mathlib-initiative/mathlib-tactics huggingface dataset:
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:
rw [show (1 : B ⧸ f.range) = (1 : B) from QuotientGroup.mk_one _, QuotientGroup.eq, inv_one,
      one_mul]

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
⊢ f x ∈ f.range
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:
← cancel_mono e₃.hom,

Goal state after:
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) ≫ e₃.hom = (S₂.g ≫ e₃.inv) ≫ e₃.hom
Module: Mathlib.Algebra.Category.Grp.EpiMono

Goal state before:
pp: A B : CommGrpCat
f : A ⟶ B
⊢ Epi f ↔ Function.Surjective ⇑(ConcreteCategory.hom f)

Tactic:
rw [epi_iff_range_eq_top, MonoidHom.range_eq_top]

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
⊢ S.rightHomologyData.g'.op ≫ S.rightHomologyData.op.cyclesIso.inv ≫ S.op.iCycles = S.op.f

Tactic:
LeftHomologyData.cyclesIso_inv_comp_iCycles,

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
inst✝ : S.HasRightHomology
⊢ S.rightHomologyData.g'.op ≫ S.rightHomologyData.op.i = S.op.f
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝¹ : S₁.HasRightHomology
inst✝ : S₂.HasRightHomology
⊢ (opcyclesMap φ).op ≫ S₁.pOpcycles.op = S₂.cyclesOpIso.inv ≫ S₂.op.iCycles ≫ (opMap φ).τ₂

Tactic:
cyclesOpIso_inv_op_iCycles_assoc,

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝¹ : S₁.HasRightHomology
inst✝ : S₂.HasRightHomology
⊢ (opcyclesMap φ).op ≫ S₁.pOpcycles.op = S₂.pOpcycles.op ≫ (opMap φ).τ₂
Module: Mathlib.Algebra.Homology.ShortComplex.Homology

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
h : Epi (cyclesMap φ)
this : Epi (S₁.homologyπ ≫ homologyMap φ)
⊢ Epi (homologyMap φ)

Tactic:
exact epi_of_epi S₁.homologyπ (homologyMap φ)

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

Goal state before:
pp: S : ShortComplex Ab
x : ↑S.X₁
⊢ (ConcreteCategory.hom S.g) ((ConcreteCategory.hom S.f) x) = 0

Tactic:
← ConcreteCategory.comp_apply,

Goal state after:
S : ShortComplex Ab
x : ↑S.X₁
⊢ (ConcreteCategory.hom (S.f ≫ S.g)) x = 0
Module: Mathlib.Algebra.Homology.ShortComplex.Ab

Goal state before:
pp: S : ShortComplex Ab
x : ↥(AddCommGrpCat.Hom.hom S.g).ker
⊢ (ConcreteCategory.hom S.iCycles) ((ConcreteCategory.hom S.abLeftHomologyData.cyclesIso.inv) x) = ↑x

Tactic:
← ConcreteCategory.comp_apply,

Goal state after:
S : ShortComplex Ab
x : ↥(AddCommGrpCat.Hom.hom S.g).ker
⊢ (ConcreteCategory.hom (S.abLeftHomologyData.cyclesIso.inv ≫ S.iCycles)) x = ↑x
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
e : S₁ ≅ S₂
h : S₁.HomologyData
z : IsZero h.left.H
⊢ S₂.Exact

Tactic:
exact ⟨⟨HomologyData.ofIso e h, z⟩⟩

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

Goal state before:
pp: case mp
S : ShortComplex Ab
h : (AddCommGrpCat.Hom.hom S.g).ker ≤ (AddCommGrpCat.Hom.hom S.f).range
x₁ : ↑S.X₁
⊢ (AddCommGrpCat.Hom.hom S.g) ((AddCommGrpCat.Hom.hom S.f) x₁) = 0

Tactic:
← ConcreteCategory.comp_apply,

Goal state after:
case mp
S : ShortComplex Ab
h : (AddCommGrpCat.Hom.hom S.g).ker ≤ (AddCommGrpCat.Hom.hom S.f).range
x₁ : ↑S.X₁
⊢ (ConcreteCategory.hom (S.f ≫ S.g)) x₁ = 0
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
h : S.LeftHomologyData
this : S.HasHomology
⊢ IsZero h.H

Tactic:
simpa only [← h.exact_iff] using hS

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
hS : S.Exact
h : S.RightHomologyData
this : S.HasHomology
⊢ IsZero h.H

Tactic:
simpa only [← h.exact_iff] using hS

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

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
inst✝¹ : Preadditive D
S : ShortComplex C
hS : S.Exact
inst✝ : HasZeroObject C
kf : KernelFork S.g
hkf : IsLimit kf
this : S.HasHomology
⊢ ((Iso.refl (Arrow.mk S.toCycles).left).hom ≫
        (Arrow.mk ((hkf.ofIsoLimit (Fork.ext (Iso.refl kf.pt) ⋯)).lift (KernelFork.ofι S.f ⋯))).hom) ≫
      Fork.ι kf =
    ((Arrow.mk S.toCycles).hom ≫ (S.cyclesIsKernel.conePointUniqueUpToIso hkf).hom) ≫ Fork.ι kf

Tactic:
simp [IsLimit.conePointUniqueUpToIso]

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

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
Y Z : C
P Q : Subobject Y
f : Z ⟶ Y
h : P ≤ Q
x✝ : (representative.obj P).Factors f
u : Z ⟶ (representative.obj P).obj.left
hu : u ≫ (representative.obj P).arrow = f
⊢ (u ≫ P.ofLE Q h) ≫ (representative.obj Q).arrow = f

Tactic:
by simp [← hu]

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

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X : C
P : Subobject X
h : P.Factors P.arrow
⊢ P.factorThru P.arrow h = 𝟙 (underlying.obj P)

Tactic:
by
  ext
  simp

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

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X Y : C
f : X ⟶ Y
inst✝ : Mono f
⊢ (mk f).factorThru f ⋯ = (underlyingIso f).inv

Tactic:
by
  ext
  simp

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

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X Y : C
P : Subobject Y
f : X ⟶ underlying.obj P
h : P.Factors (f ≫ P.arrow)
⊢ P.factorThru (f ≫ P.arrow) h = f

Tactic:
by
  ext
  simp

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

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
Y Z : C
P Q : Subobject Y
f : Z ⟶ Y
h : P ≤ Q
w : P.Factors f
⊢ Q.factorThru f ⋯ = P.factorThru f w ≫ P.ofLE Q h

Tactic:
by
  ext
  simp

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

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
X Y : C
P : Subobject Y
f g : X ⟶ Y
w : P.Factors (f + g)
wf : P.Factors f
wg : P.Factors g
⊢ P.factorThru (f + g) w = P.factorThru f wf + P.factorThru g wg

Tactic:
by
  ext
  simp

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

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
X Y : C
P : Subobject Y
f g : X ⟶ Y
w : P.Factors (f + g)
wg : P.Factors g
⊢ P.factorThru (f + g) w - P.factorThru g wg = P.factorThru f ⋯

Tactic:
by
  ext
  simp

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

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
X Y : C
P : Subobject Y
f g : X ⟶ Y
w : P.Factors (f + g)
wf : P.Factors f
⊢ P.factorThru (f + g) w - P.factorThru f wf = P.factorThru g ⋯

Tactic:
by
  ext
  simp

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

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
inst✝¹ : Preadditive D
S : ShortComplex C
inst✝ : HasZeroObject C
s : S.Splitting
hp : IsColimit (CokernelCofork.ofπ S.g ⋯)
g' : (CokernelCofork.ofπ S.g ⋯).pt ⟶ (CokernelCofork.ofπ S.g ⋯).pt := hp.desc (CokernelCofork.ofπ S.g ⋯)
hg' : g' = 𝟙 (CokernelCofork.ofπ S.g ⋯).pt
⊢ S.RightHomologyData

Tactic:
have wι : (0 : 0 ⟶ S.X₃) ≫ g' = 0 := zero_comp

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 : Preadditive C
inst✝¹ : Preadditive D
S : ShortComplex C
inst✝ : HasZeroObject C
s : S.Splitting
hp : IsColimit (CokernelCofork.ofπ S.g ⋯)
g' : (CokernelCofork.ofπ S.g ⋯).pt ⟶ (CokernelCofork.ofπ S.g ⋯).pt := hp.desc (CokernelCofork.ofπ S.g ⋯)
hg' : g' = 𝟙 (CokernelCofork.ofπ S.g ⋯).pt
wι : 0 ≫ g' = 0
⊢ S.RightHomologyData
Module: Mathlib.CategoryTheory.Subobject.Basic

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X Y Z : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
A : C
B : D
e : MonoOver A ≌ MonoOver B
⊢ 𝟭 (Subobject A) = lower e.functor ⋙ lower e.inverse

Tactic:
convert! ThinSkeleton.map_iso_eq e.unitIso

Goal state after:
case e'_2
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X Y Z : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
A : C
B : D
e : MonoOver A ≌ MonoOver B
⊢ 𝟭 (Subobject A) = ThinSkeleton.map (𝟭 (MonoOver A))

case e'_3
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X Y Z : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
A : C
B : D
e : MonoOver A ≌ MonoOver B
⊢ lower e.functor ⋙ lower e.inverse = ThinSkeleton.map (e.functor ⋙ e.inverse)
Module: Mathlib.CategoryTheory.Subobject.MonoOver

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
X Y Z✝ : C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasImages C
f : X ⟶ Y
inst✝ : Mono f
Z : MonoOver X
this : (forget Y).obj ((«exists» f).obj Z) ≅ (forget Y).obj ((map f).obj Z)
⊢ («exists» f).obj Z ≅ (map f).obj Z

Tactic:
apply (forget _).preimageIso this

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✝ : Abelian C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hf₁ : S₁.f = 0
hg₁ : S₁.g = 0
hf₂ : S₂.f = 0
⊢ QuasiIso φ ↔ { X₁ := S₁.X₂, X₂ := S₂.X₂, X₃ := S₂.X₃, f := φ.τ₂, g := S₂.g, zero := ⋯ }.Exact ∧ Mono φ.τ₂

Tactic:
by
  have w : φ.τ₂ ≫ S₂.g = 0 := by rw [φ.comm₂₃, hg₁, zero_comp]
  rw [quasiIso_iff_isIso_liftCycles φ hf₁ hg₁ hf₂]
  constructor
  · intro h
    have : Mono φ.τ₂ := by
      rw [← S₂.liftCycles_i φ.τ₂ w]
      apply mono_comp
    refine ⟨?_, this⟩
    apply exact_of_f_is_kernel
    exact IsLimit.ofIsoLimit S₂.cyclesIsKernel
      (Fork.ext (asIso (S₂.liftCycles φ.τ₂ w)).symm (by simp))
  · rintro ⟨h₁, h₂⟩
    refine ⟨⟨h₁.lift S₂.iCycles (by simp), ?_, ?_⟩⟩
    · rw [← cancel_mono φ.τ₂, assoc, h₁.lift_f, liftCycles_i, id_comp]
    · rw [← cancel_mono S₂.iCycles, assoc, liftCycles_i, h₁.lift_f, id_comp]

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

Goal state before:
pp: case hg₁
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hg₁ : S₁.g = 0
hf₂ : S₂.f = 0
hg₂ : S₂.g = 0
⊢ S₂.f.op = 0

Tactic:
rw [hf₂, op_zero]

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

Goal state before:
pp: case hg₁
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hg₁ : S₁.g = 0
hf₂ : S₂.f = 0
hg₂ : S₂.g = 0
⊢ S₂.op.g = 0

Tactic:
dsimp
    rw [hf₂, op_zero]

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

Goal state before:
pp: V : Type u
inst✝³ : Category.{v, u} V
inst✝² : HasZeroMorphisms V
A B C : V
f : A ⟶ B
inst✝¹ : HasImage f
g : B ⟶ C
inst✝ : HasKernel g
w : f ≫ g = 0
⊢ factorThruImageSubobject f ≫ imageToKernel f g w = factorThruKernelSubobject g f w

Tactic:
by
  ext
  simp

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

Goal state before:
pp: V : Type u
inst✝³ : Category.{v, u} V
inst✝² : HasZeroMorphisms V
A B C : V
g : B ⟶ C
inst✝¹ : HasKernels V
inst✝ : HasZeroObject V
w : 0 ≫ g = 0
⊢ imageToKernel 0 g w = 0

Tactic:
by
  ext
  simp

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

Goal state before:
pp: V : Type u
inst✝³ : Category.{v, u} V
inst✝² : HasZeroMorphisms V
A B C : V
f : A ⟶ B
g : B ⟶ C
inst✝¹ : HasKernels V
inst✝ : HasImages V
D : V
h : C ⟶ D
w : f ≫ g = 0
⊢ imageToKernel f (g ≫ h) ⋯ = imageToKernel f g w ≫ (kernelSubobject g).ofLE (kernelSubobject (g ≫ h)) ⋯

Tactic:
by
  ext
  simp

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

Goal state before:
pp: V : Type u
inst✝³ : Category.{v, u} V
inst✝² : HasZeroMorphisms V
A B C : V
f : A ⟶ B
g : B ⟶ C
inst✝¹ : HasKernels V
inst✝ : HasImages V
Z : V
h : Z ⟶ A
w : f ≫ g = 0
⊢ imageToKernel (h ≫ f) g ⋯ = (imageSubobject (h ≫ f)).ofLE (imageSubobject f) ⋯ ≫ imageToKernel f g w

Tactic:
by
  ext
  simp

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

Goal state before:
pp: V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : HasZeroMorphisms V
A B C : V
f : A ⟶ B
g : B ⟶ C
inst✝² : HasKernels V
inst✝¹ : HasImages V
D : V
h : C ⟶ D
inst✝ : Mono h
w : f ≫ g ≫ h = 0
⊢ imageToKernel f (g ≫ h) w = imageToKernel f g ⋯ ≫ ((kernelSubobject (g ≫ h)).isoOfEq (kernelSubobject g) ⋯).inv

Tactic:
by
  ext
  simp

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

Goal state before:
pp: V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : HasZeroMorphisms V
A B C : V
f : A ⟶ B
g : B ⟶ C
inst✝² : HasKernels V
inst✝¹ : HasImages V
Z : V
h : Z ⟶ A
inst✝ : Epi h
w : (h ≫ f) ≫ g = 0
⊢ imageToKernel (h ≫ f) g w = (imageSubobject (h ≫ f)).ofLE (imageSubobject f) ⋯ ≫ imageToKernel f g ⋯

Tactic:
by
  ext
  simp

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

Goal state before:
pp: V : Type u
inst✝³ : Category.{v, u} V
inst✝² : HasZeroMorphisms V
A B C : V
f : A ⟶ B
g : B ⟶ C
inst✝¹ : HasEqualizers V
inst✝ : HasImages V
Z : V
i : B ≅ Z
w : (f ≫ i.hom) ≫ i.inv ≫ g = 0
⊢ imageToKernel (f ≫ i.hom) (i.inv ≫ g) w =
    (imageSubobjectCompIso f i.hom).hom ≫ imageToKernel f g ⋯ ≫ (kernelSubobjectIsoComp i.inv g).inv

Tactic:
by
  ext
  simp

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

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Abelian C
S : ShortComplex C
⊢ S.Exact ↔ Epi (imageToKernel S.f S.g ⋯)

Tactic:
by
  rw [S.exact_iff_epi_imageToKernel']
  apply (MorphismProperty.epimorphisms C).arrow_mk_iso_iff
  exact Arrow.isoMk (imageSubobjectIso S.f).symm (kernelSubobjectIso S.g).symm

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
X Y : C
f : X ⟶ Y
inst✝ : HasImage f
W : C
k : W ⟶ X
h : (imageSubobject f).Factors (k ≫ f)
⊢ (imageSubobject f).factorThru (k ≫ f) h = k ≫ factorThruImageSubobject f

Tactic:
by
  ext
  simp

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
X Y : C
f : X ⟶ Y
inst✝ : HasImage f
W W' : C
k : W ⟶ W'
k' : W' ⟶ X
h : (imageSubobject f).Factors (k ≫ k' ≫ f)
⊢ (imageSubobject f).factorThru (k ≫ k' ≫ f) h = k ≫ k' ≫ factorThruImageSubobject f

Tactic:
by
  ext
  simp

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

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
Z : C
tfae_2_to_1 : cokernel.π f = 0 → Epi f
tfae_1_to_2 : Epi f → cokernel.π f = 0
⊢ [Epi f, cokernel.π f = 0, { X₁ := X, X₂ := Y, X₃ := Z, f := f, g := 0, zero := ⋯ }.Exact].TFAE

Tactic:
tfae_have 3 ↔ 1 := ShortComplex.exact_iff_epi _ (by simp)

Goal state after:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
Z : C
tfae_2_to_1 : cokernel.π f = 0 → Epi f
tfae_1_to_2 : Epi f → cokernel.π f = 0
tfae_3_iff_1 : { X₁ := X, X₂ := Y, X₃ := Z, f := f, g := 0, zero := ⋯ }.Exact ↔ Epi f
⊢ [Epi f, cokernel.π f = 0, { X₁ := X, X₂ := Y, X₃ := Z, f := f, g := 0, zero := ⋯ }.Exact].TFAE
Module: Mathlib.CategoryTheory.Abelian.FunctorCategory

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
D : Type w
inst✝¹ : Category.{z, w} D
inst✝ : Abelian D
F G : C ⥤ D
α : F ⟶ G
X : C
this : ∀ (X : C), IsIso ((coimageImageComparison α).app X)
⊢ IsIso (coimageImageComparison α)

Tactic:
apply NatIso.isIso_of_isIso_app

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

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X✝ Y✝ Z✝ : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
X : C
Y : Subobject X
Z : ↑(Set.Iic Y)
⊢ (underlyingIso ((↑Z).ofLE Y ⋯)).hom ≫ (↑Z).arrow = ((fun Z ↦ mk ((↑Z).ofLE Y ⋯)) Z).arrow ≫ Y.arrow

Tactic:
by simp [← Iso.eq_inv_comp]

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products

Goal state before:
pp: case h₁
C : Type u₁
inst✝² : Category.{v₁, u₁} C
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
X : Type v₂
A B : C
inst✝ : HasBinaryProduct A B
⊢ coprod.inl ≫ coprod.desc prod.fst.op prod.snd.op ≫ (prod.lift coprod.inl.unop coprod.inr.unop).op =
    coprod.inl ≫ 𝟙 (op A ⨿ op B)

Tactic:
· simp only [colimit.ι_desc_assoc]
      apply Quiver.Hom.unop_inj
      simp

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products

Goal state before:
pp: case h₂
C : Type u₁
inst✝² : Category.{v₁, u₁} C
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
X : Type v₂
A B : C
inst✝ : HasBinaryProduct A B
⊢ coprod.inr ≫ coprod.desc prod.fst.op prod.snd.op ≫ (prod.lift coprod.inl.unop coprod.inr.unop).op =
    coprod.inr ≫ 𝟙 (op A ⨿ op B)

Tactic:
· simp only [colimit.ι_desc_assoc]
      apply Quiver.Hom.unop_inj
      simp

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasFilteredColimitsOfSize.{w, w, v, u} C
α : Type w
⊢ HasColimitsOfShape (Discrete α) C

Tactic:
by
  exact ⟨fun F => HasColimit.mk (liftToFinsetColimitCocone F)⟩

Goal state after:
Module: Mathlib.CategoryTheory.Functor.KanExtension.Basic

Goal state before:
pp: case mp
C : Type u_1
H : Type u_3
D : Type u_4
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_3, u_3} H
inst✝¹ : Category.{v_4, u_4} D
F' F'' : D ⥤ H
φ : F' ⟶ F''
L : C ⥤ D
F : C ⥤ H
α : F ⟶ L ⋙ F'
α' : F ⟶ L ⋙ F''
comm : α ≫ L.whiskerLeft φ = α'
inst✝ : F'.IsLeftKanExtension α
a✝ : F''.IsLeftKanExtension α'
⊢ IsIso φ

Tactic:
F'.hom_ext_of_isLeftKanExtension α φ (leftKanExtensionUnique _ α _ α').hom
      (by simp [comm])

Goal state after:
case mp
C : Type u_1
H : Type u_3
D : Type u_4
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_3, u_3} H
inst✝¹ : Category.{v_4, u_4} D
F' F'' : D ⥤ H
φ : F' ⟶ F''
L : C ⥤ D
F : C ⥤ H
α : F ⟶ L ⋙ F'
α' : F ⟶ L ⋙ F''
comm : α ≫ L.whiskerLeft φ = α'
inst✝ : F'.IsLeftKanExtension α
a✝ : F''.IsLeftKanExtension α'
⊢ IsIso (F'.leftKanExtensionUnique α F'' α').hom
Module: Mathlib.CategoryTheory.Functor.KanExtension.Basic

Goal state before:
pp: C : Type u_1
H : Type u_3
D : Type u_4
D' : Type u_5
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_3, u_3} H
inst✝² : Category.{v_4, u_4} D
inst✝¹ : Category.{v_5, u_5} D'
L : C ⥤ D
L' : C ⥤ D'
G : D ⥤ D'
inst✝ : G.IsEquivalence
e : L ⋙ G ≅ L'
F : C ⥤ H
F' : D' ⥤ H
α : F ⟶ L' ⋙ F'
⊢ F'.IsLeftKanExtension α ↔ (G ⋙ F').IsLeftKanExtension (α ≫ whiskerRight e.inv F' ≫ (L.associator G F').hom)

Tactic:
by
  let eq : (LeftExtension.mk _ α).IsUniversal ≃
      (LeftExtension.mk _
        (α ≫ whiskerRight e.inv _ ≫ (associator _ _ _).hom)).IsUniversal :=
    (LeftExtension.isUniversalPostcomp₁Equiv G e F _).trans
    (IsInitial.equivOfIso (StructuredArrow.isoMk (Iso.refl _)))
  constructor
  · exact fun _ => ⟨⟨eq (isUniversalOfIsLeftKanExtension _ _)⟩⟩
  · exact fun _ => ⟨⟨eq.symm (isUniversalOfIsLeftKanExtension _ _)⟩⟩

Goal state after:
Module: Mathlib.CategoryTheory.Functor.KanExtension.Basic

Goal state before:
pp: case mp
C : Type u_1
H : Type u_3
D : Type u_4
D' : Type u_5
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_3, u_3} H
inst✝² : Category.{v_4, u_4} D
inst✝¹ : Category.{v_5, u_5} D'
L : C ⥤ D
L' : C ⥤ D'
G : D ⥤ D'
inst✝ : G.IsEquivalence
e : L ⋙ G ≅ L'
F : C ⥤ H
F' : D' ⥤ H
α : L' ⋙ F' ⟶ F
eq : CostructuredArrow.IsUniversal (RightExtension.mk F' α) ≃
  CostructuredArrow.IsUniversal (RightExtension.mk (G ⋙ F') ((L.associator G F').inv ≫ whiskerRight e.hom F' ≫ α)) := ⋯
⊢ F'.IsRightKanExtension α → (G ⋙ F').IsRightKanExtension ((L.associator G F').inv ≫ whiskerRight e.hom F' ≫ α)

Tactic:
exact fun _ => ⟨⟨eq (isUniversalOfIsRightKanExtension _ _)⟩⟩

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Preserves.FunctorCategory

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v₁, u} C
D : Type u₂
inst✝⁴ : Category.{u, u₂} D
inst✝³ : HasBinaryProducts D
inst✝² : HasColimits D
inst✝¹ : ∀ (X : D), PreservesColimits (prod.functor.obj X)
F : C ⥤ D
J : Type u
inst✝ : Category.{u, u} J
K : J ⥤ C ⥤ D
c : Cocone K
t : IsColimit c
k : C
this : IsColimit (((evaluation C D).obj k ⋙ prod.functor.obj (F.obj k)).mapCocone c) := ⋯
⊢ ∀ {X Y : C ⥤ D} (f : X ⟶ Y),
    (prod.functor.obj F ⋙ (evaluation C D).obj k).map f ≫ (asIso (prodComparison ((evaluation C D).obj k) F Y)).hom =
      (asIso (prodComparison ((evaluation C D).obj k) F X)).hom ≫
        ((evaluation C D).obj k ⋙ prod.functor.obj (F.obj k)).map f

Tactic:
intro G G'

Goal state after:
C : Type u
inst✝⁵ : Category.{v₁, u} C
D : Type u₂
inst✝⁴ : Category.{u, u₂} D
inst✝³ : HasBinaryProducts D
inst✝² : HasColimits D
inst✝¹ : ∀ (X : D), PreservesColimits (prod.functor.obj X)
F : C ⥤ D
J : Type u
inst✝ : Category.{u, u} J
K : J ⥤ C ⥤ D
c : Cocone K
t : IsColimit c
k : C
this : IsColimit (((evaluation C D).obj k ⋙ prod.functor.obj (F.obj k)).mapCocone c) :=
  isColimitOfPreserves ((evaluation C D).obj k ⋙ prod.functor.obj (F.obj k)) t
G G' : C ⥤ D
⊢ ∀ (f : G ⟶ G'),
    (prod.functor.obj F ⋙ (evaluation C D).obj k).map f ≫ (asIso (prodComparison ((evaluation C D).obj k) F G')).hom =
      (asIso (prodComparison ((evaluation C D).obj k) F G)).hom ≫
        ((evaluation C D).obj k ⋙ prod.functor.obj (F.obj k)).map f
Module: Mathlib.CategoryTheory.Functor.KanExtension.Basic

Goal state before:
pp: case mp
C : Type u_1
C' : Type u_2
H : Type u_3
D : Type u_4
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} C'
inst✝² : Category.{v_3, u_3} H
inst✝¹ : Category.{v_4, u_4} D
L : C ⥤ D
F : C ⥤ H
F' : D ⥤ H
G : C' ⥤ C
inst✝ : G.IsEquivalence
α : L ⋙ F' ⟶ F
eq : CostructuredArrow.IsUniversal (RightExtension.mk F' α) ≃
  CostructuredArrow.IsUniversal (RightExtension.mk F' ((G.associator L F').hom ≫ G.whiskerLeft α)) := ⋯
⊢ F'.IsRightKanExtension α → F'.IsRightKanExtension ((G.associator L F').hom ≫ G.whiskerLeft α)

Tactic:
exact fun _ => ⟨⟨eq (isUniversalOfIsRightKanExtension _ _)⟩⟩

Goal state after: