mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: case neg
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
hp : ∀ (x : M), x ≠ 0 → 0 < (B x) x
x y : M
he : (B x) y * (B y) x = (B x) x * (B y) y
hz : (B x) y • x - (B x) x • y = 0
hL : LinearIndependent R ![x, y]
hx : ¬x = 0
⊢ False

Tactic:
have h := sub_eq_zero.mpr (sub_eq_zero.mp hz).symm

Goal state after:
case neg
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
hp : ∀ (x : M), x ≠ 0 → 0 < (B x) x
x y : M
he : (B x) y * (B y) x = (B x) x * (B y) y
hz : (B x) y • x - (B x) x • y = 0
hL : LinearIndependent R ![x, y]
hx : ¬x = 0
h : (B x) x • y - (B x) y • x = 0
⊢ False
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'
a : Dual K (V₁ ⧸ W)
b : Dual K (V₁ ⧸ W')
⊢ ∃ y ∈ dualAnnihilator W,
    ∃ z ∈ dualAnnihilator W', y + z = F.dualMap ((dualProdDualEquivDual K (V₁ ⧸ W) (V₁ ⧸ W')) (a, b))

Tactic:
obtain ⟨a', rfl⟩ := (dualQuotEquivDualAnnihilator W).symm.surjective a

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'
b : Dual K (V₁ ⧸ W')
a' : ↥(dualAnnihilator W)
⊢ ∃ y ∈ dualAnnihilator W,
    ∃ z ∈ dualAnnihilator W',
      y + z = F.dualMap ((dualProdDualEquivDual K (V₁ ⧸ W) (V₁ ⧸ W')) ((dualQuotEquivDualAnnihilator W).symm a', b))
Module: Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Basic

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
X✝ Y✝ : C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₁, u₂} D
inst✝¹ : HasZeroMorphisms D
F : C ⥤ D
inst✝ : F.IsEquivalence
X Y : C
f : X ⟶ Y
hf : NormalMono (F.map f)
⊢ (Iso.refl (KernelFork.ofι (Fork.ι (KernelFork.ofι (F.map f) ⋯)) ⋯).pt).hom ≫
      Fork.ι ((Cone.postcomposeEquivalence (compNatIso F)).functor.obj (F.mapCone (KernelFork.ofι f ⋯))) =
    Fork.ι (KernelFork.ofι (Fork.ι (KernelFork.ofι (F.map f) ⋯)) ⋯)

Tactic:
simp [compNatIso, Fork.ι]

Goal state after:
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']
  exact IsIso.comp_isIso

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

Goal state before:

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

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
X Y Z : C
f : X ⟶ Y
g : X ⟶ Z
inst✝ : Mono f
s : PushoutCocone f g
hs : IsColimit s
this : Mono ((colimit.cocone (span f g)).ι.app WalkingCospan.right)
⊢ Mono s.inr

Tactic:
apply
    mono_of_mono_fac (IsColimit.comp_coconePointUniqueUpToIso_hom hs (colimit.isColimit _) _)

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
X Y Z : C
f : X ⟶ Y
g : X ⟶ Z
inst✝ : Mono g
s : PushoutCocone f g
hs : IsColimit s
this : Mono ((colimit.cocone (span f g)).ι.app WalkingCospan.left)
⊢ Mono s.inl

Tactic:
apply
    mono_of_mono_fac (IsColimit.comp_coconePointUniqueUpToIso_hom hs (colimit.isColimit _) _)

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 S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ LeftHomologyMapData φ h₁ h₂

Tactic:
let φK : h₁.K ⟶ h₂.K := h₂.liftK (h₁.i ≫ φ.τ₂)
    (by rw [assoc, φ.comm₂₃, h₁.wi_assoc, zero_comp])

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.LeftHomologyData
h₂ : S₂.LeftHomologyData
φK : h₁.K ⟶ h₂.K := h₂.liftK (h₁.i ≫ φ.τ₂) ⋯
⊢ LeftHomologyMapData 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 S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
⊢ (-φ₁).τ₂ = S₁.g ≫ (-h.h₂) + (-h.h₁) ≫ S₂.f + (-φ₂).τ₂

Tactic:
neg_τ₂,

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
⊢ -φ₁.τ₂ = S₁.g ≫ (-h.h₂) + (-h.h₁) ≫ S₂.f + (-φ₂).τ₂
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
⊢ -φ₁.τ₂ = S₁.g ≫ (-h.h₂) + (-h.h₁) ≫ S₂.f + (-φ₂).τ₂

Tactic:
neg_τ₂,

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
⊢ -φ₁.τ₂ = S₁.g ≫ (-h.h₂) + (-h.h₁) ≫ S₂.f + -φ₂.τ₂
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
ψ : S₃ ⟶ S₁
⊢ S₃.g ≫ ψ.τ₃ ≫ h.h₃ = 0

Tactic:
rw [← ψ.comm₂₃_assoc, h.g_h₃, comp_zero]

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
⊢ (opMap φ₁).τ₃.unop = (h.h₀.op + h.h₁.op ≫ S₁.op.g + (opMap φ₂).τ₃).unop

Tactic:
by dsimp; rw [h.comm₁]; abel

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✝
S₁ S₂ : ShortComplex Cᵒᵖ
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
⊢ (unopMap φ₁).τ₃.op = (h.h₀.unop + h.h₁.unop ≫ S₁.unop.g + (unopMap φ₂).τ₃).op

Tactic:
by dsimp; rw [h.comm₁]; abel

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

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.ShortExact
⊢ IsLimit (KernelFork.ofι S.f ⋯)

Tactic:
by
  have := hS.mono_f
  exact hS.exact.fIsKernel

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
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : F.PreservesLeftHomologyOf S
inst✝ : (S.map F).HasHomology
this : S.HasHomology
h : 𝟙 S.leftHomologyData.H = 0
⊢ IsZero (F.obj S.leftHomologyData.H)

Tactic:
IsZero.iff_id_eq_zero,

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 : 𝟙 S.leftHomologyData.H = 0
⊢ 𝟙 (F.obj S.leftHomologyData.H) = 0
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
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : F.PreservesRightHomologyOf S
inst✝ : (S.map F).HasHomology
this : S.HasHomology
h : 𝟙 S.rightHomologyData.H = 0
⊢ IsZero (F.obj S.rightHomologyData.H)

Tactic:
IsZero.iff_id_eq_zero,

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.PreservesRightHomologyOf S
inst✝ : (S.map F).HasHomology
this : S.HasHomology
h : 𝟙 S.rightHomologyData.H = 0
⊢ 𝟙 (F.obj S.rightHomologyData.H) = 0
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 : IsZero (S.leftHomologyData.map F).H
⊢ 𝟙 S.leftHomologyData.H = 0

Tactic:
IsZero.iff_id_eq_zero,

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 : 𝟙 (S.leftHomologyData.map F).H = 0
⊢ 𝟙 S.leftHomologyData.H = 0
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
i : D ⥤ C
inst✝¹ : Reflective i
A : C
inst✝ : IsSplitMono ((reflectorAdjunction i).unit.app A)
η : 𝟭 C ⟶ reflector i ⋙ i := (reflectorAdjunction i).unit
this✝¹ : IsIso (η.app (i.obj ((reflector i).obj A)))
this✝ : Epi (η.app A)
this : IsIso (η.app A)
⊢ i.essImage A

Tactic:
exact (reflectorAdjunction i).mem_essImage_of_unit_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
hS : S.Exact
hf : S.f = 0
hg : S.g = 0
this : Mono S.g
⊢ IsZero S.X₂

Tactic:
IsZero.iff_id_eq_zero,

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

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X : C
p : Subobject X → Prop
h : ∀ ⦃A : C⦄ (f : A ⟶ X) [inst : Mono f], p (mk f)
P : Subobject X
⊢ p P

Tactic:
induction P using Quotient.inductionOn' with | _ a

Goal state after:
case h
C : Type u₁
inst✝ : Category.{v₁, u₁} C
X : C
p : Subobject X → Prop
h : ∀ ⦃A : C⦄ (f : A ⟶ X) [inst : Mono f], p (mk f)
a : MonoOver X
⊢ p (Quotient.mk'' a)
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

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 : Preadditive C
inst✝² : Preadditive D
S : ShortComplex C
inst✝¹ : Balanced C
hS : S.Exact
inst✝ : Epi S.g
this : S.HasHomology
⊢ IsColimit (CokernelCofork.ofπ S.g ⋯)

Tactic:
have := hS.isIso_fromOpcycles

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 : Preadditive C
inst✝² : Preadditive D
S : ShortComplex C
inst✝¹ : Balanced C
hS : S.Exact
inst✝ : Epi S.g
this✝ : S.HasHomology
this : IsIso S.fromOpcycles
⊢ IsColimit (CokernelCofork.ofπ S.g ⋯)
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

Goal state before:
pp: case mp
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hf₁ : S₁.f = 0
hg₁ : S₁.g = 0
hf₂ : S₂.f = 0
w : φ.τ₂ ≫ S₂.g = 0
h : IsIso (S₂.liftCycles φ.τ₂ ⋯)
this : Mono φ.τ₂
⊢ { X₁ := S₁.X₂, X₂ := S₂.X₂, X₃ := S₂.X₃, f := φ.τ₂, g := S₂.g, zero := ⋯ }.Exact

Tactic:
apply exact_of_f_is_kernel

Goal state after:
case mp
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
w : φ.τ₂ ≫ S₂.g = 0
h : IsIso (S₂.liftCycles φ.τ₂ ⋯)
this : Mono φ.τ₂
⊢ IsLimit (KernelFork.ofι { X₁ := S₁.X₂, X₂ := S₂.X₂, X₃ := S₂.X₃, f := φ.τ₂, g := S₂.g, zero := ⋯ }.f ⋯)
Module: Mathlib.CategoryTheory.Subobject.Limits

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
X Y W : C
f : X ⟶ Y
inst✝ : HasPullbacks C
y : Subobject Y
h : W ⟶ X
hf : ((Subobject.pullback f).obj y).Factors h
⊢ y.Factors (h ≫ f)

Tactic:
factors_iff

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
X Y W : C
f : X ⟶ Y
inst✝ : HasPullbacks C
y : Subobject Y
h : W ⟶ X
hf : ((Subobject.pullback f).obj y).Factors h
⊢ (representative.obj y).Factors (h ≫ f)
Module: Mathlib.Algebra.Homology.ShortComplex.ExactFunctor

Goal state before:
pp: 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 : Preadditive C
inst✝⁵ : Preadditive D
F : C ⥤ D
inst✝⁴ : F.Additive
inst✝³ : F.PreservesHomology
inst✝² : HasZeroObject C
inst✝¹ : HasFiniteProducts C
inst✝ : HasKernels C
this✝¹ : ∀ {X Y : C} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) F
this✝ : HasBinaryBiproducts C
this : HasEqualizers C
⊢ IsZero (F.obj 0)

Tactic:
IsZero.iff_id_eq_zero,

Goal state after:
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 : Preadditive C
inst✝⁵ : Preadditive D
F : C ⥤ D
inst✝⁴ : F.Additive
inst✝³ : F.PreservesHomology
inst✝² : HasZeroObject C
inst✝¹ : HasFiniteProducts C
inst✝ : HasKernels C
this✝¹ : ∀ {X Y : C} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) F
this✝ : HasBinaryBiproducts C
this : HasEqualizers C
⊢ 𝟙 (F.obj 0) = 0
Module: Mathlib.Algebra.Homology.ShortComplex.ExactFunctor

Goal state before:
pp: 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 : Preadditive C
inst✝⁵ : Preadditive D
F : C ⥤ D
inst✝⁴ : F.Additive
inst✝³ : F.PreservesHomology
inst✝² : HasZeroObject C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasCokernels C
this✝¹ : ∀ {X Y : C} (f : X ⟶ Y), PreservesColimit (parallelPair f 0) F
this✝ : HasBinaryBiproducts C
this : HasCoequalizers C
⊢ IsZero (F.obj 0)

Tactic:
IsZero.iff_id_eq_zero,

Goal state after:
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 : Preadditive C
inst✝⁵ : Preadditive D
F : C ⥤ D
inst✝⁴ : F.Additive
inst✝³ : F.PreservesHomology
inst✝² : HasZeroObject C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasCokernels C
this✝¹ : ∀ {X Y : C} (f : X ⟶ Y), PreservesColimit (parallelPair f 0) F
this✝ : HasBinaryBiproducts C
this : HasCoequalizers C
⊢ 𝟙 (F.obj 0) = 0
Module: Mathlib.CategoryTheory.Limits.Constructions.Filtered

Goal state before:
pp: 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
⊢ (Sigma.ι (fun x ↦ f (↑x).as) j ≫ colimit.ι (liftToFinsetObj (Discrete.functor f)) S) ≫
      ((liftToFinsetColimitCocone (Discrete.functor f)).isColimit.coconePointUniqueUpToIso
          (colimit.isColimit (Discrete.functor f))).hom =
    Sigma.ι (fun x ↦ f (↑x).as) j ≫ Sigma.desc fun s ↦ Sigma.ι f (↑s).as

Tactic:
convert!
      IsColimit.comp_coconePointUniqueUpToIso_hom
        (liftToFinsetColimitCocone (Discrete.functor f)).isColimit (colimit.isColimit _) j

Goal state after:
case e'_2.e'_6
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)
e_3✝ : f (↑j).as = (Discrete.functor f).obj ↑j
e_4✝ : colimit (liftToFinsetObj (Discrete.functor f)) = (liftToFinsetColimitCocone (Discrete.functor f)).cocone.pt
⊢ Sigma.ι (fun x ↦ f (↑x).as) j ≫ colimit.ι (liftToFinsetObj (Discrete.functor f)) S =
    (liftToFinsetColimitCocone (Discrete.functor f)).cocone.ι.app ↑j

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
Module: Mathlib.CategoryTheory.Functor.KanExtension.Pointwise

Goal state before:
pp: C : Type u_1
D : Type u_2
D' : Type u_3
H : Type u_4
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Category.{v_3, u_3} D'
inst✝¹ : Category.{v_4, u_4} H
L : C ⥤ D
L' : C ⥤ D'
F : C ⥤ H
inst✝ : L.HasPointwiseLeftKanExtension F
Y : D
j : CostructuredArrow L Y
⊢ colimit.ι (CostructuredArrow.proj L Y ⋙ F) j ≫
      colimit.desc (CostructuredArrow.proj L Y ⋙ F)
        { pt := colimit (CostructuredArrow.proj L Y ⋙ F),
          ι :=
            { app := fun g ↦ colimit.ι (CostructuredArrow.proj L Y ⋙ F) ((CostructuredArrow.map (𝟙 Y)).obj g),
              naturality := ⋯ } } =
    colimit.ι (CostructuredArrow.proj L Y ⋙ F) j ≫ 𝟙 (colimit (CostructuredArrow.proj L Y ⋙ F))

Tactic:
by
    dsimp
    simp only [colimit.ι_desc, comp_id]
    congr
    apply CostructuredArrow.map_id

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

Goal state before:
pp: C : Type u_1
C' : Type u_2
H : Type u_3
D : Type u_4
D' : Type u_5
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
inst✝ : Category.{v_5, u_5} D'
L : C ⥤ D
L' : D ⥤ D'
F₀ : C ⥤ H
F₁ : D ⥤ H
F₂ : D' ⥤ H
α : F₀ ⟶ L ⋙ F₁
hα : StructuredArrow.IsUniversal (mk F₁ α)
b : L'.LeftExtension F₁
hb : StructuredArrow.IsUniversal ((precomp₂ L' α).obj b)
y : L'.LeftExtension F₁
u : (precomp₂ L' α).obj b ⟶ (precomp₂ L' α).obj y := IsInitial.to hb ((precomp₂ L' α).obj y)
⊢ Unique (b ⟶ y)

Tactic:
refine
      ⟨⟨StructuredArrow.homMk u.right <| by
          apply hα.hom_ext
          ext t
          have := congr_app u.w t
          dsimp at this
          simp only [precomp₂_obj_hom_app, assoc] at this
          simp [this]⟩, fun a => ?_⟩

Goal state after:
C : Type u_1
C' : Type u_2
H : Type u_3
D : Type u_4
D' : Type u_5
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
inst✝ : Category.{v_5, u_5} D'
L : C ⥤ D
L' : D ⥤ D'
F₀ : C ⥤ H
F₁ : D ⥤ H
F₂ : D' ⥤ H
α : F₀ ⟶ L ⋙ F₁
hα : StructuredArrow.IsUniversal (mk F₁ α)
b : L'.LeftExtension F₁
hb : StructuredArrow.IsUniversal ((precomp₂ L' α).obj b)
y : L'.LeftExtension F₁
u : (precomp₂ L' α).obj b ⟶ (precomp₂ L' α).obj y := IsInitial.to hb ((precomp₂ L' α).obj y)
a : b ⟶ y
⊢ a = default
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.Basic

Goal state before:
pp: case refine_1
C : Type u
inst✝⁴ : Category.{v, u} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
inst✝² : Abelian C
inst✝¹ : Abelian D
inst✝ : IsGrothendieckAbelian.{w, v, u} C
α : C ≌ D
hasFilteredColimits : HasFilteredColimitsOfSize.{w, w, v₂, u₂} D
⊢ LocallySmall.{w, v₂, u₂} D

Tactic:
exact locallySmall_of_faithful α.inverse

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

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : LocallySmall.{w, v₁, u₁} Cᵒᵖ
inst✝² : WellPowered.{w, v₁, u₁} Cᵒᵖ
inst✝¹ : HasColimitsOfSize.{w, w, v₁, u₁} C
P : ObjectProperty C
inst✝ : ObjectProperty.Small.{w, v₁, u₁} P
hP : P.IsSeparating
this : HasInitial Cᵒᵖ
⊢ HasTerminal C

Tactic:
exact hasTerminal_of_hasInitial_op

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

Goal state before:
pp: X'✝ X✝ : Type u
Y✝ : AddCommGrpCat
f✝ : X'✝ ⟶ X✝
g✝ : X✝ ⟶ (forget AddCommGrpCat).obj Y✝
x✝ : ↑(free.obj X'✝)
⊢ (FreeAbelianGroup.lift ⇑(ConcreteCategory.hom (f✝ ≫ g✝))) x✝ =
    (FreeAbelianGroup.lift ⇑(ConcreteCategory.hom g✝)) ((FreeAbelianGroup.map ⇑(ConcreteCategory.hom f✝)) x✝)

Tactic:
rw [← FreeAbelianGroup.lift_comp]

Goal state after:
X'✝ X✝ : Type u
Y✝ : AddCommGrpCat
f✝ : X'✝ ⟶ X✝
g✝ : X✝ ⟶ (forget AddCommGrpCat).obj Y✝
x✝ : ↑(free.obj X'✝)
⊢ (FreeAbelianGroup.lift ⇑(ConcreteCategory.hom (f✝ ≫ g✝))) x✝ =
    (FreeAbelianGroup.lift (⇑(ConcreteCategory.hom g✝) ∘ ⇑(ConcreteCategory.hom f✝))) x✝
Module: Mathlib.CategoryTheory.Limits.Types.Coproducts

Goal state before:
pp: C : Type u
F : C → Type v
c : CofanTypes F
⊢ Functor.CoconeTypes.IsColimit c ↔ Function.Bijective (fromSigma F c)

Tactic:
rw [(isColimit_sigma F).iff_bijective]
  aesop

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

Goal state before:
pp: α : Type u_1
β : Type u_2
f : α → β
c x : α
inst✝¹ : AddGroup α
inst✝ : SubtractionMonoid β
h : Antiperiodic f c
n : ℤ
⊢ f (x - n • c) = ↑n.negOnePow • f x

Tactic:
by
  simpa only [sub_eq_add_neg, neg_zsmul, Int.negOnePow_neg] using h.add_zsmul_eq (-n)

Goal state after:
Module: Mathlib.Data.Nat.Factorization.Induction

Goal state before:
pp: case convert_1
a b m n✝ p✝ : ℕ
motive : ℕ → Sort u_1
zero : motive 0
one : motive 1
prime_pow_mul : (a p n : ℕ) → Prime p → ¬p ∣ a → 0 < n → motive a → motive (p ^ n * a)
n k : ℕ
hk : (m : ℕ) → m < k + 2 → motive m
p : ℕ := (k + 2).minFac
hp : Prime p
t : ℕ := (k + 2).factorization p
hpt : p ^ t ∣ k + 2
htp : 0 < t
⊢ ¬p ^ t.succ ∣ k + 2

Tactic:
exact pow_succ_factorization_not_dvd (k + 1).succ_ne_zero hp

Goal state after:
Module: Mathlib.Data.Nat.Count

Goal state before:
pp: p : ℕ → Prop
inst✝ : DecidablePred p
hp : (Set.ofPred p).Finite
n : ℕ
⊢ #({x ∈ range n | p x}) ≤ #hp.toFinset

Tactic:
exact Finset.card_mono fun x hx ↦ hp.mem_toFinset.2 (mem_filter.1 hx).2

Goal state after:
Module: Mathlib.Data.Nat.Factorization.Induction

Goal state before:
pp: case coprime
β : Type u_1
inst✝ : CommMonoid β
f : ℕ → β
h_mult : ∀ (x y : ℕ), x.Coprime y → f (x * y) = f x * f y
hf : f 1 = 1
a b : ℕ
a✝¹ : 1 < a
a✝ : 1 < b
hab : a.Coprime b
ha : a ≠ 0 → f a = a.factorization.prod fun p k ↦ f (p ^ k)
hb : b ≠ 0 → f b = b.factorization.prod fun p k ↦ f (p ^ k)
hab_pos : a * b ≠ 0
⊢ ((a.factorization.prod fun p k ↦ f (p ^ k)) * b.factorization.prod fun p k ↦ f (p ^ k)) =
    (a.factorization + b.factorization).prod fun p k ↦ f (p ^ k)

Tactic:
← prod_add_index_of_disjoint

Goal state after:
case coprime
β : Type u_1
inst✝ : CommMonoid β
f : ℕ → β
h_mult : ∀ (x y : ℕ), x.Coprime y → f (x * y) = f x * f y
hf : f 1 = 1
a b : ℕ
a✝¹ : 1 < a
a✝ : 1 < b
hab : a.Coprime b
ha : a ≠ 0 → f a = a.factorization.prod fun p k ↦ f (p ^ k)
hb : b ≠ 0 → f b = b.factorization.prod fun p k ↦ f (p ^ k)
hab_pos : a * b ≠ 0
⊢ ((a.factorization + b.factorization).prod fun p k ↦ f (p ^ k)) =
    (a.factorization + b.factorization).prod fun p k ↦ f (p ^ k)

case coprime.hd
β : Type u_1
inst✝ : CommMonoid β
f : ℕ → β
h_mult : ∀ (x y : ℕ), x.Coprime y → f (x * y) = f x * f y
hf : f 1 = 1
a b : ℕ
a✝¹ : 1 < a
a✝ : 1 < b
hab : a.Coprime b
ha : a ≠ 0 → f a = a.factorization.prod fun p k ↦ f (p ^ k)
hb : b ≠ 0 → f b = b.factorization.prod fun p k ↦ f (p ^ k)
hab_pos : a * b ≠ 0
⊢ _root_.Disjoint a.factorization.support b.factorization.support
Module: Mathlib.Data.Nat.Factorization.Induction

Goal state before:
pp: β : Type u_1
inst✝ : CommMonoid β
f : ℕ → β
h_mult : ∀ (x y : ℕ), x.Coprime y → f (x * y) = f x * f y
hf : f 1 = 1
⊢ ∀ {n : ℕ}, n ≠ 0 → f n = n.factorization.prod fun p k ↦ f (p ^ k)

Tactic:
apply Nat.recOnPosPrimePosCoprime
  · rintro p k hp - -
    simp [Prime.factorization_pow hp, Finsupp.prod_single_index _, hf]
  · simp
  · rintro -
    rw [factorization_one, hf]
    simp
  · intro a b _ _ hab ha hb hab_pos
    rw [h_mult a b hab, ha (left_ne_zero_of_mul hab_pos), hb (right_ne_zero_of_mul hab_pos),
      factorization_mul_of_coprime hab, ← prod_add_index_of_disjoint]
    exact hab.disjoint_primeFactors

Goal state after: