mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Polynomial.GroupRingAction

Goal state before:
pp: G : Type u_2
inst✝³ : Group G
inst✝² : Fintype G
R : Type u_3
inst✝¹ : CommRing R
inst✝ : MulSemiringAction G R
x : R
g : G
this : DecidableEq R := Classical.decEq R
g' : G ⧸ stabilizer G x
⊢ g • (X - C (ofQuotientStabilizer G x g')) = X - C (ofQuotientStabilizer G x (g • g'))

Tactic:
by
      rw [ofQuotientStabilizer_smul, smul_sub, Polynomial.smul_X, Polynomial.smul_C]

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

Goal state before:
pp: R : Type u
L : Type v
M : Type w
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : Nontrivial M
inst✝ : IsNilpotent L M
⊢ Nontrivial ↥(lowerCentralSeriesLast R L M)

Tactic:
rw [LieSubmodule.nontrivial_iff_ne_bot, lowerCentralSeriesLast]

Goal state after:
R : Type u
L : Type v
M : Type w
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : Nontrivial M
inst✝ : IsNilpotent L M
⊢ (match nilpotencyLength L M with
    | 0 => ⊥
    | k.succ => lowerCentralSeries R L M k) ≠
    ⊥
Module: Mathlib.Algebra.Lie.Nilpotent

Goal state before:
pp: case succ
R : Type u
L : Type v
M : Type w
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✝ : IsNilpotent L M
k : ℕ
h : 1 < k + 1
hk : nilpotencyLength L M = k + 1
⊢ (match k + 1 with
    | 0 => ⊥
    | k.succ => lowerCentralSeries R L M k) ≤
    lowerCentralSeries R L M 1

Tactic:
· exact antitone_lowerCentralSeries _ _ _ (Nat.le_of_lt_succ h)

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

Goal state before:
pp: R : Type u
L : Type v
M : Type w
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
⊢ IsNilpotent (↥(toEnd R L M).range) M ↔ IsNilpotent L M

Tactic:
simp only [isNilpotent_iff R _ M]

Goal state after:
R : Type u
L : Type v
M : Type w
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, lowerCentralSeries R (↥(toEnd R L M).range) M k = ⊥) ↔ ∃ k, lowerCentralSeries R L M k = ⊥
Module: Mathlib.FieldTheory.Finiteness

Goal state before:
pp: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : IsNoetherian K V
⊢ Set.range ⇑(finsetBasis K V) = Basis.ofVectorSpaceIndex K V

Tactic:
finsetBasis,

Goal state after:
K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : IsNoetherian K V
⊢ Set.range ⇑((Basis.ofVectorSpace K V).reindex (⋯.mpr (Equiv.refl ↑(Basis.ofVectorSpaceIndex K V)))) =
    Basis.ofVectorSpaceIndex K V
Module: Mathlib.Algebra.Lie.Nilpotent

Goal state before:
pp: case succ
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
I : LieIdeal R L
k : ℕ
ih : ↑(lowerCentralSeries R (↥I) (↥I) k) ≤ ↑(lowerCentralSeries R L (↥I) k)
y z : ↥I
hz : z ∈ lowerCentralSeries R (↥I) (↥I) k
⊢ ⁅y, z⁆ ∈ {x | ∃ x_1 n, ⁅↑x_1, ↑n⁆ = x}

Tactic:
exact ⟨⟨y.val, LieSubmodule.mem_top _⟩, ⟨z, ih hz⟩, rfl⟩

Goal state after:
Module: Mathlib.FieldTheory.Normal.Defs

Goal state before:
pp: F : Type u_1
K : Type u_2
inst✝¹⁶ : Field F
inst✝¹⁵ : Field K
inst✝¹⁴ : Algebra F K
K₁ : Type u_3
K₂ : Type u_4
K₃ : Type u_5
inst✝¹³ : Field K₁
inst✝¹² : Field K₂
inst✝¹¹ : Field K₃
inst✝¹⁰ : Algebra F K₁
inst✝⁹ : Algebra F K₂
inst✝⁸ : Algebra F K₃
ϕ : K₁ →ₐ[F] K₂
χ : K₁ ≃ₐ[F] K₂
ψ : K₂ →ₐ[F] K₃
ω : K₂ ≃ₐ[F] K₃
E : Type u_6
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
inst✝⁵ : Algebra E K₁
inst✝⁴ : Algebra E K₂
inst✝³ : Algebra E K₃
inst✝² : IsScalarTower F E K₁
inst✝¹ : IsScalarTower F E K₂
inst✝ : IsScalarTower F E K₃
h : Normal F E
x✝ : ↥(toAlgHom F E K₁).range
x : K₂
y : K₁
hx : ↑ϕ y = x
z : E
hy : (toAlgHom F E K₁).toRingHom z = y
⊢ (aeval (↑ϕ ((toAlgHom F E K₁).toRingHom z))) (map (algebraMap F E) (minpoly F z)) = 0

Tactic:
simp [aeval_algHom_apply]

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

Goal state before:
pp: k : Type u
inst✝² : Field k
k' : Type u
inst✝¹ : Field k'
e : k ≃+* k'
inst✝ : IsAlgClosed k
p : k'[X]
hmp : p.Monic
hp : Irreducible p
⊢ (map e.symm.toRingHom p).degree ≠ 0

Tactic:
degree_map

Goal state after:
k : Type u
inst✝² : Field k
k' : Type u
inst✝¹ : Field k'
e : k ≃+* k'
inst✝ : IsAlgClosed k
p : k'[X]
hmp : p.Monic
hp : Irreducible p
⊢ p.degree ≠ 0
Module: Mathlib.FieldTheory.IntermediateField.Adjoin.Basic

Goal state before:
pp: K : Type u_3
L : Type u_4
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
E1 E2 : IntermediateField K L
halg : Algebra.IsAlgebraic K ↥E1 ∨ Algebra.IsAlgebraic K ↥E2
this :
  Module.rank K ↥(E1.toSubalgebra ⊔ E2.toSubalgebra) ≤ Module.rank K ↥E1.toSubalgebra * Module.rank K ↥E2.toSubalgebra
⊢ Module.rank K ↥(E1 ⊔ E2) ≤ Module.rank K ↥E1 * Module.rank K ↥E2

Tactic:
← sup_toSubalgebra_of_isAlgebraic E1 E2 halg

Goal state after:
K : Type u_3
L : Type u_4
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
E1 E2 : IntermediateField K L
halg : Algebra.IsAlgebraic K ↥E1 ∨ Algebra.IsAlgebraic K ↥E2
this : Module.rank K ↥(E1 ⊔ E2).toSubalgebra ≤ Module.rank K ↥E1.toSubalgebra * Module.rank K ↥E2.toSubalgebra
⊢ Module.rank K ↥(E1 ⊔ E2) ≤ Module.rank K ↥E1 * Module.rank K ↥E2
Module: Mathlib.FieldTheory.IsAlgClosed.Basic

Goal state before:
pp: k : Type u
inst✝¹⁶ : Field k
K : Type u
inst✝¹⁵ : Field K
L : Type v
M : Type w
inst✝¹⁴ : Field L
inst✝¹³ : Algebra K L
inst✝¹² : Field M
inst✝¹¹ : Algebra K M
inst✝¹⁰ : IsAlgClosed M
inst✝⁹ : Algebra.IsAlgebraic K L
R : Type u
inst✝⁸ : CommRing R
inst✝⁷ : IsDomain R
S : Type v
inst✝⁶ : CommRing S
inst✝⁵ : IsDomain S
inst✝⁴ : Algebra R S
inst✝³ : Algebra R M
inst✝² : IsTorsionFree R S
inst✝¹ : IsTorsionFree R M
inst✝ : Algebra.IsAlgebraic R S
this✝¹ : IsDomain R := Function.Injective.isDomain (algebraMap R S) (FaithfulSMul.algebraMap_injective R S)
this✝ : Algebra (FractionRing R) M := FractionRing.liftAlgebra R M
this : Algebra (FractionRing R) (FractionRing S) := FractionRing.liftAlgebra R (FractionRing S)
⊢ S →ₐ[R] M

Tactic:
have := FractionRing.isScalarTower_liftAlgebra R M

Goal state after:
k : Type u
inst✝¹⁶ : Field k
K : Type u
inst✝¹⁵ : Field K
L : Type v
M : Type w
inst✝¹⁴ : Field L
inst✝¹³ : Algebra K L
inst✝¹² : Field M
inst✝¹¹ : Algebra K M
inst✝¹⁰ : IsAlgClosed M
inst✝⁹ : Algebra.IsAlgebraic K L
R : Type u
inst✝⁸ : CommRing R
inst✝⁷ : IsDomain R
S : Type v
inst✝⁶ : CommRing S
inst✝⁵ : IsDomain S
inst✝⁴ : Algebra R S
inst✝³ : Algebra R M
inst✝² : IsTorsionFree R S
inst✝¹ : IsTorsionFree R M
inst✝ : Algebra.IsAlgebraic R S
this✝² : IsDomain R := Function.Injective.isDomain (algebraMap R S) (FaithfulSMul.algebraMap_injective R S)
this✝¹ : Algebra (FractionRing R) M := FractionRing.liftAlgebra R M
this✝ : Algebra (FractionRing R) (FractionRing S) := FractionRing.liftAlgebra R (FractionRing S)
this : IsScalarTower R (FractionRing R) M
⊢ S →ₐ[R] M
Module: Mathlib.FieldTheory.IsAlgClosed.Basic

Goal state before:
pp: F : Type u_1
F' : Type u_2
E : Type u_3
inst✝⁶ : Field F
inst✝⁵ : Field F'
inst✝⁴ : Field E
inst✝³ : Algebra F F'
inst✝² : Algebra F E
inst✝¹ : Algebra.IsAlgebraic F E
inst✝ : IsAlgClosed F'
h : finrank F F' = 2
emb : E →ₐ[F] F'
e : E ≃ₐ[F] ↥emb.range
this : IsSimpleOrder (Subalgebra F F')
⊢ Nonempty (E ≃ₐ[F] F) ∨ Nonempty (E ≃ₐ[F] F')

Tactic:
obtain h | h := IsSimpleOrder.eq_bot_or_eq_top emb.range

Goal state after:
case inl
F : Type u_1
F' : Type u_2
E : Type u_3
inst✝⁶ : Field F
inst✝⁵ : Field F'
inst✝⁴ : Field E
inst✝³ : Algebra F F'
inst✝² : Algebra F E
inst✝¹ : Algebra.IsAlgebraic F E
inst✝ : IsAlgClosed F'
h✝ : finrank F F' = 2
emb : E →ₐ[F] F'
e : E ≃ₐ[F] ↥emb.range
this : IsSimpleOrder (Subalgebra F F')
h : emb.range = ⊥
⊢ Nonempty (E ≃ₐ[F] F) ∨ Nonempty (E ≃ₐ[F] F')

case inr
F : Type u_1
F' : Type u_2
E : Type u_3
inst✝⁶ : Field F
inst✝⁵ : Field F'
inst✝⁴ : Field E
inst✝³ : Algebra F F'
inst✝² : Algebra F E
inst✝¹ : Algebra.IsAlgebraic F E
inst✝ : IsAlgClosed F'
h✝ : finrank F F' = 2
emb : E →ₐ[F] F'
e : E ≃ₐ[F] ↥emb.range
this : IsSimpleOrder (Subalgebra F F')
h : emb.range = ⊤
⊢ Nonempty (E ≃ₐ[F] F) ∨ Nonempty (E ≃ₐ[F] F')
Module: Mathlib.LinearAlgebra.Eigenspace.Pi

Goal state before:
pp: ι : Type u_1
R : Type u_2
M : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
f : ι → End R M
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M
h : ∀ (l : ι) (χ : ι → R), MapsTo (⇑(f l)) (⨅ i, ↑((f i).maxGenEigenspace (χ i))) (⨅ i, ↑((f i).maxGenEigenspace (χ i)))
χ₁ : ι → R
⊢ ∀ (s : Finset (ι → R)),
    χ₁ ∉ s → Disjoint (⨅ i, (f i).maxGenEigenspace (χ₁ i)) (s.sup fun χ ↦ ⨅ i, (f i).maxGenEigenspace (χ i))

Tactic:
s

Goal state after:
ι : Type u_1
R : Type u_2
M : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
f : ι → End R M
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M
h : ∀ (l : ι) (χ : ι → R), MapsTo (⇑(f l)) (⨅ i, ↑((f i).maxGenEigenspace (χ i))) (⨅ i, ↑((f i).maxGenEigenspace (χ i)))
χ₁ : ι → R
s : Finset (ι → R)
⊢ χ₁ ∉ s → Disjoint (⨅ i, (f i).maxGenEigenspace (χ₁ i)) (s.sup fun χ ↦ ⨅ i, (f i).maxGenEigenspace (χ i))
Module: Mathlib.FieldTheory.Extension

Goal state before:
pp: F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Field K
inst✝¹ : Algebra F E
inst✝ : Algebra F K
x : Lifts F E K
s : E
h1 : IsIntegral (↥x.carrier) s
h2 : (Polynomial.map x.emb.toRingHom (minpoly (↥x.carrier) s)).Splits
I2 : (minpoly (↥x.carrier) s).degree ≠ 0
this✝ : Algebra (↥x.carrier) K := x.emb.toAlgebra
carrier : IntermediateField F E := restrictScalars F (↥x.carrier)⟮s⟯
this : Algebra ↥x.carrier ↥carrier := (↥x.carrier)⟮s⟯.algebra
⊢ (Polynomial.map x.emb.toRingHom (minpoly (↥x.carrier) s)).degree ≠ 0

Tactic:
degree_map

Goal state after:
F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Field K
inst✝¹ : Algebra F E
inst✝ : Algebra F K
x : Lifts F E K
s : E
h1 : IsIntegral (↥x.carrier) s
h2 : (Polynomial.map x.emb.toRingHom (minpoly (↥x.carrier) s)).Splits
I2 : (minpoly (↥x.carrier) s).degree ≠ 0
this✝ : Algebra (↥x.carrier) K := x.emb.toAlgebra
carrier : IntermediateField F E := restrictScalars F (↥x.carrier)⟮s⟯
this : Algebra ↥x.carrier ↥carrier := (↥x.carrier)⟮s⟯.algebra
⊢ (minpoly (↥x.carrier) s).degree ≠ 0
Module: Mathlib.Algebra.Lie.Weights.Basic

Goal state before:
pp: case h.inr
R : Type u_2
L : Type u_3
inst✝¹⁴ : CommRing R
inst✝¹³ : LieRing L
inst✝¹² : LieAlgebra R L
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
inst✝¹¹ : AddCommGroup M₁
inst✝¹⁰ : Module R M₁
inst✝⁹ : LieRingModule L M₁
inst✝⁸ : LieModule R L M₁
inst✝⁷ : AddCommGroup M₂
inst✝⁶ : Module R M₂
inst✝⁵ : LieRingModule L M₂
inst✝⁴ : LieModule R L M₂
inst✝³ : AddCommGroup M₃
inst✝² : Module R M₃
inst✝¹ : LieRingModule L M₃
inst✝ : LieModule R L M₃
g : M₁ ⊗[R] M₂ →ₗ⁅R,L⁆ M₃
χ₁ χ₂ : R
x : L
t : ↥𝕎(M₁, χ₁, x) ⊗[R] ↥𝕎(M₂, χ₂, x)
F : Module.End R M₃ := (toEnd R L M₃) x - (χ₁ + χ₂) • 1
m₁ : M₁
m₂ : M₂
f₁ : Module.End R (M₁ ⊗[R] M₂) := LinearMap.rTensor M₂ ((toEnd R L M₁) x - χ₁ • 1)
f₂ : Module.End R (M₁ ⊗[R] M₂) := LinearMap.lTensor M₁ ((toEnd R L M₂) x - χ₂ • 1)
h_comm_square : F ∘ₗ ↑g = ↑g ∘ₗ (f₁ + f₂)
k₁ : ℕ
hk₁ : (((toEnd R L M₁) x - χ₁ • 1) ^ k₁) m₁ = 0
k₂ : ℕ
hk₂ : (((toEnd R L M₂) x - χ₂ • 1) ^ k₂) m₂ = 0
hf₁ : (f₁ ^ k₁) (m₁ ⊗ₜ[R] m₂) = 0
hf₂ : (f₂ ^ k₂) (m₁ ⊗ₜ[R] m₂) = 0
hf_comm : Commute f₁ f₂
i j : ℕ
hij : (i, j) ∈ Finset.antidiagonal (k₁ + k₂ - 1)
hj : k₂ ≤ (i, j).2
⊢ (f₁ ^ i * f₂ ^ j) (m₁ ⊗ₜ[R] m₂) = 0

Tactic:
rw [Module.End.mul_apply, Module.End.pow_map_zero_of_le hj hf₂, map_zero]

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

Goal state before:
pp: K : Type u_1
R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : LieRing L
inst✝⁵ : LieAlgebra R L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
inst✝¹ : LieModule R L M
inst✝ : LieRing.IsNilpotent L
x : L
⊢ ∀ {x_1 : L} {m : M},
    m ∈ (⨅ k, ((toEnd R L M) x ^ k).range).carrier → ⁅x_1, m⁆ ∈ (⨅ k, ((toEnd R L M) x ^ k).range).carrier

Tactic:
by
      set φ := toEnd R L M x
      intro y m hm
      simp only [AddSubsemigroup.mem_carrier, AddSubmonoid.mem_toSubsemigroup,
        Submodule.mem_toAddSubmonoid, Submodule.mem_iInf, LinearMap.mem_range] at hm ⊢
      intro k
      obtain ⟨N, hN⟩ := LieAlgebra.nilpotent_ad_of_nilpotent_algebra R L
      obtain ⟨m, rfl⟩ := hm (N + k)
      let f₁ : Module.End R (L ⊗[R] M) := (LieAlgebra.ad R L x).rTensor M
      let f₂ : Module.End R (L ⊗[R] M) := φ.lTensor L
      replace hN : f₁ ^ N = 0 := by ext; simp [f₁, hN]
      have h₁ : Commute f₁ f₂ := by ext; simp [f₁, f₂]
      have h₂ : φ ∘ₗ toModuleHom R L M = toModuleHom R L M ∘ₗ (f₁ + f₂) := by ext; simp [φ, f₁, f₂]
      obtain ⟨q, hq⟩ := h₁.add_pow_dvd_pow_of_pow_eq_zero_right (N + k).le_succ hN
      use toModuleHom R L M (q (y ⊗ₜ m))
      change (φ ^ k).comp ((toModuleHom R L M : L ⊗[R] M →ₗ[R] M)) _ = _
      simp [φ, f₁, f₂, Module.End.commute_pow_left_of_commute h₂,
        LinearMap.comp_apply (g := (f₁ + f₂) ^ k), ← LinearMap.comp_apply (g := q),
        ← Module.End.mul_eq_comp, ← hq]

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

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
B : BilinForm R M
r : S
x y : M
⊢ (B x) ((algebraMap S R) r • y) = r • (B x) y

Tactic:
smul_right,

Goal state after:
R : Type u_1
M : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
B : BilinForm R M
r : S
x y : M
⊢ (algebraMap S R) r * (B x) y = r • (B x) y
Module: Mathlib.LinearAlgebra.BilinearForm.Hom

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
B : BilinForm R M
ι : Type u_9
b : Basis ι R M
x y : M
⊢ ∑ a ∈ (b.repr x).support, ∑ x_1 ∈ (b.repr y).support, (b.repr x) a • (b.repr y) x_1 • (B (b a)) (b x_1) =
    ∑ x_1 ∈ (b.repr x).support, ∑ x_2 ∈ (b.repr y).support, (b.repr x) x_1 * (B (b x_1)) ((b.repr y) x_2 • b x_2)

Tactic:
smul_right,

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
B : BilinForm R M
ι : Type u_9
b : Basis ι R M
x y : M
⊢ ∑ a ∈ (b.repr x).support, ∑ x_1 ∈ (b.repr y).support, (b.repr x) a • (b.repr y) x_1 • (B (b a)) (b x_1) =
    ∑ x_1 ∈ (b.repr x).support, ∑ x_2 ∈ (b.repr y).support, (b.repr x) x_1 * ((b.repr y) x_2 * (B (b x_1)) (b x_2))
Module: Mathlib.LinearAlgebra.BilinearForm.Orthogonal

Goal state before:
pp: V : Type u_5
K : Type u_6
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
B : BilinForm K V
hB : B.Nondegenerate
hB₀ : B.IsRefl
W : Submodule K V
⊢ B.orthogonal (B.orthogonal W) = W

Tactic:
apply (eq_of_le_of_finrank_le (LinearMap.BilinForm.le_orthogonal_orthogonal hB₀) _).symm

Goal state after:
V : Type u_5
K : Type u_6
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
B : BilinForm K V
hB : B.Nondegenerate
hB₀ : B.IsRefl
W : Submodule K V
⊢ finrank K ↥(B.orthogonal (B.orthogonal W)) ≤ finrank K ↥W
Module: Mathlib.Algebra.Lie.Semisimple.Basic

Goal state before:
pp: case right
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
I : LieIdeal R L
hI : IsAtom I
J : LieIdeal R ↥I
J' : LieIdeal R L :=
  let __spread.0 := Submodule.map ↑I.incl ↑J;
  { toSubmodule := __spread.0, lie_mem := ⋯ }
hJ : J' = I
⊢ ⊤ ≤ J

Tactic:
rintro ⟨x, hx⟩ -

Goal state after:
case right
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
I : LieIdeal R L
hI : IsAtom I
J : LieIdeal R ↥I
J' : LieIdeal R L :=
  let __spread.0 := Submodule.map ↑I.incl ↑J;
  { toSubmodule := __spread.0, lie_mem := ⋯ }
hJ : J' = I
x : L
hx : x ∈ I
⊢ ⟨x, hx⟩ ∈ J
Module: Mathlib.Data.Multiset.Fintype

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
m : Multiset α
⊢ map (fun x ↦ x.fst) Finset.univ.val = m

Tactic:
have := m.map_toEnumFinset_fst
  rw [← m.map_univ_coeEmbedding] at this
  simpa only [Finset.map_val, Multiset.coeEmbedding_apply, Multiset.map_map,
    Function.comp_apply] using this

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
χ₁ χ₂ : L → R
inst✝³ : IsAddTorsionFree R
inst✝² : IsDomain R
inst✝¹ : IsTorsionFree R M
inst✝ : IsNoetherian R M
hχ₁ : χ₁ ≠ 0
f : ℕ → L → R := fun k ↦ k • χ₁ + χ₂
⊢ ∀ᶠ (k : ℕ) in Filter.atTop, genWeightSpace M (k • χ₁ + χ₂) = ⊥

Tactic:
suffices Injective f by
    rw [← Nat.cofinite_eq_atTop, Filter.eventually_cofinite, ← finite_image_iff this.injOn]
    apply (finite_genWeightSpace_ne_bot R L M).subset
    simp [f]

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
χ₁ χ₂ : L → R
inst✝³ : IsAddTorsionFree R
inst✝² : IsDomain R
inst✝¹ : IsTorsionFree R M
inst✝ : IsNoetherian R M
hχ₁ : χ₁ ≠ 0
f : ℕ → L → R := fun k ↦ k • χ₁ + χ₂
⊢ Injective f
Module: Mathlib.Algebra.Lie.TraceForm

Goal state before:
pp: case zero
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
x y : L
hx : x ∈ ⊤.lcs L 0
hy : y ∈ LieSubmodule.ucs 0 ⊥
⊢ ((traceForm R L M) x) y = 0

Tactic:
replace hy : y = 0 := by simpa using hy

Goal state after:
case zero
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
x y : L
hx : x ∈ ⊤.lcs L 0
hy : y = 0
⊢ ((traceForm R L M) x) y = 0
Module: Mathlib.Algebra.Lie.Weights.Killing

Goal state before:
pp: K : Type u_2
L : Type u_3
inst✝⁵ : LieRing L
inst✝⁴ : Field K
inst✝³ : LieAlgebra K L
inst✝² : FiniteDimensional K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsKilling K L
x : ↥H
hx : x ∈ corootSpace 0
⊢ {x | ∃ y ∈ H, ∃ z ∈ H, ⁅y, z⁆ = x} = {0}

Tactic:
refine eq_singleton_iff_unique_mem.mpr ⟨⟨0, H.zero_mem, 0, H.zero_mem, zero_lie 0⟩, ?_⟩

Goal state after:
K : Type u_2
L : Type u_3
inst✝⁵ : LieRing L
inst✝⁴ : Field K
inst✝³ : LieAlgebra K L
inst✝² : FiniteDimensional K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsKilling K L
x : ↥H
hx : x ∈ corootSpace 0
⊢ ∀ x ∈ {x | ∃ y ∈ H, ∃ z ∈ H, ⁅y, z⁆ = x}, x = 0
Module: Mathlib.Algebra.Polynomial.Sequence

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
S : Sequence R
inst✝ : IsDomain R
s : Finset ℕ
g : ℕ → R
eqzero : ∑ i ∈ s, g i • ↑S i = 0
i : ℕ
hi : i ∈ s
⊢ g i = 0

Tactic:
by
  by_cases hsupzero : s.sup (fun i ↦ (g i • S i).degree) = ⊥
  · have le_sup := Finset.le_sup hi (f := fun i ↦ (g i • S i).degree)
    exact (smul_eq_zero_iff_left (S.ne_zero i)).mp <| degree_eq_bot.mp (eq_bot_mono le_sup hsupzero)
  have hpairwise : {i | i ∈ s ∧ g i • S i ≠ 0}.Pairwise (Ne on fun i ↦ (g i • S i).degree) := by
    intro x ⟨_, hx⟩ y ⟨_, hy⟩ xney
    have zgx : g x ≠ 0 := (smul_ne_zero_iff.mp hx).1
    have zgy : g y ≠ 0 := (smul_ne_zero_iff.mp hy).1
    have rx : IsRightRegular (S x).leadingCoeff := IsRegular.of_ne_zero (by simp) |>.right
    have ry : IsRightRegular (S y).leadingCoeff := IsRegular.of_ne_zero (by simp) |>.right
    simp [degree_smul_of_isRightRegular_leadingCoeff, rx, ry, zgx, zgy, xney]
  obtain ⟨n, hn⟩ : ∃ n, (s.sup fun i ↦ (g i • S i).degree) = n := exists_eq'
  refine degree_ne_bot.mp ?_ eqzero |>.elim
  have hsum := degree_sum_eq_of_disjoint _ s hpairwise
  exact hsum.trans hn |>.trans_ne <| (ne_of_ne_of_eq (hsupzero ·.symm) hn).symm

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

Goal state before:
pp: R : Type u_1
L : Type u_3
M : Type u_4
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 : LieRing.IsNilpotent L
inst✝² : IsDomain R
inst✝¹ : Free R M
inst✝ : Module.Finite R M
z : L
hz : z ∈ lowerCentralSeries R L L 1
hzc : z ∈ LieAlgebra.center R L
x : L
A : Type u_1 := AlgebraicClosure (FractionRing R)
⊢ 1 ⊗ₜ[R] z ∈ lowerCentralSeries A (A ⊗[R] L) (A ⊗[R] L) 1

Tactic:
simp only [lowerCentralSeries_succ, lowerCentralSeries_zero] at hz ⊢

Goal state after:
R : Type u_1
L : Type u_3
M : Type u_4
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 : LieRing.IsNilpotent L
inst✝² : IsDomain R
inst✝¹ : Free R M
inst✝ : Module.Finite R M
z : L
hzc : z ∈ LieAlgebra.center R L
x : L
A : Type u_1 := AlgebraicClosure (FractionRing R)
hz : z ∈ ⁅⊤, ⊤⁆
⊢ 1 ⊗ₜ[R] z ∈ ⁅⊤, ⊤⁆
Module: Mathlib.Algebra.Lie.TraceForm

Goal state before:
pp: R : Type u_1
L : Type u_3
M : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
inst✝⁴ : Free R M
inst✝³ : Module.Finite R M
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
N : LieSubmodule R L M
I : LieIdeal R L
h : I ≤ N.idealizer
x y : L
hy : y ∈ I
inst✝ : LieModule.IsTrivial ↥I ↥N
hy' : ∀ m ∈ N, (φ x ∘ₗ φ y) m ∈ N := fun m x_1 ↦ N.lie_mem ((mem_idealizer N).mp (h hy) m)
this : (φ x ∘ₗ φ y).restrict hy' = 0
⊢ (trace R M) (φ x ∘ₗ φ y) = 0

Tactic:
simp [this, N.trace_eq_trace_restrict_of_le_idealizer I h x hy]

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

Goal state before:
pp: K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁰ : LieRing L
inst✝⁹ : AddCommGroup M
inst✝⁸ : LieRingModule L M
inst✝⁷ : Field K
inst✝⁶ : LieAlgebra K L
inst✝⁵ : Module K M
inst✝⁴ : LieModule K L M
inst✝³ : FiniteDimensional K M
inst✝² : LieRing.IsNilpotent L
inst✝¹ : LinearWeights K L M
inst✝ : IsTriangularizable K L M
x y : L
⊢ ((traceForm K L M) x) y = ∑ χ, finrank K ↥(genWeightSpace M ⇑χ) • (χ x * χ y)

Tactic:
have hxy : ∀ χ : Weight K L M, MapsTo (toEnd K L M x ∘ₗ toEnd K L M y)
      (genWeightSpace M χ) (genWeightSpace M χ) :=
    fun χ m hm ↦ LieSubmodule.lie_mem _ <| LieSubmodule.lie_mem _ hm

Goal state after:
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁰ : LieRing L
inst✝⁹ : AddCommGroup M
inst✝⁸ : LieRingModule L M
inst✝⁷ : Field K
inst✝⁶ : LieAlgebra K L
inst✝⁵ : Module K M
inst✝⁴ : LieModule K L M
inst✝³ : FiniteDimensional K M
inst✝² : LieRing.IsNilpotent L
inst✝¹ : LinearWeights K L M
inst✝ : IsTriangularizable K L M
x y : L
hxy : ∀ (χ : Weight K L M), MapsTo ⇑((toEnd K L M) x ∘ₗ (toEnd K L M) y) ↑(genWeightSpace M ⇑χ) ↑(genWeightSpace M ⇑χ)
⊢ ((traceForm K L M) x) y = ∑ χ, finrank K ↥(genWeightSpace M ⇑χ) • (χ x * χ y)