mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hi : IsLimit (KernelFork.ofι (X.kernelSequenceCycles f₁ f₂ n₁ n₂ hn₂).f ⋯) := ⋯.fIsKernel
this : hi.lift (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯).f ⋯) = X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯
⊢ (parallelPair (hi.lift (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f ⋯)) 0).map
        WalkingParallelPairHom.right ≫
      (Iso.refl
          ((parallelPair (hi.lift (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f ⋯)) 0).obj
            WalkingParallelPair.one)).hom =
    (Iso.refl
          ((parallelPair (hi.lift (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f ⋯)) 0).obj
            WalkingParallelPair.zero)).hom ≫
      (parallelPair (X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯) 0).map WalkingParallelPairHom.right

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ (Iso.refl (X.cokernelSequenceCyclesE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).X₁).hom ≫
      { X₁ := (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).X₁, X₂ := (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).cycles,
          X₃ := (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).homology,
          f := (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).toCycles,
          g := (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).homologyπ, zero := ⋯ }.f =
    (X.cokernelSequenceCyclesE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f ≫ (X.cyclesIso f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯).symm.hom

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hp : IsColimit (CokernelCofork.ofπ (X.cokernelSequenceOpcycles f₂ f₃ n₀ n₁ hn₁).g ⋯) := ⋯.gIsCokernel
this : hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯) = X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂
⊢ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f ≫ X.pOpcycles f₂ f₃ n₁ = 0

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hp : IsColimit (CokernelCofork.ofπ (X.cokernelSequenceOpcycles f₂ f₃ n₀ n₁ hn₁).g ⋯) := ⋯.gIsCokernel
this : hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯) = X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂
⊢ (parallelPair (hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯)) 0).map
        WalkingParallelPairHom.right ≫
      (Iso.refl
          ((parallelPair (hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯)) 0).obj
            WalkingParallelPair.one)).hom =
    (Iso.refl
          ((parallelPair (hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯)) 0).obj
            WalkingParallelPair.zero)).hom ≫
      (parallelPair (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯) 0).map WalkingParallelPairHom.right

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ (X.opcyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).hom ≫ (X.kernelSequenceOpcyclesE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g =
    { X₁ := (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).homology,
          X₂ := (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).opcycles,
          X₃ := (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).X₃,
          f := (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).homologyι,
          g := (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).fromOpcycles, zero := ⋯ }.g ≫
      (Iso.refl
          { X₁ := (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).homology,
              X₂ := (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).opcycles,
              X₃ := (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).X₃,
              f := (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).homologyι,
              g := (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).fromOpcycles, zero := ⋯ }.X₃).hom

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
pq : ℤ × ℤ
⊢ WithBotTop.coe pq.2 ≤ WithBotTop.coe (pq.2 + 1)

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
pq : ℕ × ℕ
⊢ WithBotTop.coe ↑pq.2 ≤ WithBotTop.coe (↑pq.2 + 1)

Tactic:
by simp

Goal state after:
Module: Mathlib.CategoryTheory.Square

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
X✝ Y✝ : (Square C)ᵒᵖ
φ : X✝ ⟶ Y✝
⊢ ((Opposite.unop X✝).op.f₁₂ ≫ φ.unop.τ₂.op).unop = (φ.unop.τ₄.op ≫ (Opposite.unop Y✝).op.f₁₂).unop

Tactic:
by simp

Goal state after:
Module: Mathlib.CategoryTheory.Square

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
X✝ Y✝ : (Square C)ᵒᵖ
φ : X✝ ⟶ Y✝
⊢ ((Opposite.unop X✝).op.f₁₃ ≫ φ.unop.τ₃.op).unop = (φ.unop.τ₄.op ≫ (Opposite.unop Y✝).op.f₁₃).unop

Tactic:
by simp

Goal state after:
Module: Mathlib.CategoryTheory.Square

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
X✝ Y✝ : (Square C)ᵒᵖ
φ : X✝ ⟶ Y✝
⊢ ((Opposite.unop X✝).op.f₂₄ ≫ φ.unop.τ₁.op).unop = (φ.unop.τ₂.op ≫ (Opposite.unop Y✝).op.f₂₄).unop

Tactic:
by simp

Goal state after:
Module: Mathlib.CategoryTheory.Square

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
X✝ Y✝ : (Square C)ᵒᵖ
φ : X✝ ⟶ Y✝
⊢ ((Opposite.unop X✝).op.f₃₄ ≫ φ.unop.τ₁.op).unop = (φ.unop.τ₃.op ≫ (Opposite.unop Y✝).op.f₃₄).unop

Tactic:
by simp

Goal state after:
Module: Mathlib.CategoryTheory.Square

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
X✝ Y✝ : (Square Cᵒᵖ)ᵒᵖ
φ : X✝ ⟶ Y✝
⊢ ((Opposite.unop X✝).unop.f₁₂ ≫ φ.unop.τ₂.unop).op = (φ.unop.τ₄.unop ≫ (Opposite.unop Y✝).unop.f₁₂).op

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
A : C
x₂ : A ⟶ (X.H n₁).obj (mk₁ f₂₃)
hx₂ : x₂ ≫ biprod.lift ((X.H n₁).map (twoδ₁Toδ₀ f₂ f₃ f₂₃ h₂₃)) (X.δ f₁ f₂₃ n₁ n₂ ⋯) = 0
⊢ X.liftOpcycles f₂ f₃ f₂₃ h₂₃ x₂ ⋯ ≫ X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂ = 0

Tactic:
rw [← X.fromOpcyles_δ f₁ f₂ f₃ f₂₃ h₂₃ n₁ n₂,
          X.liftOpcycles_fromOpcycles_assoc]

Goal state after:
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
A : C
x₂ : A ⟶ (X.H n₁).obj (mk₁ f₂₃)
hx₂ : x₂ ≫ biprod.lift ((X.H n₁).map (twoδ₁Toδ₀ f₂ f₃ f₂₃ h₂₃)) (X.δ f₁ f₂₃ n₁ n₂ ⋯) = 0
⊢ x₂ ≫ X.δ f₁ f₂₃ n₁ n₂ ⋯ = 0
Module: Mathlib.CategoryTheory.Square

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
X✝ Y✝ : (Square Cᵒᵖ)ᵒᵖ
φ : X✝ ⟶ Y✝
⊢ ((Opposite.unop X✝).unop.f₁₃ ≫ φ.unop.τ₃.unop).op = (φ.unop.τ₄.unop ≫ (Opposite.unop Y✝).unop.f₁₃).op

Tactic:
by simp

Goal state after:
Module: Mathlib.CategoryTheory.Square

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
X✝ Y✝ : (Square Cᵒᵖ)ᵒᵖ
φ : X✝ ⟶ Y✝
⊢ ((Opposite.unop X✝).unop.f₂₄ ≫ φ.unop.τ₁.unop).op = (φ.unop.τ₂.unop ≫ (Opposite.unop Y✝).unop.f₂₄).op

Tactic:
by simp

Goal state after:
Module: Mathlib.CategoryTheory.Square

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
X✝ Y✝ : (Square Cᵒᵖ)ᵒᵖ
φ : X✝ ⟶ Y✝
⊢ ((Opposite.unop X✝).unop.f₃₄ ≫ φ.unop.τ₁.unop).op = (φ.unop.τ₃.unop ≫ (Opposite.unop Y✝).unop.f₃₄).op

Tactic:
by simp

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.Square

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
sq : Square C
h : sq.IsPushout
inst✝ : Epi sq.f₁₂
⊢ Epi sq.f₃₄

Tactic:
by
  have : Epi sq.flip.f₁₃ := by dsimp; infer_instance
  exact h.flip.epi_f₂₄

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
l : ℕ
⊢ ∀ (r r' : ℤ) (pq : ℤ × Fin l) (hr : autoParam (2 ≤ r) SpectralSequenceDataCore._auto_24)
    (hrr' : autoParam (r ≤ r') SpectralSequenceDataCore._auto_26),
    ⟨(↑↑pq.2 - (r' - 2)).toNat, ⋯⟩ ≤ ⟨(↑↑pq.2 - (r - 2)).toNat, ⋯⟩

Tactic:
by
    rintro r r' ⟨a, ⟨a', _⟩⟩ hr hrr'
    rw [Fin.mk_le_mk]
    lia

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
pq : ℕ × ℕ
⊢ WithBotTop.coe (-↑pq.2) ≤ WithBotTop.coe (-↑pq.2 + 1)

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ i₂ i₃ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₁₂ : i₀ ⟶ i₂
f₂₃ : i₁ ⟶ i₃
h₁₂ : f₁ ≫ f₂ = f₁₂
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ X.δ f₁₂ f₃ n₀ n₁ hn₁ ≫ X.toCycles f₁ f₂ f₁₂ h₁₂ n₁ ≫ X.πE f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯ = 0

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝ : X.HasSpectralSequence data
E : SpectralObject C EInt
r r' : ℤ
pq : ℤ × ℤ
hpq : ∀ (pq' : ℤ × ℤ), ¬(ComplexShape.up' (r, 1 - r)).Rel pq' pq
n : ℤ
hn : n = coreE₂Cohomological.deg pq - 1
hrr' : r + 1 = r'
hr : 2 ≤ r
⊢ (ComplexShape.up' (r, 1 - r)).Rel (pq - (r, 1 - r)) pq

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ i₂ i₃ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₁₂ : i₀ ⟶ i₂
f₂₃ : i₁ ⟶ i₃
h₁₂ : f₁ ≫ f₂ = f₁₂
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ (X.ιE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂ ≫ X.fromOpcycles f₂ f₃ f₂₃ h₂₃ n₁) ≫ X.δ f₁ f₂₃ n₁ n₂ hn₂ = 0

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j : ι
f : i ⟶ j
i' j' : ι
f' : i' ⟶ j'
α : mk₁ f ⟶ mk₁ f'
β : mk₃ (𝟙 i) f (𝟙 j) ⟶ mk₃ (𝟙 i') f' (𝟙 j')
n₀ n₁ n₂ : ℤ
⊢ (mk₃ (𝟙 i) f (𝟙 j)).map' 0 1 homMk₁._proof_4 _proof_298✝ ≫ α.app 0 =
    α.app 0 ≫ (mk₃ (𝟙 i') f' (𝟙 j')).map' 0 1 homMk₁._proof_4 _proof_298✝

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ : ι
f : i₀ ⟶ i₁
n₀ n₁ : ℤ
hn₁ : autoParam (n₀ + 1 = n₁) cyclesIsoH_inv._auto_1
⊢ 𝟙 i₀ ≫ f = f

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ : ι
f : i₀ ⟶ i₁
n₀ n₁ : ℤ
hn₁ : autoParam (n₀ + 1 = n₁) cyclesIsoH_hom_inv_id._auto_1
⊢ 𝟙 i₀ ≫ f = f

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ : ι
f : i₀ ⟶ i₁
n₀ n₁ : ℤ
hn₁ : autoParam (n₀ + 1 = n₁) cyclesIsoH_inv_hom_id._auto_1
⊢ 𝟙 i₀ ≫ f = f

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ : ι
f : i₀ ⟶ i₁
n₀ n₁ : ℤ
hn₁ : autoParam (n₀ + 1 = n₁) opcyclesIsoH_hom._auto_1
⊢ f ≫ 𝟙 i₁ = f

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ : ι
f : i₀ ⟶ i₁
n₀ n₁ : ℤ
hn₁ : autoParam (n₀ + 1 = n₁) opcyclesIsoH_hom_inv_id._auto_1
⊢ f ≫ 𝟙 i₁ = f

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ : ι
f : i₀ ⟶ i₁
n₀ n₁ : ℤ
hn₁ : autoParam (n₀ + 1 = n₁) opcyclesIsoH_inv_hom_id._auto_1
⊢ f ≫ 𝟙 i₁ = f

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
i j : ι
f : i ⟶ j
h : (X.shortComplex (𝟙 i) f (𝟙 j) n₀ n₁ n₂ hn₁ hn₂).LeftHomologyData := (X.homologyDataIdId f n₀ n₁ n₂ hn₁ hn₂).left
⊢ 𝟙 i ≫ f = f

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
i j : ι
f : i ⟶ j
h : (X.shortComplex (𝟙 i) f (𝟙 j) n₀ n₁ n₂ hn₁ hn₂).HomologyData := X.homologyDataIdId f n₀ n₁ n₂ hn₁ hn₂
⊢ f ≫ 𝟙 j = f

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ i₂ i₃ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₁₂ : i₀ ⟶ i₂
f₂₃ : i₁ ⟶ i₃
h₁₂ : f₁ ≫ f₂ = f₁₂
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
A : C
x : A ⟶ X.opcycles f₁ f₂₃ n₁
hx :
  x ≫ X.opcyclesMap f₁ f₂₃ f₁₂ f₃ (threeδ₂Toδ₁ f₁ f₂ f₃ f₁₂ f₂₃ h₁₂ h₂₃) n₁ ≫ X.fromOpcycles f₁₂ f₃ (f₁₂ ≫ f₃) ⋯ n₁ = 0
⊢ (𝟙 (mk₁ (f₁₂ ≫ f₃))).app 0 = (threeδ₂Toδ₁ f₁ f₂ f₃ f₁₂ f₂₃ h₁₂ h₂₃).app 0

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
h : (c r).Rel pq pq'
⊢ n₀ - 1 + 1 = n₀

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Lie.Basic

Goal state before:
pp: R : Type u
L : Type v
M : Type w
N : Type w₁
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : LieRingModule L N
inst✝ : LieModule R L N
t : R
x✝ y✝ z : L
m n✝ : M
x y : L
f : M →ₗ[R] N
n : M
⊢ { toFun := fun m ↦ ⁅x + y, f m⁆ - f ⁅x + y, m⁆, map_add' := ⋯, map_smul' := ⋯ } n =
    ({ toFun := fun m ↦ ⁅x, f m⁆ - f ⁅x, m⁆, map_add' := ⋯, map_smul' := ⋯ } +
        { toFun := fun m ↦ ⁅y, f m⁆ - f ⁅y, m⁆, map_add' := ⋯, map_smul' := ⋯ })
      n

Tactic:
simp only [add_lie, coe_mk, AddHom.coe_mk, add_apply, map_add]

Goal state after:
R : Type u
L : Type v
M : Type w
N : Type w₁
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : LieRingModule L N
inst✝ : LieModule R L N
t : R
x✝ y✝ z : L
m n✝ : M
x y : L
f : M →ₗ[R] N
n : M
⊢ ⁅x, f n⁆ + ⁅y, f n⁆ - (f ⁅x, n⁆ + f ⁅y, n⁆) = ⁅x, f n⁆ - f ⁅x, n⁆ + (⁅y, f n⁆ - f ⁅y, n⁆)
Module: Mathlib.RingTheory.Artinian.Module

Goal state before:
pp: R : Type u_1
P : Type u_3
inst✝² : Ring R
inst✝¹ : AddCommGroup P
inst✝ : Module R P
S : Submodule R P
x✝ : IsArtinian R ↥S ∧ IsArtinian R (P ⧸ S)
left✝ : IsArtinian R ↥S
right✝ : IsArtinian R (P ⧸ S)
⊢ S.subtype.range = S.mkQ.ker

Tactic:
rw [Submodule.ker_mkQ, Submodule.range_subtype]

Goal state after:
Module: Mathlib.Algebra.Lie.Basic

Goal state before:
pp: R : Type u
L : Type v
M : Type w
N : Type w₁
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : LieRingModule L N
inst✝ : LieModule R L N
t : R
x✝ y z : L
m n : M
x : L
f : M →ₗ[R] R
⊢ ∀ (m : R) (x_1 : M), -f ⁅x, m • x_1⁆ = (RingHom.id R) m • -f ⁅x, x_1⁆

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Lie.Basic

Goal state before:
pp: R : Type u
L₁ : Type v
L₂ : Type w
L₃ : Type w₁
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L₁
inst✝⁴ : LieAlgebra R L₁
inst✝³ : LieRing L₂
inst✝² : LieAlgebra R L₂
inst✝¹ : LieRing L₃
inst✝ : LieAlgebra R L₃
⊢ ∀ {x y : L₁}, (LinearMap.toAddHom 0).toFun ⁅x, y⁆ = ⁅(LinearMap.toAddHom 0).toFun x, (LinearMap.toAddHom 0).toFun y⁆

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Lie.Basic

Goal state before:
pp: R : Type u
L : Type v
M : Type w
N : Type w₁
P : Type w₂
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : AddCommGroup P
inst✝⁵ : Module R M
inst✝⁴ : Module R N
inst✝³ : Module R P
inst✝² : LieRingModule L M
inst✝¹ : LieRingModule L N
inst✝ : LieRingModule L P
⊢ ∀ {x : L} {m : M}, (LinearMap.toAddHom 0).toFun ⁅x, m⁆ = ⁅x, (LinearMap.toAddHom 0).toFun m⁆

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Lie.Basic

Goal state before:
pp: R : Type u
L : Type v
M : Type w
N : Type w₁
P : Type w₂
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : AddCommGroup P
inst✝⁵ : Module R M
inst✝⁴ : Module R N
inst✝³ : Module R P
inst✝² : LieRingModule L M
inst✝¹ : LieRingModule L N
inst✝ : LieRingModule L P
f g : M →ₗ⁅R,L⁆ N
⊢ ∀ {x : L} {m : M}, (↑f + ↑g).toFun ⁅x, m⁆ = ⁅x, (↑f + ↑g).toFun m⁆

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Lie.Basic

Goal state before:
pp: R : Type u
L : Type v
M : Type w
N : Type w₁
P : Type w₂
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : AddCommGroup P
inst✝⁵ : Module R M
inst✝⁴ : Module R N
inst✝³ : Module R P
inst✝² : LieRingModule L M
inst✝¹ : LieRingModule L N
inst✝ : LieRingModule L P
f g : M →ₗ⁅R,L⁆ N
⊢ ∀ {x : L} {m : M}, (↑f - ↑g).toFun ⁅x, m⁆ = ⁅x, (↑f - ↑g).toFun m⁆

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Lie.Basic

Goal state before:
pp: R : Type u
L : Type v
M : Type w
N : Type w₁
P : Type w₂
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : AddCommGroup P
inst✝⁵ : Module R M
inst✝⁴ : Module R N
inst✝³ : Module R P
inst✝² : LieRingModule L M
inst✝¹ : LieRingModule L N
inst✝ : LieRingModule L P
f : M →ₗ⁅R,L⁆ N
⊢ ∀ {x : L} {m : M}, (-↑f).toFun ⁅x, m⁆ = ⁅x, (-↑f).toFun m⁆

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Lie.Basic

Goal state before:
pp: R : Type u
L : Type v
M : Type w
N : Type w₁
P : Type w₂
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : AddCommGroup P
inst✝⁵ : Module R M
inst✝⁴ : Module R N
inst✝³ : Module R P
inst✝² : LieRingModule L M
inst✝¹ : LieRingModule L N
inst✝ : LieRingModule L P
n : ℕ
f : M →ₗ⁅R,L⁆ N
⊢ ∀ {x : L} {m : M}, (n • ↑f).toFun ⁅x, m⁆ = ⁅x, (n • ↑f).toFun m⁆

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Lie.Basic

Goal state before:
pp: R : Type u
L : Type v
M : Type w
N : Type w₁
P : Type w₂
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : AddCommGroup P
inst✝⁵ : Module R M
inst✝⁴ : Module R N
inst✝³ : Module R P
inst✝² : LieRingModule L M
inst✝¹ : LieRingModule L N
inst✝ : LieRingModule L P
z : ℤ
f : M →ₗ⁅R,L⁆ N
⊢ ∀ {x : L} {m : M}, (z • ↑f).toFun ⁅x, m⁆ = ⁅x, (z • ↑f).toFun m⁆

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Lie.Basic

Goal state before:
pp: R : Type u
L : Type v
M : Type w
N : Type w₁
P : Type w₂
inst✝¹² : CommRing R
inst✝¹¹ : LieRing L
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : AddCommGroup N
inst✝⁸ : AddCommGroup P
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : LieRingModule L M
inst✝³ : LieRingModule L N
inst✝² : LieRingModule L P
inst✝¹ : LieAlgebra R L
inst✝ : LieModule R L N
t : R
f : M →ₗ⁅R,L⁆ N
⊢ ∀ {x : L} {m : M}, (t • ↑f).toFun ⁅x, m⁆ = ⁅x, (t • ↑f).toFun m⁆

Tactic:
by simp

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq pq' : κ
hpq : (c r).prev pq' = pq
i₀ i₁ i₂ i₃ i₃' : ι
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
hi₃' : i₃' = data.i₃ r' pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
inst✝ : X.HasSpectralSequence data
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
h : (c r).Rel pq pq'
⊢ n₀ - 1 + 1 = n₀

Tactic:
by simp

Goal state after:
Module: Mathlib.RingTheory.Artinian.Module

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsArtinian R M
f : Module.End R M
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → range (f ^ n) = range (f ^ m)
m : ℕ
hm : n ≤ m
x y : M
hy : (f ^ m) ((f ^ n) y) = (f ^ m) x
⊢ (f ^ n) y ∈ range (f ^ n) ∧ x - (f ^ n) y + (f ^ n) y = x

Tactic:
by simp

Goal state after: