mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.ObjectProperty.ColimitsOfShape

Goal state before:
pp: case refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : ObjectProperty C
J : Type u'
inst✝ : Category.{v', u'} J
X : C
x✝ : (P.op.limitsOfShape Jᵒᵖ).unop X
h : P.op.LimitOfShape Jᵒᵖ (Opposite.op X)
⊢ P.ColimitOfShape J X

Tactic:
· exact
      { diag := h.diag.unop
        ι := NatTrans.unop h.π
        isColimit := isColimitOfOp h.isLimit
        prop_diag_obj _ := h.prop_diag_obj _ }

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.ColimitsOfShape

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
P : ObjectProperty C
J : Type u'
inst✝² : Category.{v', u'} J
J' : Type u''
inst✝¹ : Category.{v'', u''} J'
Q : ObjectProperty Cᵒᵖ
inst✝ : Q.IsClosedUnderLimitsOfShape J
⊢ Q.unop.IsClosedUnderColimitsOfShape Jᵒᵖ

Tactic:
← isClosedUnderLimitsOfShape_iff_unop

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
P : ObjectProperty C
J : Type u'
inst✝² : Category.{v', u'} J
J' : Type u''
inst✝¹ : Category.{v'', u''} J'
Q : ObjectProperty Cᵒᵖ
inst✝ : Q.IsClosedUnderLimitsOfShape J
⊢ Q.IsClosedUnderLimitsOfShape J
Module: Mathlib.Algebra.Category.CommAlgCat.Monoidal

Goal state before:
pp: R : Type u
inst✝ : CommRing R
A B C D : CommAlgCat R
S T : (CommAlgCat R)ᵒᵖ
⊢ ((unop S).binaryCofan (unop T)).inl.op = S ◁ (terminalOpOfInitial isInitialSelf).from T ≫ (ρ_ S).hom

Tactic:
ext x; change x ⊗ₜ 1 = x ⊗ₜ algebraMap R T.unop 1; simp

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Grp

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : CartesianMonoidalCategory C
X : C
h₁ h₂ : GrpObj X
e : h₁.toMonObj = h₂.toMonObj
⊢ lift ι (𝟙 X) ≫ μ = toUnit X ≫ η

Tactic:
convert! @left_inv _ _ _ _ h₁ using 2

Goal state after:
case e'_2
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : CartesianMonoidalCategory C
X : C
h₁ h₂ : GrpObj X
e : h₁.toMonObj = h₂.toMonObj
⊢ μ = μ

case e'_3
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : CartesianMonoidalCategory C
X : C
h₁ h₂ : GrpObj X
e : h₁.toMonObj = h₂.toMonObj
⊢ η = η
Module: Mathlib.LinearAlgebra.Dual.Basis

Goal state before:
pp: R : Type uR
M : Type uM
ι : Type uι
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : DecidableEq ι
b : Basis ι R M
f : ι →₀ R
i : ι
⊢ (∑ x ∈ f.support, f x • if x = i then 1 else 0) = f i

Tactic:
smul_eq_mul,

Goal state after:
R : Type uR
M : Type uM
ι : Type uι
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : DecidableEq ι
b : Basis ι R M
f : ι →₀ R
i : ι
⊢ (∑ x ∈ f.support, f x * if x = i then 1 else 0) = f i
Module: Mathlib.LinearAlgebra.Dual.Basis

Goal state before:
pp: R : Type uR
M : Type uM
ι : Type uι
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : DecidableEq ι
b : Basis ι R M
f : ι →₀ R
i : ι
⊢ (∑ x ∈ f.support, f x • if i = x then 1 else 0) = f i

Tactic:
smul_eq_mul,

Goal state after:
R : Type uR
M : Type uM
ι : Type uι
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : DecidableEq ι
b : Basis ι R M
f : ι →₀ R
i : ι
⊢ (∑ x ∈ f.support, f x * if i = x then 1 else 0) = f i
Module: Mathlib.LinearAlgebra.Dual.Basis

Goal state before:
pp: R : Type uR
M : Type uM
ι : Type uι
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : DecidableEq ι
b : Basis ι R M
x y : M
h : b.toDual x = b.toDual y
i : ι
⊢ (b.repr x) i = (b.repr y) i

Tactic:
by
  simp_rw [← toDual_eq_repr]; exact DFunLike.congr_fun h _

Goal state after:
Module: Mathlib.LinearAlgebra.Dual.Basis

Goal state before:
pp: R : Type uR
M : Type uM
ι : Type uι
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
b : Basis ι R M
inst✝ : Fintype ι
f : Dual R M
m : M
⊢ ∑ x, f (b x) • (b.coord x) m = f m

Tactic:
smul_eq_mul,

Goal state after:
R : Type uR
M : Type uM
ι : Type uι
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
b : Basis ι R M
inst✝ : Fintype ι
f : Dual R M
m : M
⊢ ∑ x, f (b x) * (b.coord x) m = f m
Module: Mathlib.LinearAlgebra.Dual.Basis

Goal state before:
pp: case intro
R : Type uR
M : Type uM
ι : Type uι
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq ι
b : Basis ι R M
inst✝ : Finite ι
f : ι →₀ R
i : ι
val✝ : Fintype ι
⊢ ∑ x, f x • (b.dualBasis x) (b i) = f i

Tactic:
smul_eq_mul,

Goal state after:
case intro
R : Type uR
M : Type uM
ι : Type uι
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq ι
b : Basis ι R M
inst✝ : Finite ι
f : ι →₀ R
i : ι
val✝ : Fintype ι
⊢ ∑ x, f x * (b.dualBasis x) (b i) = f i
Module: Mathlib.LinearAlgebra.FiniteDimensional.Lemmas

Goal state before:
pp: K : Type u
V : Type v
inst✝⁴ : DivisionRing K
inst✝³ : AddCommGroup V
inst✝² : Module K V
s t : Submodule K V
inst✝¹ : FiniteDimensional K ↥s
inst✝ : FiniteDimensional K ↥t
⊢ finrank K ↥(s ⊔ t) ≤ finrank K ↥s + finrank K ↥t

Tactic:
← finrank_sup_add_finrank_inf_eq

Goal state after:
K : Type u
V : Type v
inst✝⁴ : DivisionRing K
inst✝³ : AddCommGroup V
inst✝² : Module K V
s t : Submodule K V
inst✝¹ : FiniteDimensional K ↥s
inst✝ : FiniteDimensional K ↥t
⊢ finrank K ↥(s ⊔ t) ≤ finrank K ↥(s ⊔ t) + finrank K ↥(s ⊓ t)
Module: Mathlib.LinearAlgebra.FiniteDimensional.Lemmas

Goal state before:
pp: K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
f : V →ₗ[K] V₂
⊢ finrank K ↥f.range + finrank K ↥f.ker = finrank K V

Tactic:
rw [← f.quotKerEquivRange.finrank_eq]
  exact Submodule.finrank_quotient_add_finrank _

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: case mpr.inl
K : Type u_13
K₁ : Type u_14
K₂ : Type u_15
V : Type u_16
V₁ : Type u_17
V₂ : Type u_18
inst✝⁸ : Field K
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module K V
inst✝⁵ : Field K₁
inst✝⁴ : AddCommGroup V₁
inst✝³ : Module K₁ V₁
inst✝² : Field K₂
inst✝¹ : AddCommGroup V₂
inst✝ : Module K₂ V₂
I₁ : K₁ →+* K
I₂ : K₂ →+* K
B : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] V
x : V₁
y : V₂
a : K₁
ha : a ≠ 0
H : I₁ a = 0
⊢ (B x) y = 0

Tactic:
rw [map_eq_zero I₁] at H

Goal state after:
case mpr.inl
K : Type u_13
K₁ : Type u_14
K₂ : Type u_15
V : Type u_16
V₁ : Type u_17
V₂ : Type u_18
inst✝⁸ : Field K
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module K V
inst✝⁵ : Field K₁
inst✝⁴ : AddCommGroup V₁
inst✝³ : Module K₁ V₁
inst✝² : Field K₂
inst✝¹ : AddCommGroup V₂
inst✝ : Module K₂ V₂
I₁ : K₁ →+* K
I₂ : K₂ →+* K
B : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] V
x : V₁
y : V₂
a : K₁
ha : a ≠ 0
H : a = 0
⊢ (B x) y = 0
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
M : Type u_5
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
N : Type u_20
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
B : LinearMap.BilinMap R M N
⊢ (∀ (x y : M), (B x) y = (B y) x) ↔ B = flip B

Tactic:
by
  simp [LinearMap.ext_iff₂]

Goal state after:
Module: Mathlib.LinearAlgebra.Contraction

Goal state before:
pp: R : Type u_2
M : Type u_3
P : Type u_5
Q : Type u_6
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid P
inst✝⁵ : AddCommMonoid Q
inst✝⁴ : Module R M
inst✝³ : Module R P
inst✝² : Module R Q
inst✝¹ : Projective R M
inst✝ : Module.Finite R M
⊢ ↑(lTensorHomEquivHomLTensor R M P Q) = lTensorHomToHomLTensor (RingHom.id R) M P Q

Tactic:
by
  let e := congr (LinearEquiv.refl R P) (dualTensorHomEquiv R M Q)
  have h : Function.Surjective e.toLinearMap := e.surjective
  refine (cancel_right h).1 ?_
  ext f q m
  simp [e, lTensorHomEquivHomLTensor]

Goal state after:
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
this : (G.mapCocone (coequalizerCocone K)).ι.app WalkingParallelPair.one ≫ m = s.ι.app WalkingParallelPair.one
⊢ (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) ⋯

Tactic:
dsimp [coequalizerCocone] at this

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
this : G.map (𝟙 (K.obj WalkingParallelPair.one) ≫ pushoutInl K) ≫ m = s.ι.app WalkingParallelPair.one
⊢ (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.CategoryTheory.Limits.Preserves.Shapes.Equalizers

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
X Y Z : C
f g : X ⟶ Y
h : Y ⟶ Z
w : f ≫ h = g ≫ h
inst✝² : HasCoequalizer f g
inst✝¹ : HasCoequalizer (G.map f) (G.map g)
inst✝ : PreservesColimit (parallelPair f g) G
⊢ IsIso (coequalizerComparison f g G)

Tactic:
← PreservesCoequalizer.iso_hom

Goal state after:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
X Y Z : C
f g : X ⟶ Y
h : Y ⟶ Z
w : f ≫ h = g ≫ h
inst✝² : HasCoequalizer f g
inst✝¹ : HasCoequalizer (G.map f) (G.map g)
inst✝ : PreservesColimit (parallelPair f g) G
⊢ IsIso (PreservesCoequalizer.iso G f g).hom
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: n : Type u_19
R : Type u_20
M : Type u_21
M₁ : Type u_22
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid M₁
inst✝³ : Module R M
inst✝² : Module R M₁
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M₁
B : M →ₗ[R] M →ₗ[R] M₁
v : Basis n R M
hO : B.flip.IsOrthoᵢ ⇑v
h : ∀ (i : n), (B (v i)) (v i) ≠ 0
⊢ B.SeparatingRight

Tactic:
← flip_separatingLeft

Goal state after:
n : Type u_19
R : Type u_20
M : Type u_21
M₁ : Type u_22
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid M₁
inst✝³ : Module R M
inst✝² : Module R M₁
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M₁
B : M →ₗ[R] M →ₗ[R] M₁
v : Basis n R M
hO : B.flip.IsOrthoᵢ ⇑v
h : ∀ (i : n), (B (v i)) (v i) ≠ 0
⊢ B.flip.SeparatingLeft
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
⊢ f = g

Tactic:
have hf : f ≠ 0 := by aesop

Goal state after:
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
⊢ f = g
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: K : Type u_1
V₁ : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V₁
inst✝ : Module K V₁
W W' : Subspace K V₁
F : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := (mkQ W).prod (mkQ W')
this : F.ker = W ⊓ W'
⊢ dualAnnihilator (W ⊓ W') ≤ dualAnnihilator W ⊔ dualAnnihilator W'

Tactic:
← this,

Goal state after:
K : Type u_1
V₁ : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V₁
inst✝ : Module K V₁
W W' : Subspace K V₁
F : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := (mkQ W).prod (mkQ W')
this : F.ker = W ⊓ W'
⊢ F.ker.dualAnnihilator ≤ dualAnnihilator W ⊔ dualAnnihilator W'
Module: Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels

Goal state before:
pp: C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : HasZeroMorphisms C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
inst✝³ : HasZeroMorphisms D
G : C ⥤ D
inst✝² : G.PreservesZeroMorphisms
X Y Z : C
f : X ⟶ Y
h : Z ⟶ X
w : h ≫ f = 0
inst✝¹ : HasKernel f
inst✝ : PreservesLimit (parallelPair f 0) G
⊢ G.map (kernel.ι f) ≫ G.map f = G.map (kernel.ι f) ≫ 0

Tactic:
simp only [← G.map_comp, kernel.condition, comp_zero, Functor.map_zero]

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels

Goal state before:
pp: C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : HasZeroMorphisms C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : HasZeroMorphisms D
G : C ⥤ D
inst✝³ : G.PreservesZeroMorphisms
X Y Z : C
f : X ⟶ Y
h : Z ⟶ X
w : h ≫ f = 0
inst✝² : HasKernel f
inst✝¹ : HasKernel (G.map f)
inst✝ : PreservesLimit (parallelPair f 0) G
⊢ IsIso (kernelComparison f G)

Tactic:
← PreservesKernel.iso_hom

Goal state after:
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : HasZeroMorphisms C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : HasZeroMorphisms D
G : C ⥤ D
inst✝³ : G.PreservesZeroMorphisms
X Y Z : C
f : X ⟶ Y
h : Z ⟶ X
w : h ≫ f = 0
inst✝² : HasKernel f
inst✝¹ : HasKernel (G.map f)
inst✝ : PreservesLimit (parallelPair f 0) G
⊢ IsIso (PreservesKernel.iso G f).hom
Module: Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels

Goal state before:
pp: C : Type u₁
inst✝¹⁰ : Category.{v₁, u₁} C
inst✝⁹ : HasZeroMorphisms C
D : Type u₂
inst✝⁸ : Category.{v₂, u₂} D
inst✝⁷ : HasZeroMorphisms D
G : C ⥤ D
inst✝⁶ : G.PreservesZeroMorphisms
X Y : C
f : X ⟶ Y
inst✝⁵ : HasKernel f
inst✝⁴ : HasKernel (G.map f)
inst✝³ : PreservesLimit (parallelPair f 0) G
X' Y' : C
g : X' ⟶ Y'
inst✝² : HasKernel g
inst✝¹ : HasKernel (G.map g)
inst✝ : PreservesLimit (parallelPair g 0) G
p : X ⟶ X'
q : Y ⟶ Y'
hpq : f ≫ q = p ≫ g
⊢ kernel.map (G.map f) (G.map g) (G.map p) (G.map q) ⋯ ≫ (PreservesKernel.iso G g).inv =
    (PreservesKernel.iso G f).inv ≫ G.map (kernel.map f g p q hpq)

Tactic:
by
  rw [Iso.comp_inv_eq, Category.assoc, PreservesKernel.iso_hom, Iso.eq_inv_comp,
    PreservesKernel.iso_hom, kernelComparison_comp_kernel_map]

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
G : C ⥤ D
inst✝ : G.PreservesZeroMorphisms
c : Cocone (parallelPair 0 0)
hc : IsColimit c
this : IsIso (Cofork.π c)
⊢ IsColimit (CokernelCofork.map c G)

Tactic:
refine IsColimit.ofIsoColimit (CokernelCofork.IsColimit.ofId _ (G.map_zero _ _)) ?_

Goal state after:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
G : C ⥤ D
inst✝ : G.PreservesZeroMorphisms
c : Cocone (parallelPair 0 0)
hc : IsColimit c
this : IsIso (Cofork.π c)
⊢ CokernelCofork.ofπ (𝟙 (G.obj Y)) ⋯ ≅ CokernelCofork.map c G
Module: Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Basic

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X Y : C
inst✝ : HasZeroMorphisms C
P Q R S : C
f : P ⟶ Q
g : P ⟶ R
h : Q ⟶ S
k : R ⟶ S
gn : NormalEpi g
comm : f ≫ h = g ≫ k
t : IsColimit (PushoutCocone.mk h k comm)
hn : RegularEpi h := regularOfIsPushoutSndOfRegular (NormalEpi.regularEpi g) comm t
⊢ IsColimit (CokernelCofork.ofπ h ⋯)

Tactic:
have q := (@zero_comp _ _ _ gn.W _ _ f).symm

Goal state after:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X Y : C
inst✝ : HasZeroMorphisms C
P Q R S : C
f : P ⟶ Q
g : P ⟶ R
h : Q ⟶ S
k : R ⟶ S
gn : NormalEpi g
comm : f ≫ h = g ≫ k
t : IsColimit (PushoutCocone.mk h k comm)
hn : RegularEpi h := regularOfIsPushoutSndOfRegular (NormalEpi.regularEpi g) comm t
q : 0 = 0 ≫ f
⊢ IsColimit (CokernelCofork.ofπ h ⋯)
Module: Mathlib.Algebra.Category.Grp.EpiMono

Goal state before:
pp: A B : GrpCat
f : A ⟶ B
a : ↑A
x✝ : X'
⊢ ((Hom.hom (f ≫ ofHom g)) a) x✝ = ((Hom.hom (f ≫ ofHom h)) a) x✝

Tactic:
simp only [hom_comp, hom_ofHom, MonoidHom.coe_comp, Function.comp_apply]

Goal state after:
A B : GrpCat
f : A ⟶ B
a : ↑A
x✝ : X'
⊢ (g ((Hom.hom f) a)) x✝ = (h ((Hom.hom f) a)) x✝
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 : ShortComplex C
h : S.LeftHomologyData
hf : S.f = 0
⊢ h.f' ≫ 𝟙 h.K = 0

Tactic:
rw [← cancel_mono h.i, comp_id, f'_i, zero_comp, hf]

Goal state after:
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 : ShortComplex C
h : S.LeftHomologyData
hf : S.f = 0
⊢ IsIso h.π

Tactic:
by
  have ⟨φ, hφ⟩ := CokernelCofork.IsColimit.desc' h.hπ' (𝟙 _)
    (by rw [← cancel_mono h.i, comp_id, f'_i, zero_comp, hf])
  dsimp at hφ
  exact ⟨φ, hφ, by rw [← cancel_epi h.π, reassoc_of% hφ, comp_id]⟩

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
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
⊢ S₁.leftHomologyIso.hom ≫ homologyMap φ = leftHomologyMap φ ≫ S₂.leftHomologyIso.hom

Tactic:
simpa only [LeftHomologyData.homologyIso_leftHomologyData, Iso.symm_inv] using!
    LeftHomologyData.leftHomologyIso_inv_naturality φ S₁.leftHomologyData S₂.leftHomologyData

Goal state after:
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
⊢ S.rightHomologyIso.hom ≫ S.homologyι = S.rightHomologyι

Tactic:
dsimp only [homologyι]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
inst✝ : S.HasHomology
⊢ S.rightHomologyIso.hom ≫ S.rightHomologyIso.inv ≫ S.rightHomologyι = S.rightHomologyι
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
h : S.RightHomologyData
inst✝ : S.HasRightHomology
⊢ S.pOpcycles ≫ h.opcyclesIso.hom = h.p

Tactic:
by
  simp only [← h.p_comp_opcyclesIso_inv, assoc, Iso.inv_hom_id, comp_id]

Goal state after:
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
⊢ S.homologyι ≫ S.fromOpcycles = 0

Tactic:
dsimp only [homologyι]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
inst✝ : S.HasHomology
⊢ (S.rightHomologyIso.inv ≫ S.rightHomologyι) ≫ S.fromOpcycles = 0
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.HasLeftHomology
⊢ S.op.pOpcycles ≫ S.leftHomologyData.op.opcyclesIso.hom ≫ S.leftHomologyData.f'.op = S.op.pOpcycles ≫ S.op.fromOpcycles

Tactic:
p_fromOpcycles,

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
inst✝ : S.HasLeftHomology
⊢ S.op.pOpcycles ≫ S.leftHomologyData.op.opcyclesIso.hom ≫ S.leftHomologyData.f'.op = S.op.g
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
A : C
inst✝ : S.HasHomology
k : S.X₂ ⟶ A
x : S.X₃ ⟶ A
hx : k = S.g ≫ x
⊢ S.homologyι ≫ S.descOpcycles k ⋯ = 0

Tactic:
dsimp only [homologyι]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
A : C
inst✝ : S.HasHomology
k : S.X₂ ⟶ A
x : S.X₃ ⟶ A
hx : k = S.g ≫ x
⊢ (S.rightHomologyIso.inv ≫ S.rightHomologyι) ≫ S.descOpcycles k ⋯ = 0
Module: Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
S S₁ S₂ : ShortComplex C
h : S.LeftHomologyData
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : h.IsPreservedBy F
this✝ : PreservesLimit (parallelPair S.g 0) F
this : PreservesColimit (parallelPair h.f' 0) F
wi : F.map h.i ≫ F.map S.g = 0
⊢ (S.map F).LeftHomologyData

Tactic:
have hi := KernelFork.mapIsLimit _ h.hi F

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
S S₁ S₂ : ShortComplex C
h : S.LeftHomologyData
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : h.IsPreservedBy F
this✝ : PreservesLimit (parallelPair S.g 0) F
this : PreservesColimit (parallelPair h.f' 0) F
wi : F.map h.i ≫ F.map S.g = 0
hi : IsLimit ((KernelFork.ofι h.i ⋯).map F)
⊢ (S.map F).LeftHomologyData
Module: Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
S S₁ S₂ : ShortComplex C
h : S.LeftHomologyData
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : h.IsPreservedBy F
this✝ : PreservesLimit (parallelPair S.g 0) F
this : PreservesColimit (parallelPair h.f' 0) F
wi : F.map h.i ≫ F.map S.g = 0
hi : IsLimit ((KernelFork.ofι h.i ⋯).map F)
f' : F.obj S.X₁ ⟶ F.obj h.K := hi.lift (KernelFork.ofι (S.map F).f ⋯)
hf' : f' = F.map h.f'
⊢ f' ≫ F.map h.π = 0

Tactic:
rw [hf', ← F.map_comp, f'_π, F.map_zero]

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

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 : S₁.RightHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
this : Epi (opMap φ).τ₁
⊢ S₂.RightHomologyData

Tactic:
haveI : IsIso (opMap φ).τ₂ := by dsimp; infer_instance

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 : S₁.RightHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
this✝ : Epi (opMap φ).τ₁
this : IsIso (opMap φ).τ₂
⊢ S₂.RightHomologyData
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

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.RightHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
this : Epi (opMap φ).τ₁
⊢ S₁.RightHomologyData

Tactic:
haveI : IsIso (opMap φ).τ₂ := by dsimp; infer_instance

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.RightHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
this✝ : Epi (opMap φ).τ₁
this : IsIso (opMap φ).τ₂
⊢ S₁.RightHomologyData
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
hg : S.g = 0
inst✝ : S.HasHomology
this : IsIso S.rightHomologyι
⊢ IsIso S.homologyι

Tactic:
dsimp only [homologyι]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
hg : S.g = 0
inst✝ : S.HasHomology
this : IsIso S.rightHomologyι
⊢ IsIso (S.rightHomologyIso.inv ≫ S.rightHomologyι)
Module: Mathlib.CategoryTheory.Preadditive.Injective.Basic

Goal state before:
pp: 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
z : Y✝
⊢ X

Tactic:
classical
      exact
          if h : z ∈ Set.range f then g (Classical.choose h) else Nonempty.some inferInstance

Goal state after: