mathlib-initiative/mathlib-tactics huggingface dataset:
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
l : ℕ
E : SpectralObject C (Fin (l + 1))
n : ℤ
k : ℕ
hr : 2 ≤ 2 + ↑k
p : ℤ
t : ℕ
hq : k + 1 + t < l
hn : n = (coreE₂CohomologicalFin l).deg (p, ⟨k + 1 + t, hq⟩) + 1
this : k < k + 1 + t
hpq : ∀ (a : ℤ) (b : Fin l), p + (2 + ↑k) = a → ¬↑(k + 1 + t) + (1 - (2 + ↑k)) = ↑↑b
⊢ ↑(k + 1 + t) + (1 - (2 + ↑k)) = ↑↑⟨t, ⋯⟩

Tactic:
simp

Goal state after:
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
l : ℕ
E : SpectralObject C (Fin (l + 1))
n : ℤ
k : ℕ
hr : 2 ≤ 2 + ↑k
p : ℤ
t : ℕ
hq : k + 1 + t < l
hn : n = (coreE₂CohomologicalFin l).deg (p, ⟨k + 1 + t, hq⟩) + 1
this : k < k + 1 + t
hpq : ∀ (a : ℤ) (b : Fin l), p + (2 + ↑k) = a → ¬↑(k + 1 + t) + (1 - (2 + ↑k)) = ↑↑b
⊢ ↑k + 1 + ↑t + (1 - (2 + ↑k)) = ↑t
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: case h₀
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
h₁₂ : f₁ ≫ f₂ = f₁₂
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
A : C
x₂ : A ⟶ X.opcycles f₁₂ f₃ n₁
hx₂ : x₂ ≫ X.opcyclesToE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂ = 0
A₁ : C
π₁ : A₁ ⟶ A
w✝¹ : Epi π₁
y₂ : A₁ ⟶ (X.H n₁).obj (mk₁ f₁₂)
hy₂ : π₁ ≫ x₂ = y₂ ≫ X.pOpcycles f₁₂ f₃ n₁
A₂ : C
π₂ : A₂ ⟶ A₁
w✝ : Epi π₂
y₁ : A₂ ⟶ (X.H n₁).obj (mk₁ f₁) ⊞ (X.H n₀).obj (mk₁ f₃)
hy₁ : π₂ ≫ y₂ = y₁ ≫ biprod.desc ((X.H n₁).map (twoδ₂Toδ₁ f₁ f₂ f₁₂ h₁₂)) (X.δ f₁₂ f₃ n₀ n₁ ⋯)
⊢ y₁ ≫ biprod.fst = ((y₁ ≫ biprod.fst) ≫ biprod.inl + (y₁ ≫ biprod.snd) ≫ biprod.inr) ≫ biprod.fst

Tactic:
simp

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

Goal state before:
pp: case h₁
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
h₁₂ : f₁ ≫ f₂ = f₁₂
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
A : C
x₂ : A ⟶ X.opcycles f₁₂ f₃ n₁
hx₂ : x₂ ≫ X.opcyclesToE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂ = 0
A₁ : C
π₁ : A₁ ⟶ A
w✝¹ : Epi π₁
y₂ : A₁ ⟶ (X.H n₁).obj (mk₁ f₁₂)
hy₂ : π₁ ≫ x₂ = y₂ ≫ X.pOpcycles f₁₂ f₃ n₁
A₂ : C
π₂ : A₂ ⟶ A₁
w✝ : Epi π₂
y₁ : A₂ ⟶ (X.H n₁).obj (mk₁ f₁) ⊞ (X.H n₀).obj (mk₁ f₃)
hy₁ : π₂ ≫ y₂ = y₁ ≫ biprod.desc ((X.H n₁).map (twoδ₂Toδ₁ f₁ f₂ f₁₂ h₁₂)) (X.δ f₁₂ f₃ n₀ n₁ ⋯)
⊢ y₁ ≫ biprod.snd = ((y₁ ≫ biprod.fst) ≫ biprod.inl + (y₁ ≫ biprod.snd) ≫ biprod.inr) ≫ biprod.snd

Tactic:
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:
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:
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:
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:
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:
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:
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 : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).Rel pq pq'
hpq' : (c r).Rel pq' pq''
n₀ n₁ n₂ n₃ n₄ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
hn₄ : n₃ + 1 = n₄
hn₂' : n₂ = data.deg pq'
⊢ data.i₂ pq'' = data.i₀ r pq' ⋯

Tactic:
rw [data.hc₀₂ r pq' pq'' hpq']

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:
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:
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:
simp

Goal state after:
Module: Mathlib.Order.Filter.Subsingleton

Goal state before:
pp: case refine_2.inl
α : Type u_1
⊢ ⊥.Subsingleton

Tactic:
simp

Goal state after:
Module: Mathlib.Order.Filter.Subsingleton

Goal state before:
pp: case refine_2.inr
α : Type u_1
a : α
⊢ (pure a).Subsingleton

Tactic:
simp

Goal state after:
Module: Mathlib.Order.Filter.Subsingleton

Goal state before:
pp: case inl
α : Type u_1
inst✝ : Nonempty α
⊢ ⊥.Subsingleton ↔ ∃ a, ⊥ ≤ pure a

Tactic:
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:
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.H n₁).map (twoδ₂Toδ₁ f₁ f₂ f₁₂ h₁₂) ≫ X.pOpcycles f₁₂ f₃ n₁ ≫ X.opcyclesToE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂ = 0

Tactic:
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:
simp

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

Goal state before:
pp: case pos.refine_1
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).next pq' = pq''
i₀' i₀ i₁ i₂ i₃ : ι
hi₀' : i₀' = data.i₀ r' pq' ⋯
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ 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''
⊢ (Iso.refl (kfSc X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂).X₁).hom ≫
      (X.dKernelSequence (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯ ≫ homOfLE ⋯) ⋯ n₀ n₁ n₂
          (n₂ + 1) ⋯ ⋯ ⋯).f =
    (kfSc X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂).f ≫
      (pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯).hom

Tactic:
simp

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

Goal state before:
pp: L : Type v
M : Type w
inst✝² : LieRing L
inst✝¹ : AddCommGroup M
inst✝ : LieRingModule L M
x : L
m : M
⊢ ⁅-x + x, m⁆ = 0

Tactic:
simp

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

Goal state before:
pp: L : Type v
M : Type w
inst✝² : LieRing L
inst✝¹ : AddCommGroup M
inst✝ : LieRingModule L M
x : L
m : M
⊢ ⁅x, -m + m⁆ = 0

Tactic:
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:
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 : 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:
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 m : L
n : M →ₗ[R] R
p : M
⊢ { toFun := fun m_1 ↦ -{ toFun := fun m_2 ↦ -n ⁅m, m_2⁆, map_add' := ⋯, map_smul' := ⋯ } ⁅x, m_1⁆, map_add' := ⋯,
        map_smul' := ⋯ }
      p =
    ({ toFun := fun m_1 ↦ -n ⁅⁅x, m⁆, m_1⁆, map_add' := ⋯, map_smul' := ⋯ } +
        { toFun := fun m_1 ↦ -{ toFun := fun m ↦ -n ⁅x, m⁆, map_add' := ⋯, map_smul' := ⋯ } ⁅m, m_1⁆, map_add' := ⋯,
          map_smul' := ⋯ })
      p

Tactic:
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
t : R
x : L
m : M →ₗ[R] R
n : M
⊢ ⁅t • x, m⁆ n = (t • ⁅x, m⁆) n

Tactic:
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
t : R
x : L
m : M →ₗ[R] R
n : M
⊢ ⁅x, t • m⁆ n = (t • ⁅x, m⁆) n

Tactic:
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:
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
f : L₂ →ₗ⁅R⁆ L₃
g : L₁ →ₗ⁅R⁆ L₂
⊢ ∀ {x y : L₁}, (↑f ∘ₗ ↑g).toFun ⁅x, y⁆ = ⁅(↑f ∘ₗ ↑g).toFun x, (↑f ∘ₗ ↑g).toFun y⁆

Tactic:
simp

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

Goal state before:
pp: R : Type u
L₁ : Type v
L₂ : Type w
inst✝⁴ : CommRing R
inst✝³ : LieRing L₁
inst✝² : LieRing L₂
inst✝¹ : LieAlgebra R L₁
inst✝ : LieAlgebra R L₂
f_inv : L₂ → L₁
f : L₁ → L₂
map_add'✝¹ : ∀ (x y : L₁), f (x + y) = f x + f y
map_smul'✝¹ :
  ∀ (m : R) (x : L₁),
    { toFun := f, map_add' := map_add'✝¹ }.toFun (m • x) =
      (RingHom.id R) m • { toFun := f, map_add' := map_add'✝¹ }.toFun x
map_lie'✝¹ :
  ∀ {x y : L₁},
    { toFun := f, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹ }.toFun ⁅x, y⁆ =
      ⁅{ toFun := f, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹ }.toFun x,
        { toFun := f, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹ }.toFun y⁆
left_inv✝¹ :
  LeftInverse f_inv (↑{ toFun := f, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹, map_lie' := map_lie'✝¹ }).toFun
right_inv✝¹ :
  RightInverse f_inv (↑{ toFun := f, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹, map_lie' := map_lie'✝¹ }).toFun
g_inv : L₂ → L₁
g : L₁ → L₂
map_add'✝ : ∀ (x y : L₁), g (x + y) = g x + g y
map_smul'✝ :
  ∀ (m : R) (x : L₁),
    { toFun := g, map_add' := map_add'✝ }.toFun (m • x) =
      (RingHom.id R) m • { toFun := g, map_add' := map_add'✝ }.toFun x
map_lie'✝ :
  ∀ {x y : L₁},
    { toFun := g, map_add' := map_add'✝, map_smul' := map_smul'✝ }.toFun ⁅x, y⁆ =
      ⁅{ toFun := g, map_add' := map_add'✝, map_smul' := map_smul'✝ }.toFun x,
        { toFun := g, map_add' := map_add'✝, map_smul' := map_smul'✝ }.toFun y⁆
left_inv✝ :
  LeftInverse g_inv (↑{ toFun := g, map_add' := map_add'✝, map_smul' := map_smul'✝, map_lie' := map_lie'✝ }).toFun
right_inv✝ :
  RightInverse g_inv (↑{ toFun := g, map_add' := map_add'✝, map_smul' := map_smul'✝, map_lie' := map_lie'✝ }).toFun
⊢ { toFun := f, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹, map_lie' := map_lie'✝¹, invFun := f_inv,
          left_inv := left_inv✝¹, right_inv := right_inv✝¹ }.toLinearEquiv =
      { toFun := g, map_add' := map_add'✝, map_smul' := map_smul'✝, map_lie' := map_lie'✝, invFun := g_inv,
          left_inv := left_inv✝, right_inv := right_inv✝ }.toLinearEquiv →
    { toFun := f, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹, map_lie' := map_lie'✝¹, invFun := f_inv,
        left_inv := left_inv✝¹, right_inv := right_inv✝¹ } =
      { toFun := g, map_add' := map_add'✝, map_smul' := map_smul'✝, map_lie' := map_lie'✝, invFun := g_inv,
        left_inv := left_inv✝, right_inv := right_inv✝ }

Tactic:
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:
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 : N →ₗ⁅R,L⁆ P
g : M →ₗ⁅R,L⁆ N
⊢ ∀ {x : L} {m : M}, (↑f ∘ₗ ↑g).toFun ⁅x, m⁆ = ⁅x, (↑f ∘ₗ ↑g).toFun m⁆

Tactic:
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:
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:
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:
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:
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:
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:
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:
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✝⁷ : CommRing R
inst✝⁶ : LieRing L
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : LieRingModule L M
inst✝ : LieRingModule L N
f_inv : N → M
f : M → N
map_add'✝¹ : ∀ (x y : M), f (x + y) = f x + f y
map_smul'✝¹ :
  ∀ (m : R) (x : M),
    { toFun := f, map_add' := map_add'✝¹ }.toFun (m • x) =
      (RingHom.id R) m • { toFun := f, map_add' := map_add'✝¹ }.toFun x
map_lie'✝¹ :
  ∀ {x : L} {m : M},
    { toFun := f, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹ }.toFun ⁅x, m⁆ =
      ⁅x, { toFun := f, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹ }.toFun m⁆
left_inv✝¹ :
  LeftInverse f_inv (↑{ toFun := f, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹, map_lie' := map_lie'✝¹ }).toFun
right_inv✝¹ :
  RightInverse f_inv (↑{ toFun := f, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹, map_lie' := map_lie'✝¹ }).toFun
g_inv : N → M
g : M → N
map_add'✝ : ∀ (x y : M), g (x + y) = g x + g y
map_smul'✝ :
  ∀ (m : R) (x : M),
    { toFun := g, map_add' := map_add'✝ }.toFun (m • x) =
      (RingHom.id R) m • { toFun := g, map_add' := map_add'✝ }.toFun x
map_lie'✝ :
  ∀ {x : L} {m : M},
    { toFun := g, map_add' := map_add'✝, map_smul' := map_smul'✝ }.toFun ⁅x, m⁆ =
      ⁅x, { toFun := g, map_add' := map_add'✝, map_smul' := map_smul'✝ }.toFun m⁆
left_inv✝ :
  LeftInverse g_inv (↑{ toFun := g, map_add' := map_add'✝, map_smul' := map_smul'✝, map_lie' := map_lie'✝ }).toFun
right_inv✝ :
  RightInverse g_inv (↑{ toFun := g, map_add' := map_add'✝, map_smul' := map_smul'✝, map_lie' := map_lie'✝ }).toFun
⊢ { toFun := f, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹, map_lie' := map_lie'✝¹, invFun := f_inv,
          left_inv := left_inv✝¹, right_inv := right_inv✝¹ }.toEquiv =
      { toFun := g, map_add' := map_add'✝, map_smul' := map_smul'✝, map_lie' := map_lie'✝, invFun := g_inv,
          left_inv := left_inv✝, right_inv := right_inv✝ }.toEquiv →
    { toFun := f, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹, map_lie' := map_lie'✝¹, invFun := f_inv,
        left_inv := left_inv✝¹, right_inv := right_inv✝¹ } =
      { toFun := g, map_add' := map_add'✝, map_smul' := map_smul'✝, map_lie' := map_lie'✝, invFun := g_inv,
        left_inv := left_inv✝, right_inv := right_inv✝ }

Tactic:
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:
simp

Goal state after: