine @mono_of_mono _ _ _ _ _ (ε.app A) (section_ (ε.app A)) ?_
    rw [show ε.app A ≫ section_ _ = _ from (ε.naturality (section_ (ε.app A))).symm]
    apply mono_comp _ (ε.app (j.obj ((coreflector j).obj A)))
  have := isIso_of_mono_of_isSplitEpi (ε.app A)
  exact (coreflectorAdjunction j).mem_essImage_of_counit_isIso A

Goal state after:
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
s : S.Splitting
⊢ Mono S.f

Tactic:
by
  have := s.isSplitMono_f
  infer_instance

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

Goal state before:
pp: case of
C : Type u₁
inst✝² : Category.{v₁, u₁} C
X✝ Y Z : C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J : Type u₃
inst✝ : Category.{v₃, u₃} J
X : C
F : Over X
x✝ : (Over.isMono X).limitsOfShape J F
p : LimitPresentation J F
hp : ∀ (j : J), Over.isMono X (p.diag.obj j)
Z✝ : C
g h : Z✝ ⟶ F.left
e : g ≫ F.hom = h ≫ F.hom
j : J
⊢ g ≫ (WithTerminal.coneEquiv.functor.obj { pt := F, π := p.π }).π.app (WithTerminal.of j) =
    h ≫ (WithTerminal.coneEquiv.functor.obj { pt := F, π := p.π }).π.app (WithTerminal.of j)

Tactic:
have := hp j; rw [← cancel_mono ((p.diag.obj j).hom)]; simpa

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

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
X Y : C
inst✝¹ : HasPullbacks C
f : X ⟶ Y
inst✝ : Mono f
g : Subobject X
⊢ (pullback f).obj ((map f).obj g) = g

Tactic:
revert g
  exact Quotient.ind (fun g' => Quotient.sound ⟨(MonoOver.pullbackMapSelf f).app _⟩)

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
X Y Z✝ : C
inst✝³ : HasZeroMorphisms C
f✝ : X ⟶ Y
inst✝² : HasKernel f✝
f : X ⟶ Y
inst✝¹ : HasKernel f
Z : C
h : Y ⟶ Z
inst✝ : Mono h
⊢ (kernelCompMono f h).inv ≫ kernel.ι (f ≫ h) = kernel.ι f

Tactic:
· simp

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
X Y Z : C
f✝ : X ⟶ Y
inst✝⁴ : HasImage f✝
inst✝³ : HasEqualizers C
X' : C
h : X' ⟶ X
inst✝² : Epi h
f : X ⟶ Y
inst✝¹ : HasImage f
inst✝ : HasImage (h ≫ f)
⊢ image.preComp h f ≫ image.ι f = image.ι (h ≫ f)

Tactic:
· simp

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.Filtered

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
α : Type w
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasColimitsOfShape (Finset (Discrete α)) C
F : Discrete α ⥤ C
s : Cocone F
m :
  { pt := colimit (liftToFinsetObj F),
        ι := Discrete.natTrans fun j ↦ Sigma.ι (fun x ↦ F.obj ↑x) ⟨j, ⋯⟩ ≫ colimit.ι (liftToFinsetObj F) {j} }.pt ⟶
    s.pt
h :
  ∀ (j : Discrete α),
    { pt := colimit (liftToFinsetObj F),
                ι :=
                  Discrete.natTrans fun j ↦
                    Sigma.ι (fun x ↦ F.obj ↑x) ⟨j, ⋯⟩ ≫ colimit.ι (liftToFinsetObj F) {j} }.ι.app
          j ≫
        m =
      s.ι.app j
⊢ m =
    colimit.desc (liftToFinsetObj F)
      { pt := s.pt, ι := { app := fun x ↦ Sigma.desc fun x_1 ↦ s.ι.app ↑x_1, naturality := ⋯ } }

Tactic:
apply colimit.hom_ext

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
α : Type w
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasColimitsOfShape (Finset (Discrete α)) C
F : Discrete α ⥤ C
s : Cocone F
m :
  { pt := colimit (liftToFinsetObj F),
        ι := Discrete.natTrans fun j ↦ Sigma.ι (fun x ↦ F.obj ↑x) ⟨j, ⋯⟩ ≫ colimit.ι (liftToFinsetObj F) {j} }.pt ⟶
    s.pt
h :
  ∀ (j : Discrete α),
    { pt := colimit (liftToFinsetObj F),
                ι :=
                  Discrete.natTrans fun j ↦
                    Sigma.ι (fun x ↦ F.obj ↑x) ⟨j, ⋯⟩ ≫ colimit.ι (liftToFinsetObj F) {j} }.ι.app
          j ≫
        m =
      s.ι.app j
⊢ ∀ (j : Finset (Discrete α)),
    colimit.ι (liftToFinsetObj F) j ≫ m =
      colimit.ι (liftToFinsetObj F) j ≫
        colimit.desc (liftToFinsetObj F)
          { pt := s.pt, ι := { app := fun x ↦ Sigma.desc fun x_1 ↦ s.ι.app ↑x_1, naturality := ⋯ } }
Module: Mathlib.CategoryTheory.Limits.Constructions.Filtered

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
α : Type w
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasColimitsOfShape (Finset (Discrete α)) C
F : Discrete α ⥤ C
s : Cocone F
m :
  { pt := colimit (liftToFinsetObj F),
        ι := Discrete.natTrans fun j ↦ Sigma.ι (fun x ↦ F.obj ↑x) ⟨j, ⋯⟩ ≫ colimit.ι (liftToFinsetObj F) {j} }.pt ⟶
    s.pt
h :
  ∀ (j : Discrete α),
    { pt := colimit (liftToFinsetObj F),
                ι :=
                  Discrete.natTrans fun j ↦
                    Sigma.ι (fun x ↦ F.obj ↑x) ⟨j, ⋯⟩ ≫ colimit.ι (liftToFinsetObj F) {j} }.ι.app
          j ≫
        m =
      s.ι.app j
t : Finset (Discrete α)
⊢ colimit.ι
        { obj := fun s ↦ ∐ fun x ↦ F.obj ↑x, map := fun {x Y} h ↦ Sigma.desc fun y ↦ Sigma.ι (fun x ↦ F.obj ↑x) ⟨↑y, ⋯⟩,
          map_id := ⋯, map_comp := ⋯ }
        t ≫
      m =
    colimit.ι
        { obj := fun s ↦ ∐ fun x ↦ F.obj ↑x, map := fun {x Y} h ↦ Sigma.desc fun y ↦ Sigma.ι (fun x ↦ F.obj ↑x) ⟨↑y, ⋯⟩,
          map_id := ⋯, map_comp := ⋯ }
        t ≫
      colimit.desc
        { obj := fun s ↦ ∐ fun x ↦ F.obj ↑x, map := fun {x Y} h ↦ Sigma.desc fun y ↦ Sigma.ι (fun x ↦ F.obj ↑x) ⟨↑y, ⋯⟩,
          map_id := ⋯, map_comp := ⋯ }
        { pt := s.pt, ι := { app := fun x ↦ Sigma.desc fun x_1 ↦ s.ι.app ↑x_1, naturality := ⋯ } }

Tactic:
apply colimit.hom_ext

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
α : Type w
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasColimitsOfShape (Finset (Discrete α)) C
F : Discrete α ⥤ C
s : Cocone F
m :
  { pt := colimit (liftToFinsetObj F),
        ι := Discrete.natTrans fun j ↦ Sigma.ι (fun x ↦ F.obj ↑x) ⟨j, ⋯⟩ ≫ colimit.ι (liftToFinsetObj F) {j} }.pt ⟶
    s.pt
h :
  ∀ (j : Discrete α),
    { pt := colimit (liftToFinsetObj F),
                ι :=
                  Discrete.natTrans fun j ↦
                    Sigma.ι (fun x ↦ F.obj ↑x) ⟨j, ⋯⟩ ≫ colimit.ι (liftToFinsetObj F) {j} }.ι.app
          j ≫
        m =
      s.ι.app j
t : Finset (Discrete α)
⊢ ∀ (j : Discrete ↥t),
    colimit.ι (Discrete.functor fun x ↦ F.obj ↑x) j ≫
        colimit.ι
            { obj := fun s ↦ ∐ fun x ↦ F.obj ↑x,
              map := fun {x Y} h ↦ Sigma.desc fun y ↦ Sigma.ι (fun x ↦ F.obj ↑x) ⟨↑y, ⋯⟩, map_id := ⋯, map_comp := ⋯ }
            t ≫
          m =
      colimit.ι (Discrete.functor fun x ↦ F.obj ↑x) j ≫
        colimit.ι
            { obj := fun s ↦ ∐ fun x ↦ F.obj ↑x,
              map := fun {x Y} h ↦ Sigma.desc fun y ↦ Sigma.ι (fun x ↦ F.obj ↑x) ⟨↑y, ⋯⟩, map_id := ⋯, map_comp := ⋯ }
            t ≫
          colimit.desc
            { obj := fun s ↦ ∐ fun x ↦ F.obj ↑x,
              map := fun {x Y} h ↦ Sigma.desc fun y ↦ Sigma.ι (fun x ↦ F.obj ↑x) ⟨↑y, ⋯⟩, map_id := ⋯, map_comp := ⋯ }
            { pt := s.pt, ι := { app := fun x ↦ Sigma.desc fun x_1 ↦ s.ι.app ↑x_1, naturality := ⋯ } }
Module: Mathlib.CategoryTheory.Limits.Constructions.Filtered

Goal state before:
pp: case e'_3
C : Type u
inst✝² : Category.{v, u} C
α : Type w
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasColimitsOfShape (Finset (Discrete α)) C
F : Discrete α ⥤ C
s : Cocone F
m :
  { pt := colimit (liftToFinsetObj F),
        ι := Discrete.natTrans fun j ↦ Sigma.ι (fun x ↦ F.obj ↑x) ⟨j, ⋯⟩ ≫ colimit.ι (liftToFinsetObj F) {j} }.pt ⟶
    s.pt
h :
  ∀ (j : Discrete α),
    { pt := colimit (liftToFinsetObj F),
                ι :=
                  Discrete.natTrans fun j ↦
                    Sigma.ι (fun x ↦ F.obj ↑x) ⟨j, ⋯⟩ ≫ colimit.ι (liftToFinsetObj F) {j} }.ι.app
          j ≫
        m =
      s.ι.app j
t : Finset (Discrete α)
j : Discrete α
hj : j ∈ t
e_1✝ : ((Discrete.functor fun x ↦ F.obj ↑x).obj { as := ⟨j, hj⟩ } ⟶ s.pt) = (F.obj j ⟶ s.pt)
⊢ colimit.ι (Discrete.functor fun x ↦ F.obj ↑x) { as := ⟨j, hj⟩ } ≫
      colimit.ι
          { obj := fun s ↦ ∐ fun x ↦ F.obj ↑x,
            map := fun {x Y} h ↦ Sigma.desc fun y ↦ Sigma.ι (fun x ↦ F.obj ↑x) ⟨↑y, ⋯⟩, map_id := ⋯, map_comp := ⋯ }
          t ≫
        colimit.desc
          { obj := fun s ↦ ∐ fun x ↦ F.obj ↑x,
            map := fun {x Y} h ↦ Sigma.desc fun y ↦ Sigma.ι (fun x ↦ F.obj ↑x) ⟨↑y, ⋯⟩, map_id := ⋯, map_comp := ⋯ }
          { pt := s.pt, ι := { app := fun x ↦ Sigma.desc fun x_1 ↦ s.ι.app ↑x_1, naturality := ⋯ } } =
    s.ι.app j

Tactic:
· simp

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.Filtered

Goal state before:
pp: case e'_3
C : Type u
inst✝³ : Category.{v, u} C
α : Type w
inst✝² : HasFiniteCoproducts C
f : α → C
inst✝¹ : HasColimitsOfShape (Finset (Discrete α)) C
inst✝ : HasCoproduct f
S : Finset (Discrete α)
j : ↥S
e_1✝ : (f (↑j).as ⟶ ∐ f) = ((Discrete.functor f).obj ↑j ⟶ (colimit.cocone (Discrete.functor f)).pt)
⊢ (Sigma.ι (fun x ↦ f (↑x).as) j ≫ Sigma.desc fun s ↦ Sigma.ι f (↑s).as) =
    (colimit.cocone (Discrete.functor f)).ι.app ↑j

Tactic:
· simp

Goal state after: