mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Lie.Sl2

Goal state before:
pp: R : Type u_1
L : Type u_2
M : Type u_3
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
h e f : L
m : M
μ : R
t : IsSl2Triple h e f
P : t.HasPrimitiveVectorWith m μ
inst✝³ : IsNoetherian R M
inst✝² : IsTorsionFree R M
inst✝¹ : IsDomain R
inst✝ : CharZero R
this✝ : ∃ n, ((toEnd R L M) f ^ n) m = 0
n : ℕ
hn₁ : ¬((toEnd R L M) f ^ n) m = 0
hn₂ : ((toEnd R L M) f ^ (n + 1)) m = 0
this : ↑n + 1 = 0 ∨ μ = ↑n
⊢ μ = ↑n

Tactic:
exact this.resolve_left <| Nat.cast_add_one_ne_zero n

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

Goal state before:
pp: case inr.hcomp
R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁹ : CommRing R
inst✝⁸ : LieRing L
inst✝⁷ : LieAlgebra R L
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : LieRingModule L M
inst✝³ : LieModule R L M
inst✝² : LieRing.IsNilpotent L
inst✝¹ : IsNoetherian R M
inst✝ : IsArtinian R M
h : ∀ N < ⊤, IsCompl (genWeightSpace (↥N) 0) (posFittingComp R L ↥N)
M₀ : LieSubmodule R L M := ⋯
M₁ : LieSubmodule R L M := ⋯
x : L
M₀ₓ : LieSubmodule R L M := ⋯
hx : M₀ₓ ≠ ⊤
M₁ₓ : LieSubmodule R L M := ⋯
M₀ₓ₀ : LieSubmodule R L ↥M₀ₓ := ⋯
M₀ₓ₁ : LieSubmodule R L ↥M₀ₓ := ⋯
h₁ : IsCompl M₀ₓ M₁ₓ
h₂ : IsCompl M₀ₓ₀ M₀ₓ₁
h₃ : LieSubmodule.map M₀ₓ.incl M₀ₓ₀ = M₀
h₄ : LieSubmodule.map M₀ₓ.incl M₀ₓ₁ ⊔ M₁ₓ = M₁
⊢ IsCompl (LieSubmodule.map M₀ₓ.incl M₀ₓ₀ ⊔ LieSubmodule.map M₀ₓ.incl M₀ₓ₁) M₁ₓ

Tactic:
rwa [← LieSubmodule.map_sup, h₂.sup_eq_top, LieModuleHom.map_top, LieSubmodule.range_incl]

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

Goal state before:
pp: R : Type u_1
L : Type u_2
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsSemisimple R L
s : Finset (LieIdeal R L)
hs : ↑s ⊆ {I | IsAtom I}
I : LieIdeal R L
hI✝ : I ≤ s.sup id
S : Set (LieIdeal R L) := {I | IsAtom I}
hI : I < s.sup id
J : LieIdeal R L
hJs : J ∈ s
hJI : ¬J ≤ I
s' : Finset (LieIdeal R L) := s.erase J
hs' : s' ⊂ s
hs'S : ↑s' ⊆ S
K : LieIdeal R L := s'.sup id
y : L
hy : y ∈ id J
z : L
hz : z ∈ K
hx : y + z ∈ I
this : ⟨y, hy⟩ ∈ center R ↥J
⊢ y + z ∈ K

Tactic:
by
    have _inst := isSimple_of_isAtom J (hs hJs)
    simp_all

Goal state after:
Module: Mathlib.FieldTheory.SplittingField.Construction

Goal state before:
pp: case pos
K : Type v
inst✝ : Field K
f : K[X]
hf1 : ¬IsUnit f
hf2 : f = 0
⊢ f.factor ∣ f

Tactic:
rw [hf2]; exact dvd_zero _

Goal state after:
Module: Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure

Goal state before:
pp: k : Type u
inst✝ : Field k
v : Monics k × ℕ →₀ MvPolynomial (Vars k) k
left✝ : v ∈ Finsupp.supported (MvPolynomial (Vars k) k) (MvPolynomial (Vars k) k) Set.univ
hv : 0 = 1
⊢ False

Tactic:
exact zero_ne_one hv

Goal state after:
Module: Mathlib.Algebra.Lie.Weights.Chain

Goal state before:
pp: R : Type u_1
L : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
H : LieSubalgebra R L
α χ : ↥H → R
p q : ℤ
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsNoetherian R L
hp : genWeightSpace M (p • α + χ) = ⊥
hq : genWeightSpace M (q • α + χ) = ⊥
x : ↥H
hx : x ∈ corootSpace α
⊢ (LinearMap.trace R ↥(genWeightSpaceChain M α χ p q)) ((toEnd R ↥H ↥(genWeightSpaceChain M α χ p q)) x) = 0

Tactic:
LieAlgebra.mem_corootSpace'

Goal state after:
R : Type u_1
L : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
H : LieSubalgebra R L
α χ : ↥H → R
p q : ℤ
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsNoetherian R L
hp : genWeightSpace M (p • α + χ) = ⊥
hq : genWeightSpace M (q • α + χ) = ⊥
x : ↥H
hx : x ∈ Submodule.span R {x | ∃ y ∈ rootSpace H α, ∃ z ∈ rootSpace H (-α), ⁅y, z⁆ = ↑x}
⊢ (LinearMap.trace R ↥(genWeightSpaceChain M α χ p q)) ((toEnd R ↥H ↥(genWeightSpaceChain M α χ p q)) x) = 0
Module: Mathlib.Algebra.Lie.Weights.Chain

Goal state before:
pp: R : Type u_1
L : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
H : LieSubalgebra R L
α χ : ↥H → R
p q : ℤ
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsNoetherian R L
hp : genWeightSpace M (p • α + χ) = ⊥
hq : genWeightSpace M (q • α + χ) = ⊥
x u : ↥H
y : L
hy : y ∈ rootSpace H α
z : L
hz : z ∈ rootSpace H (-α)
hyz : ⁅y, z⁆ = ↑u
f : End R ↥(genWeightSpaceChain M α χ p q) :=
  {
    toFun := fun x ↦
      match x with
      | ⟨m, hm⟩ => ⟨⁅y, m⁆, ⋯⟩,
    map_add' := ⋯, map_smul' := ⋯ }
g : End R ↥(genWeightSpaceChain M α χ p q) :=
  {
    toFun := fun x ↦
      match x with
      | ⟨m, hm⟩ => ⟨⁅z, m⁆, ⋯⟩,
    map_add' := ⋯, map_smul' := ⋯ }
x✝ : ↥(genWeightSpaceChain M α χ p q)
⊢ ⁅⁅y, z⁆, ↑x✝⁆ = ↑(⁅f, g⁆ x✝)

Tactic:
simp only [lie_lie, LieHom.lie_apply, LinearMap.coe_mk, AddHom.coe_mk, Module.End.lie_apply,
        AddSubgroupClass.coe_sub, f, g]

Goal state after:
Module: Mathlib.Algebra.Lie.Weights.Chain

Goal state before:
pp: case convert_2
R : Type u_1
L : Type u_2
inst✝¹³ : CommRing R
inst✝¹² : LieRing L
inst✝¹¹ : LieAlgebra R L
M : Type u_3
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : LieRingModule L M
inst✝⁷ : LieModule R L M
H : LieSubalgebra R L
α χ : ↥H → R
inst✝⁶ : H.IsCartanSubalgebra
inst✝⁵ : IsNoetherian R L
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : CharZero R
inst✝¹ : IsTorsionFree R M
inst✝ : IsNoetherian R M
hα : α ≠ 0
hχ : genWeightSpace M χ ≠ ⊥
p : ℤ
hp₀ : p < 0
q : ℤ
hq₀ : q > 0
hp : genWeightSpace M (p • α + χ) = ⊥
hq : genWeightSpace M (q • α + χ) = ⊥
x : ↥H
hx : x ∈ corootSpace α
hb : 0 < ∑ i ∈ Finset.Ioo p q, finrank R ↥(genWeightSpace M (i • α + χ))
h₁ : iSupIndep fun i ↦ (fun k ↦ ↑(genWeightSpace M (k • α + χ))) ↑i
h₂ :
  ∀ (i : ℤ),
    MapsTo ⇑((toEnd R (↥H) M) x) ↑((fun k ↦ ↑(genWeightSpace M (k • α + χ))) i)
      ↑((fun k ↦ ↑(genWeightSpace M (k • α + χ))) i)
h₃ : ↑(genWeightSpaceChain M α χ p q) = ⨆ i ∈ Finset.Ioo p q, (fun k ↦ ↑(genWeightSpace M (k • α + χ))) i
⊢ ((∑ i ∈ Finset.Ioo p q, finrank R ↥(genWeightSpace M (i • α + χ)) • i) • α +
        ↑(∑ i ∈ Finset.Ioo p q, finrank R ↥(genWeightSpace M (i • α + χ))) • χ)
      x =
    ∑ k ∈ Finset.Ioo p q,
      (LinearMap.trace R ↥(genWeightSpace M (k • α + χ))) ((toEnd R ↥H ↥(genWeightSpace M (k • α + χ))) x)

Tactic:
simp_rw [trace_toEnd_genWeightSpace, Pi.add_apply, Pi.smul_apply, smul_add,
    ← smul_assoc, Finset.sum_add_distrib, ← Finset.sum_smul, natCast_zsmul]

Goal state after:
Module: Mathlib.Algebra.Lie.Weights.Chain

Goal state before:
pp: R : Type u_1
L : Type u_2
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
M : Type u_3
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
inst✝⁴ : LieRing.IsNilpotent L
inst✝³ : IsAddTorsionFree R
inst✝² : IsDomain R
inst✝¹ : IsTorsionFree R M
inst✝ : IsNoetherian R M
α : L → R
β : Weight R L M
hα : α ≠ 0
⊢ genWeightSpace M (α + chainTopCoeff α β • α + ⇑β) = ⊥

Tactic:
← succ_nsmul',

Goal state after:
R : Type u_1
L : Type u_2
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
M : Type u_3
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
inst✝⁴ : LieRing.IsNilpotent L
inst✝³ : IsAddTorsionFree R
inst✝² : IsDomain R
inst✝¹ : IsTorsionFree R M
inst✝ : IsNoetherian R M
α : L → R
β : Weight R L M
hα : α ≠ 0
⊢ genWeightSpace M ((chainTopCoeff α β + 1) • α + ⇑β) = ⊥
Module: Mathlib.Algebra.Lie.TraceForm

Goal state before:
pp: case succ
R : Type u_1
L : Type u_3
M : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
k : ℕ
ih : ∀ {x y : L}, x ∈ ⊤.lcs L k → y ∈ LieSubmodule.ucs k ⊥ → ((traceForm R L M) x) y = 0
x y : L
hy : ∀ (x : L), ⁅x, y⁆ ∈ LieSubmodule.ucs k ⊥
hx : x ∈ Submodule.span R {x | ∃ x_1 ∈ ⊤, ∃ n ∈ ⊤.lcs L k, ⁅x_1, n⁆ = x}
⊢ ((traceForm R L M) x) y = 0

Tactic:
LieSubmodule.mem_top,

Goal state after:
case succ
R : Type u_1
L : Type u_3
M : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
k : ℕ
ih : ∀ {x y : L}, x ∈ ⊤.lcs L k → y ∈ LieSubmodule.ucs k ⊥ → ((traceForm R L M) x) y = 0
x y : L
hy : ∀ (x : L), ⁅x, y⁆ ∈ LieSubmodule.ucs k ⊥
hx : x ∈ Submodule.span R {x | ∃ x_1, True ∧ ∃ n ∈ ⊤.lcs L k, ⁅x_1, n⁆ = x}
⊢ ((traceForm R L M) x) y = 0
Module: Mathlib.LinearAlgebra.PerfectPairing.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsReflexive R M
e : N ≃ₗ[R] Dual R M
p : Submodule R (Dual R M)
⊢ map (↑e).flip p.dualCoannihilator = (map (↑e.symm) p).dualAnnihilator

Tactic:
have : IsReflexive R N := e.isReflexive_of_equiv_dual_of_isReflexive
  suffices
      (p.map (e.symm : Dual R M →ₗ[R] N)).dualAnnihilator.map (e.flip.symm : Dual R N →ₗ[R] M) =
        (p.dualCoannihilator.map (e.flip : M →ₗ[R] Dual R N)).map (e.flip.symm : Dual R N →ₗ[R] M)
    from (Submodule.map_injective_of_injective e.flip.symm.injective this).symm
  rw [← dualCoannihilator_map_linearEquiv_flip, ← LinearEquiv.coe_toLinearMap_flip,
    LinearEquiv.flip_flip, ← map_comp, ← map_comp]
  simp [-coe_toLinearMap_flip]

Goal state after:
Module: Mathlib.Algebra.Lie.Weights.Killing

Goal state before:
pp: R : Type u_1
L : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : IsKilling R L
inst✝² : IsNoetherian R L
inst✝¹ : IsArtinian R L
H : LieSubalgebra R L
inst✝ : H.IsCartanSubalgebra
h : Codisjoint (rootSpace H 0) (LieModule.posFittingComp R (↥H) L)
⊢ LinearMap.ker ((killingForm R L).restrict H.toSubmodule) = ⊥

Tactic:
replace h : Codisjoint (H : Submodule R L) (LieModule.posFittingComp R H L : Submodule R L) := by
    rwa [codisjoint_iff, ← LieSubmodule.toSubmodule_inj, LieSubmodule.sup_toSubmodule,
      LieSubmodule.top_toSubmodule, rootSpace_zero_eq R L H, LieSubalgebra.coe_toLieSubmodule,
      ← codisjoint_iff] at h

Goal state after:
R : Type u_1
L : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : IsKilling R L
inst✝² : IsNoetherian R L
inst✝¹ : IsArtinian R L
H : LieSubalgebra R L
inst✝ : H.IsCartanSubalgebra
h : Codisjoint H.toSubmodule ↑(LieModule.posFittingComp R (↥H) L)
⊢ LinearMap.ker ((killingForm R L).restrict H.toSubmodule) = ⊥
Module: Mathlib.Algebra.Lie.TraceForm

Goal state before:
pp: R : Type u_1
L : Type u_3
M : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
k : ℕ
x : L
hk : 1 ≤ k
hx : x ∈ lowerCentralSeries R L L 1
⊢ x ∈ Submodule.span R {m | ∃ u v, ⁅u, v⁆ = m}

Tactic:
lowerCentralSeries_succ,

Goal state after:
R : Type u_1
L : Type u_3
M : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
k : ℕ
x : L
hk : 1 ≤ k
hx : x ∈ ⁅⊤, lowerCentralSeries R L L 0⁆
⊢ x ∈ Submodule.span R {m | ∃ u v, ⁅u, v⁆ = m}
Module: Mathlib.Algebra.Lie.Weights.Killing

Goal state before:
pp: K : Type u_2
L : Type u_3
inst✝⁶ : LieRing L
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝² : H.IsCartanSubalgebra
inst✝¹ : IsKilling K L
inst✝ : IsTriangularizable K (↥H) L
α : Weight K (↥H) L
e : L
he₀ : e ≠ 0
he : ∀ (x : ↥H), ⁅x, e⁆ = α x • e
heα : e ∈ rootSpace H ⇑α
⊢ (cartanEquivDual H).symm (Weight.toLinear K (↥H) L α) ∈ corootSpace ⇑α

Tactic:
obtain ⟨f, hfα, hf⟩ : ∃ f ∈ rootSpace H (-α), killingForm K L e f ≠ 0 := by
    contrapose! he₀
    simpa using mem_ker_killingForm_of_mem_rootSpace_of_forall_rootSpace_neg K L H heα he₀

Goal state after:
K : Type u_2
L : Type u_3
inst✝⁶ : LieRing L
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝² : H.IsCartanSubalgebra
inst✝¹ : IsKilling K L
inst✝ : IsTriangularizable K (↥H) L
α : Weight K (↥H) L
e : L
he₀ : e ≠ 0
he : ∀ (x : ↥H), ⁅x, e⁆ = α x • e
heα : e ∈ rootSpace H ⇑α
f : L
hfα : f ∈ rootSpace H (-⇑α)
hf : ((killingForm K L) e) f ≠ 0
⊢ (cartanEquivDual H).symm (Weight.toLinear K (↥H) L α) ∈ corootSpace ⇑α
Module: Mathlib.Algebra.Lie.Weights.Killing

Goal state before:
pp: K : Type u_2
L : Type u_3
inst✝⁶ : LieRing L
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝² : H.IsCartanSubalgebra
inst✝¹ : IsKilling K L
inst✝ : IsTriangularizable K (↥H) L
⊢ ⊤ ≤ (LinearMap.ker (traceForm K (↥H) L)).dualAnnihilator

Tactic:
ker_traceForm_eq_bot_of_isCartanSubalgebra,

Goal state after:
K : Type u_2
L : Type u_3
inst✝⁶ : LieRing L
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝² : H.IsCartanSubalgebra
inst✝¹ : IsKilling K L
inst✝ : IsTriangularizable K (↥H) L
⊢ ⊤ ≤ ⊥.dualAnnihilator
Module: Mathlib.Algebra.Lie.Weights.Killing

Goal state before:
pp: K : Type u_2
L : Type u_3
inst✝⁷ : LieRing L
inst✝⁶ : Field K
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : IsKilling K L
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : PerfectField K
x : L
hx : x ∈ H
N S : End K L
hN : _root_.IsNilpotent ((ad K L) x - S)
hS : S.IsSemisimple
hSN : (ad K L) x = N + S
hS₀ : Commute ((ad K L) x) S
x' : ↥H := ⟨x, hx⟩
α : ↥H → K
y : L
k : ℕ := ((ad K L) x).maxGenEigenspaceIndex (α x')
hy : y ∈ (((ad K L) x).genEigenspace (α x')) ↑k
⊢ S y = α x' • y

Tactic:
apply_eq_of_mem_of_comm_of_isFinitelySemisimple_of_isNil hy hS₀ hS.isFinitelySemisimple hN

Goal state after:
K : Type u_2
L : Type u_3
inst✝⁷ : LieRing L
inst✝⁶ : Field K
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : IsKilling K L
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : PerfectField K
x : L
hx : x ∈ H
N S : End K L
hN : _root_.IsNilpotent ((ad K L) x - S)
hS : S.IsSemisimple
hSN : (ad K L) x = N + S
hS₀ : Commute ((ad K L) x) S
x' : ↥H := ⟨x, hx⟩
α : ↥H → K
y : L
k : ℕ := ((ad K L) x).maxGenEigenspaceIndex (α x')
hy : y ∈ (((ad K L) x).genEigenspace (α x')) ↑k
⊢ α x' • y = α x' • y
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℤ
hn : Odd n
⊢ eval 0 (T R n) = 0

Tactic:
by
  simp [T_eval_zero, ← Int.not_odd_iff_even, hn]

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

Goal state before:
pp: case smul
ι : Type u_1
R : Type u_2
L : Type u_3
inst✝⁴ : Finite ι
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
H : LieSubalgebra R L
b : Basis ι H
inst✝ : Fintype ι
i : ι
x : L
t : R
u : L
hu : u ∈ Submodule.span R (range b.h)
hv' : ∀ (hx : u ∈ H), (b.baseSupp i) ⟨u, hx⟩ • b.e i = ⁅u, b.e i⁆
hx : t • u ∈ H
⊢ (b.baseSupp i) ⟨t • u, hx⟩ • b.e i = ⁅t • u, b.e i⁆

Tactic:
rw [← coe_cartan_eq_span, LieSubalgebra.mem_toSubmodule] at hu

Goal state after:
case smul
ι : Type u_1
R : Type u_2
L : Type u_3
inst✝⁴ : Finite ι
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
H : LieSubalgebra R L
b : Basis ι H
inst✝ : Fintype ι
i : ι
x : L
t : R
u : L
hu : u ∈ H
hv' : ∀ (hx : u ∈ H), (b.baseSupp i) ⟨u, hx⟩ • b.e i = ⁅u, b.e i⁆
hx : t • u ∈ H
⊢ (b.baseSupp i) ⟨t • u, hx⟩ • b.e i = ⁅t • u, b.e i⁆
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
⊢ U R (-2) = -1

Tactic:
by
  simpa [zero_sub, Int.reduceNeg, U_neg_one, mul_zero, U_zero] using U_sub_two R 0

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: case more
R : Type u_1
inst✝ : CommRing R
x : R
n : ℕ
ih1 : eval (-x) (U R ↑n) = ↑↑(↑n).negOnePow * eval x (U R ↑n)
ih2 : eval (-x) (U R ↑(n + 1)) = ↑↑(↑(n + 1)).negOnePow * eval x (U R ↑(n + 1))
⊢ eval (-x) (U R ↑(n + 2)) = ↑↑(↑(n + 2)).negOnePow * eval x (U R ↑(n + 2))

Tactic:
trans (n + 2 : ℤ).negOnePow * (2 * x * (U R (n + 1)).eval x - (U R n).eval x)
    · push_cast; push_cast at ih2
      rw [U_add_two, eval_sub, eval_mul, eval_mul, ih1, ih2,
        Int.negOnePow_succ, Int.negOnePow_add, Int.negOnePow_even 2 even_two]
      simp; ring
    · simp

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

Goal state before:
pp: ι : Type u_1
R : Type u_2
L : Type u_3
inst✝⁷ : Finite ι
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
H : LieSubalgebra R L
b : Basis ι H
inst✝³ : Fintype ι
inst✝² : IsDomain R
inst✝¹ : CharZero R
inst✝ : IsTorsionFree R L
this : IsLieAbelian ↥H := isLieAbelian_cartan b
U : LieSubmodule R (↥H) L := ⨆ n, ⨆ (_ : n ≠ 0), rootSpace H (∑ i, n i • ⇑((-b.baseSupp) i))
hU : U = ⨆ n, ⨆ (_ : n ≠ 0), rootSpace H (∑ i, n i • ⇑((-b.baseSupp) i))
V : LieSubmodule R (↥H) L := ⨆ n, ⨆ (_ : n ≠ 0), rootSpace H (∑ i, n i • ⇑(b.baseSupp i))
hV : V = ⨆ n, ⨆ (_ : n ≠ 0), rootSpace H (∑ i, n i • ⇑(b.baseSupp i))
s0 : Set (↥H → R) := {0}
hs0 : s0 = {0}
sU : Set (↥H → R) := {f | ∃ n, n ≠ 0 ∧ f = ⇑(∑ i, n i • (-b.baseSupp) i)}
hsU : sU = {f | ∃ n, n ≠ 0 ∧ f = ⇑(∑ i, n i • (-b.baseSupp) i)}
sV : Set (↥H → R) := {f | ∃ n, n ≠ 0 ∧ f = ⇑(∑ i, n i • b.baseSupp i)}
hsV : sV = {f | ∃ n, n ≠ 0 ∧ f = ⇑(∑ i, n i • b.baseSupp i)}
hs0' : rootSpace H 0 = ⨆ i ∈ s0, LieModule.genWeightSpace L i
hsU' : U = ⨆ i ∈ sU, LieModule.genWeightSpace L i
hsV' : V = ⨆ i ∈ sV, LieModule.genWeightSpace L i
n : ι → ℕ
hn : n ≠ 0
hg : -⇑(∑ i, n i • b.baseSupp i) ∈ sU
contra : ∑ i, n i • b.baseSupp i = 0
⊢ ∑ i, (Nat.cast ∘ n) i • b.baseSupp i = 0

Tactic:
by simpa [Nat.cast_smul_eq_nsmul]

Goal state after:
Module: Mathlib.Algebra.Lie.Weights.Killing

Goal state before:
pp: K : Type u_2
L : Type u_3
inst✝⁷ : LieRing L
inst✝⁶ : Field K
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : IsKilling K L
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : CharZero K
I : LieIdeal K L
| LieSubmodule.restr I H

Tactic:
lieIdeal_eq_inf_cartan_sup_biSup_inf_rootSpace

Goal state after:
K : Type u_2
L : Type u_3
inst✝⁷ : LieRing L
inst✝⁶ : Field K
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : IsKilling K L
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : CharZero K
I : LieIdeal K L
| LieSubmodule.restr I H ⊓ H.toLieSubmodule ⊔ ⨆ α, ⨆ (_ : α.IsNonZero), LieSubmodule.restr I H ⊓ rootSpace H ⇑α
Module: Mathlib.Algebra.Lie.Basis.Basic

Goal state before:
pp: ι : Type u_1
R : Type u_2
L : Type u_3
inst✝⁷ : Finite ι
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
H : LieSubalgebra R L
b : Basis ι H
inst✝³ : Fintype ι
inst✝² : IsDomain R
inst✝¹ : CharZero R
inst✝ : IsTorsionFree R L
this : IsLieAbelian ↥H := isLieAbelian_cartan b
U : LieSubmodule R (↥H) L := ⨆ n, ⨆ (_ : n ≠ 0), rootSpace H (∑ i, n i • ⇑((-b.baseSupp) i))
hU : U = ⨆ n, ⨆ (_ : n ≠ 0), rootSpace H (∑ i, n i • ⇑((-b.baseSupp) i))
V : LieSubmodule R (↥H) L := ⨆ n, ⨆ (_ : n ≠ 0), rootSpace H (∑ i, n i • ⇑(b.baseSupp i))
hV : V = ⨆ n, ⨆ (_ : n ≠ 0), rootSpace H (∑ i, n i • ⇑(b.baseSupp i))
s0 : Set (↥H → R) := {0}
hs0 : s0 = {0}
sU : Set (↥H → R) := {f | ∃ n, n ≠ 0 ∧ f = ⇑(∑ i, n i • (-b.baseSupp) i)}
hsU : sU = {f | ∃ n, n ≠ 0 ∧ f = ⇑(∑ i, n i • (-b.baseSupp) i)}
sV : Set (↥H → R) := {f | ∃ n, n ≠ 0 ∧ f = ⇑(∑ i, n i • b.baseSupp i)}
hsV : sV = {f | ∃ n, n ≠ 0 ∧ f = ⇑(∑ i, n i • b.baseSupp i)}
hs0' : rootSpace H 0 = ⨆ i ∈ s0, LieModule.genWeightSpace L i
hsU' : U = ⨆ i ∈ sU, LieModule.genWeightSpace L i
hsV' : V = ⨆ i ∈ sV, LieModule.genWeightSpace L i
hU0 : Disjoint s0 sU
n : ι → ℕ
hn : n ≠ 0
hg : ⇑(∑ i, n i • b.baseSupp i) ∈ sV
contra : ∑ i, n i • b.baseSupp i = 0
⊢ ∑ i, (Nat.cast ∘ n) i • b.baseSupp i = 0

Tactic:
by simpa [Nat.cast_smul_eq_nsmul]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℤ
k : ℕ
h :
  (⇑derivative)^[k + 2] (T R n) - (⇑derivative)^[k] ((⇑derivative)^[2] (T R n) * X ^ 2) =
    (⇑derivative)^[k] (X * derivative (T R n)) - (⇑derivative)^[k] (↑(n ^ 2) * T R n)
⊢ (1 - X ^ 2) * (⇑derivative)^[k + 2] (T R n) =
    (2 * ↑k + 1) * X * (⇑derivative)^[k + 1] (T R n) - (↑n ^ 2 - ↑k ^ 2) * (⇑derivative)^[k] (T R n)

Tactic:
mul_comm X,

Goal state after:
R : Type u_1
inst✝ : CommRing R
n : ℤ
k : ℕ
h :
  (⇑derivative)^[k + 2] (T R n) - (⇑derivative)^[k] ((⇑derivative)^[2] (T R n) * X ^ 2) =
    (⇑derivative)^[k] (derivative (T R n) * X) - (⇑derivative)^[k] (↑(n ^ 2) * T R n)
⊢ (1 - X ^ 2) * (⇑derivative)^[k + 2] (T R n) =
    (2 * ↑k + 1) * X * (⇑derivative)^[k + 1] (T R n) - (↑n ^ 2 - ↑k ^ 2) * (⇑derivative)^[k] (T R n)
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℤ
k : ℕ
h :
  (⇑derivative)^[k + 2] (U R n) - (⇑derivative)^[k] ((⇑derivative)^[2] (U R n) * X ^ 2) =
    ↑3 * (⇑derivative)^[k] (X * derivative (U R n)) - (⇑derivative)^[k] (↑((n + 2) * n) * U R n)
⊢ (1 - X ^ 2) * (⇑derivative)^[k + 2] (U R n) =
    (2 * ↑k + 3) * X * (⇑derivative)^[k + 1] (U R n) - ((↑n + 1) ^ 2 - (↑k + 1) ^ 2) * (⇑derivative)^[k] (U R n)

Tactic:
mul_comm X,

Goal state after:
R : Type u_1
inst✝ : CommRing R
n : ℤ
k : ℕ
h :
  (⇑derivative)^[k + 2] (U R n) - (⇑derivative)^[k] ((⇑derivative)^[2] (U R n) * X ^ 2) =
    ↑3 * (⇑derivative)^[k] (derivative (U R n) * X) - (⇑derivative)^[k] (↑((n + 2) * n) * U R n)
⊢ (1 - X ^ 2) * (⇑derivative)^[k + 2] (U R n) =
    (2 * ↑k + 3) * X * (⇑derivative)^[k + 1] (U R n) - ((↑n + 1) ^ 2 - (↑k + 1) ^ 2) * (⇑derivative)^[k] (U R n)
Module: Mathlib.RingTheory.Localization.NumDen

Goal state before:
pp: A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
⊢ (algebraMap A K) (num A x) / (algebraMap A K) ↑(den A x) = x

Tactic:
rw [← mk'_eq_div]
  apply mk'_num_den

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Range

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹ : MonoidWithZero A
inst✝ : LinearOrderedCommGroupWithZero B
f : A →*₀ B
⊢ StrictMono ⇑embedding

Tactic:
by
  intro x y hxy
  rw [← monoidWithZeroHom_strictMono.lt_iff_lt] at hxy
  simpa using! (OrderEmbedding.lt_iff_lt (OrderIso.withZeroUnits.toOrderEmbedding)).mpr hxy

Goal state after:
Module: Mathlib.RingTheory.Length

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
s : CompositionSeries (Submodule R M)
h₁ : RelSeries.head s = ⊥
h₂ : RelSeries.last s = ⊤
H : IsFiniteLength R M
this✝¹ : IsNoetherian R M
this✝ : IsArtinian R M
t : LTSeries (Submodule R M)
t' : RelSeries {(a, b) | a ⋖ b}
i : Fin (t.length + 1) ↪ Fin (t'.length + 1)
hi : t'.toFun ∘ ⇑i = t.toFun
ht₁ : t'.head = ⊥
ht₂ : t'.last = ⊤
this : Fin s.length ≃ Fin t'.length
h : t.length ≤ t'.length
⊢ t'.length = s.length

Tactic:
simpa using Fintype.card_congr this.symm

Goal state after:
Module: Mathlib.RingTheory.Length

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
⊢ length R M = ⊤ ↔ InfiniteDimensionalOrder (Submodule R M)

Tactic:
rw [← WithBot.coe_inj, WithBot.coe_top, coe_length, Order.krullDim_eq_top_iff,
      ← not_finiteDimensionalOrder_iff]

Goal state after:
Module: Mathlib.RingTheory.Length

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
⊢ length R ↥⊤ = length R M

Tactic:
by
  rw [Module.length_submodule, Module.length_eq_height]

Goal state after: