ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MvPolynomial.eval_neg
theorem eval_neg (f : σ → R) : eval f (-p) = -eval f p 
Arguments: {R : Type u} {σ : Type u_1} [CommRing R] (p : MvPolynomial σ R) (f : σ → R)
Goal: (MvPolynomial.eval f) (-p) = -(MvPolynomial.eval f) p
-- MvPolynomial.hom_C
theorem hom_C (f : MvPolynomial σ ℤ →+* S) (n : ℤ) : f (C n) = (n : S) 
Arguments: {S : Type v} {σ : Type u_1} [CommRing S] (f : MvPolynomial σ ℤ →+* S) (n : ℤ)
Goal: f (MvPolynomial.C n) = ↑n
-- MvPolynomial.eval₂Hom_X
@[simp]
theorem eval₂Hom_X {R : Type u} (c : ℤ →+* S) (f : MvPolynomial R ℤ →+* S) (x : MvPolynomial R ℤ) :
    eval₂ c (f ∘ X) x = f x 
Arguments: {S : Type v} [CommRing S] {R : Type u} (c : ℤ →+* S) (f : MvPolynomial R ℤ →+* S) (x : MvPolynomial R ℤ)
Goal: MvPolynomial.eval₂ c (⇑f ∘ MvPolynomial.X) x = f x
-- MvPolynomial.degreeOf_sub_lt
theorem degreeOf_sub_lt {x : σ} {f g : MvPolynomial σ R} {k : ℕ} (h : 0 < k)
    (hf : ∀ m : σ →₀ ℕ, m ∈ f.support → k ≤ m x → coeff m f = coeff m g)
    (hg : ∀ m : σ →₀ ℕ, m ∈ g.support → k ≤ m x → coeff m f = coeff m g) :
    degreeOf x (f - g) < k 
Arguments: {R : Type u} {σ : Type u_1} [CommRing R] {x : σ} {f g : MvPolynomial σ R} {k : ℕ} (h : 0 < k) (hf : ∀ m ∈ f.support, k ≤ m x → MvPolynomial.coeff m f = MvPolynomial.coeff m g) (hg : ∀ m ∈ g.support, k ≤ m x → MvPolynomial.coeff m f = MvPolynomial.coeff m g)
Goal: MvPolynomial.degreeOf x (f - g) < k
-- MvPolynomial.totalDegree_neg
@[simp]
theorem totalDegree_neg (a : MvPolynomial σ R) : (-a).totalDegree = a.totalDegree 
Arguments: {R : Type u} {σ : Type u_1} [CommRing R] (a : MvPolynomial σ R)
Goal: (-a).totalDegree = a.totalDegree
-- MvPolynomial.totalDegree_sub
theorem totalDegree_sub (a b : MvPolynomial σ R) :
    (a - b).totalDegree ≤ max a.totalDegree b.totalDegree 
Arguments: {R : Type u} {σ : Type u_1} [CommRing R] (a b : MvPolynomial σ R)
Goal: (a - b).totalDegree ≤ max a.totalDegree b.totalDegree
-- MvPolynomial.totalDegree_sub_C_le
theorem totalDegree_sub_C_le (p : MvPolynomial σ R) (r : R) :
    totalDegree (p - C r) ≤ totalDegree p 
Arguments: {R : Type u} {σ : Type u_1} [CommRing R] (p : MvPolynomial σ R) (r : R)
Goal: (p - MvPolynomial.C r).totalDegree ≤ p.totalDegree
-- MvPolynomial.coeff_divMonomial
@[simp]
theorem coeff_divMonomial (s : σ →₀ ℕ) (x : MvPolynomial σ R) (s' : σ →₀ ℕ) :
    coeff s' (x /ᵐᵒⁿᵒᵐⁱᵃˡ s) = coeff (s + s') x 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (s : σ →₀ ℕ) (x : MvPolynomial σ R) (s' : σ →₀ ℕ)
Goal: MvPolynomial.coeff s' (x.divMonomial s) = MvPolynomial.coeff (s + s') x
-- MvPolynomial.support_divMonomial
@[simp]
theorem support_divMonomial (s : σ →₀ ℕ) (x : MvPolynomial σ R) :
    (x /ᵐᵒⁿᵒᵐⁱᵃˡ s).support = x.support.preimage _ (add_right_injective s).injOn 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (s : σ →₀ ℕ) (x : MvPolynomial σ R)
Goal: (x.divMonomial s).support = x.support.preimage (fun x => s + x) ⋯
-- MvPolynomial.zero_divMonomial
@[simp]
theorem zero_divMonomial (s : σ →₀ ℕ) : (0 : MvPolynomial σ R) /ᵐᵒⁿᵒᵐⁱᵃˡ s = 0 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (s : σ →₀ ℕ)
Goal: MvPolynomial.divMonomial 0 s = 0
-- MvPolynomial.divMonomial_zero
theorem divMonomial_zero (x : MvPolynomial σ R) : x /ᵐᵒⁿᵒᵐⁱᵃˡ 0 = x 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (x : MvPolynomial σ R)
Goal: x.divMonomial 0 = x
-- MvPolynomial.add_divMonomial
theorem add_divMonomial (x y : MvPolynomial σ R) (s : σ →₀ ℕ) :
    (x + y) /ᵐᵒⁿᵒᵐⁱᵃˡ s = x /ᵐᵒⁿᵒᵐⁱᵃˡ s + y /ᵐᵒⁿᵒᵐⁱᵃˡ s 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (x y : MvPolynomial σ R) (s : σ →₀ ℕ)
Goal: (x + y).divMonomial s = x.divMonomial s + y.divMonomial s
-- MvPolynomial.divMonomial_add
theorem divMonomial_add (a b : σ →₀ ℕ) (x : MvPolynomial σ R) :
    x /ᵐᵒⁿᵒᵐⁱᵃˡ (a + b) = x /ᵐᵒⁿᵒᵐⁱᵃˡ a /ᵐᵒⁿᵒᵐⁱᵃˡ b 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (a b : σ →₀ ℕ) (x : MvPolynomial σ R)
Goal: x.divMonomial (a + b) = (x.divMonomial a).divMonomial b
-- MvPolynomial.divMonomial_monomial_mul
@[simp]
theorem divMonomial_monomial_mul (a : σ →₀ ℕ) (x : MvPolynomial σ R) :
    monomial a 1 * x /ᵐᵒⁿᵒᵐⁱᵃˡ a = x 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (a : σ →₀ ℕ) (x : MvPolynomial σ R)
Goal: ((MvPolynomial.monomial a) 1 * x).divMonomial a = x
-- MvPolynomial.divMonomial_mul_monomial
@[simp]
theorem divMonomial_mul_monomial (a : σ →₀ ℕ) (x : MvPolynomial σ R) :
    x * monomial a 1 /ᵐᵒⁿᵒᵐⁱᵃˡ a = x 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (a : σ →₀ ℕ) (x : MvPolynomial σ R)
Goal: (x * (MvPolynomial.monomial a) 1).divMonomial a = x
-- MvPolynomial.divMonomial_monomial
@[simp]
theorem divMonomial_monomial (a : σ →₀ ℕ) : monomial a 1 /ᵐᵒⁿᵒᵐⁱᵃˡ a = (1 : MvPolynomial σ R) 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (a : σ →₀ ℕ)
Goal: ((MvPolynomial.monomial a) 1).divMonomial a = 1
-- MvPolynomial.coeff_modMonomial_of_not_le
@[simp]
theorem coeff_modMonomial_of_not_le {s' s : σ →₀ ℕ} (x : MvPolynomial σ R) (h : ¬s ≤ s') :
    coeff s' (x %ᵐᵒⁿᵒᵐⁱᵃˡ s) = coeff s' x 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] {s' s : σ →₀ ℕ} (x : MvPolynomial σ R) (h : ¬s ≤ s')
Goal: MvPolynomial.coeff s' (x.modMonomial s) = MvPolynomial.coeff s' x
-- MvPolynomial.coeff_modMonomial_of_le
@[simp]
theorem coeff_modMonomial_of_le {s' s : σ →₀ ℕ} (x : MvPolynomial σ R) (h : s ≤ s') :
    coeff s' (x %ᵐᵒⁿᵒᵐⁱᵃˡ s) = 0 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] {s' s : σ →₀ ℕ} (x : MvPolynomial σ R) (h : s ≤ s')
Goal: MvPolynomial.coeff s' (x.modMonomial s) = 0
-- MvPolynomial.monomial_mul_modMonomial
@[simp]
theorem monomial_mul_modMonomial (s : σ →₀ ℕ) (x : MvPolynomial σ R) :
    monomial s 1 * x %ᵐᵒⁿᵒᵐⁱᵃˡ s = 0 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (s : σ →₀ ℕ) (x : MvPolynomial σ R)
Goal: ((MvPolynomial.monomial s) 1 * x).modMonomial s = 0
-- MvPolynomial.mul_monomial_modMonomial
@[simp]
theorem mul_monomial_modMonomial (s : σ →₀ ℕ) (x : MvPolynomial σ R) :
    x * monomial s 1 %ᵐᵒⁿᵒᵐⁱᵃˡ s = 0 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (s : σ →₀ ℕ) (x : MvPolynomial σ R)
Goal: (x * (MvPolynomial.monomial s) 1).modMonomial s = 0
-- MvPolynomial.monomial_modMonomial
@[simp]
theorem monomial_modMonomial (s : σ →₀ ℕ) : monomial s (1 : R) %ᵐᵒⁿᵒᵐⁱᵃˡ s = 0 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (s : σ →₀ ℕ)
Goal: ((MvPolynomial.monomial s) 1).modMonomial s = 0
-- MvPolynomial.divMonomial_add_modMonomial
theorem divMonomial_add_modMonomial (x : MvPolynomial σ R) (s : σ →₀ ℕ) :
    monomial s 1 * (x /ᵐᵒⁿᵒᵐⁱᵃˡ s) + x %ᵐᵒⁿᵒᵐⁱᵃˡ s = x 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (x : MvPolynomial σ R) (s : σ →₀ ℕ)
Goal: (MvPolynomial.monomial s) 1 * x.divMonomial s + x.modMonomial s = x
-- MvPolynomial.modMonomial_add_divMonomial
theorem modMonomial_add_divMonomial (x : MvPolynomial σ R) (s : σ →₀ ℕ) :
    x %ᵐᵒⁿᵒᵐⁱᵃˡ s + monomial s 1 * (x /ᵐᵒⁿᵒᵐⁱᵃˡ s) = x 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (x : MvPolynomial σ R) (s : σ →₀ ℕ)
Goal: x.modMonomial s + (MvPolynomial.monomial s) 1 * x.divMonomial s = x
-- MvPolynomial.monomial_one_dvd_iff_modMonomial_eq_zero
theorem monomial_one_dvd_iff_modMonomial_eq_zero {i : σ →₀ ℕ} {x : MvPolynomial σ R} :
    monomial i (1 : R) ∣ x ↔ x %ᵐᵒⁿᵒᵐⁱᵃˡ i = 0 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] {i : σ →₀ ℕ} {x : MvPolynomial σ R}
Goal: (MvPolynomial.monomial i) 1 ∣ x ↔ x.modMonomial i = 0
-- MvPolynomial.X_mul_divMonomial
@[simp]
theorem X_mul_divMonomial (i : σ) (x : MvPolynomial σ R) :
    X i * x /ᵐᵒⁿᵒᵐⁱᵃˡ Finsupp.single i 1 = x 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (i : σ) (x : MvPolynomial σ R)
Goal: (MvPolynomial.X i * x).divMonomial (Finsupp.single i 1) = x
-- MvPolynomial.X_divMonomial
@[simp]
theorem X_divMonomial (i : σ) : (X i : MvPolynomial σ R) /ᵐᵒⁿᵒᵐⁱᵃˡ Finsupp.single i 1 = 1 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (i : σ)
Goal: (MvPolynomial.X i).divMonomial (Finsupp.single i 1) = 1
-- MvPolynomial.mul_X_divMonomial
@[simp]
theorem mul_X_divMonomial (x : MvPolynomial σ R) (i : σ) :
    x * X i /ᵐᵒⁿᵒᵐⁱᵃˡ Finsupp.single i 1 = x 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (x : MvPolynomial σ R) (i : σ)
Goal: (x * MvPolynomial.X i).divMonomial (Finsupp.single i 1) = x
-- MvPolynomial.X_mul_modMonomial
@[simp]
theorem X_mul_modMonomial (i : σ) (x : MvPolynomial σ R) :
    X i * x %ᵐᵒⁿᵒᵐⁱᵃˡ Finsupp.single i 1 = 0 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (i : σ) (x : MvPolynomial σ R)
Goal: (MvPolynomial.X i * x).modMonomial (Finsupp.single i 1) = 0
-- MvPolynomial.mul_X_modMonomial
@[simp]
theorem mul_X_modMonomial (x : MvPolynomial σ R) (i : σ) :
    x * X i %ᵐᵒⁿᵒᵐⁱᵃˡ Finsupp.single i 1 = 0 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (x : MvPolynomial σ R) (i : σ)
Goal: (x * MvPolynomial.X i).modMonomial (Finsupp.single i 1) = 0
-- MvPolynomial.modMonomial_X
@[simp]
theorem modMonomial_X (i : σ) : (X i : MvPolynomial σ R) %ᵐᵒⁿᵒᵐⁱᵃˡ Finsupp.single i 1 = 0 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (i : σ)
Goal: (MvPolynomial.X i).modMonomial (Finsupp.single i 1) = 0
-- MvPolynomial.divMonomial_add_modMonomial_single
theorem divMonomial_add_modMonomial_single (x : MvPolynomial σ R) (i : σ) :
    X i * (x /ᵐᵒⁿᵒᵐⁱᵃˡ Finsupp.single i 1) + x %ᵐᵒⁿᵒᵐⁱᵃˡ Finsupp.single i 1 = x 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (x : MvPolynomial σ R) (i : σ)
Goal: MvPolynomial.X i * x.divMonomial (Finsupp.single i 1) + x.modMonomial (Finsupp.single i 1) = x
-- MvPolynomial.modMonomial_add_divMonomial_single
theorem modMonomial_add_divMonomial_single (x : MvPolynomial σ R) (i : σ) :
    x %ᵐᵒⁿᵒᵐⁱᵃˡ Finsupp.single i 1 + X i * (x /ᵐᵒⁿᵒᵐⁱᵃˡ Finsupp.single i 1) = x 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] (x : MvPolynomial σ R) (i : σ)
Goal: x.modMonomial (Finsupp.single i 1) + MvPolynomial.X i * x.divMonomial (Finsupp.single i 1) = x
-- MvPolynomial.X_dvd_iff_modMonomial_eq_zero
theorem X_dvd_iff_modMonomial_eq_zero {i : σ} {x : MvPolynomial σ R} :
    X i ∣ x ↔ x %ᵐᵒⁿᵒᵐⁱᵃˡ Finsupp.single i 1 = 0 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] {i : σ} {x : MvPolynomial σ R}
Goal: MvPolynomial.X i ∣ x ↔ x.modMonomial (Finsupp.single i 1) = 0
-- MvPolynomial.monomial_dvd_monomial
theorem monomial_dvd_monomial {r s : R} {i j : σ →₀ ℕ} :
    monomial i r ∣ monomial j s ↔ (s = 0 ∨ i ≤ j) ∧ r ∣ s 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] {r s : R} {i j : σ →₀ ℕ}
Goal: (MvPolynomial.monomial i) r ∣ (MvPolynomial.monomial j) s ↔ (s = 0 ∨ i ≤ j) ∧ r ∣ s
-- MvPolynomial.monomial_one_dvd_monomial_one
@[simp]
theorem monomial_one_dvd_monomial_one [Nontrivial R] {i j : σ →₀ ℕ} :
    monomial i (1 : R) ∣ monomial j 1 ↔ i ≤ j 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] [Nontrivial R] {i j : σ →₀ ℕ}
Goal: (MvPolynomial.monomial i) 1 ∣ (MvPolynomial.monomial j) 1 ↔ i ≤ j
-- MvPolynomial.X_dvd_X
@[simp]
theorem X_dvd_X [Nontrivial R] {i j : σ} :
    (X i : MvPolynomial σ R) ∣ (X j : MvPolynomial σ R) ↔ i = j 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] [Nontrivial R] {i j : σ}
Goal: MvPolynomial.X i ∣ MvPolynomial.X j ↔ i = j
-- MvPolynomial.X_dvd_monomial
@[simp]
theorem X_dvd_monomial {i : σ} {j : σ →₀ ℕ} {r : R} :
    (X i : MvPolynomial σ R) ∣ monomial j r ↔ r = 0 ∨ j i ≠ 0 
Arguments: {σ : Type u_1} {R : Type u_2} [CommSemiring R] {i : σ} {j : σ →₀ ℕ} {r : R}
Goal: MvPolynomial.X i ∣ (MvPolynomial.monomial j) r ↔ r = 0 ∨ j i ≠ 0
-- LieModule.weight_vector_multiplication
protected theorem weight_vector_multiplication (M₁ M₂ M₃ : Type*)
    [AddCommGroup M₁] [Module R M₁] [LieRingModule L M₁] [LieModule R L M₁] [AddCommGroup M₂]
    [Module R M₂] [LieRingModule L M₂] [LieModule R L M₂] [AddCommGroup M₃] [Module R M₃]
    [LieRingModule L M₃] [LieModule R L M₃] (g : M₁ ⊗[R] M₂ →ₗ⁅R,L⁆ M₃) (χ₁ χ₂ : R) (x : L) :
    LinearMap.range ((g : M₁ ⊗[R] M₂ →ₗ[R] M₃).comp (mapIncl 𝕎(M₁, χ₁, x) 𝕎(M₂, χ₂, x))) ≤
      𝕎(M₃, χ₁ + χ₂, x) 
Arguments: {R : Type u_2} {L : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] (M₁ : Type u_5) (M₂ : Type u_6) (M₃ : Type u_7) [AddCommGroup M₁] [Module R M₁] [LieRingModule L M₁] [LieModule R L M₁] [AddCommGroup M₂] [Module R M₂] [LieRingModule L M₂] [LieModule R L M₂] [AddCommGroup M₃] [Module R M₃] [LieRingModule L M₃] [LieModule R L M₃] (g : TensorProduct R M₁ M₂ →ₗ⁅R,L⁆ M₃) (χ₁ χ₂ : R) (x : L)
Goal: LinearMap.range
    (↑g ∘ₗ
      TensorProduct.mapIncl (((LieModule.toEnd R L M₁) x).maxGenEigenspace χ₁)
        (((LieModule.toEnd R L M₂) x).maxGenEigenspace χ₂)) ≤
  ((LieModule.toEnd R L M₃) x).maxGenEigenspace (χ₁ + χ₂)
-- LieModule.lie_mem_maxGenEigenspace_toEnd
lemma lie_mem_maxGenEigenspace_toEnd
    {χ₁ χ₂ : R} {x y : L} {m : M} (hy : y ∈ 𝕎(L, χ₁, x)) (hm : m ∈ 𝕎(M, χ₂, x)) :
    ⁅y, m⁆ ∈ 𝕎(M, χ₁ + χ₂, x) 
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {χ₁ χ₂ : R} {x y : L} {m : M} (hy : y ∈ ((LieModule.toEnd R L L) x).maxGenEigenspace χ₁) (hm : m ∈ ((LieModule.toEnd R L M) x).maxGenEigenspace χ₂)
Goal: ⁅y, m⁆ ∈ ((LieModule.toEnd R L M) x).maxGenEigenspace (χ₁ + χ₂)
-- LieModule.mem_weightSpaceOf
theorem mem_weightSpaceOf (χ : R) (x : L) (m : M) :
    m ∈ weightSpaceOf M χ x ↔ ∃ k : ℕ, ((toEnd R L M x - χ • ↑1) ^ k) m = 0 
Arguments: {R : Type u_2} {L : Type u_3} (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (χ : R) (x : L) (m : M)
Goal: m ∈ LieModule.weightSpaceOf M χ x ↔ ∃ k, (((LieModule.toEnd R L M) x - χ • 1) ^ k) m = 0
-- LieModule.coe_weightSpaceOf_zero
theorem coe_weightSpaceOf_zero (x : L) :
    ↑(weightSpaceOf M (0 : R) x) = ⨆ k, LinearMap.ker (toEnd R L M x ^ k) 
Arguments: {R : Type u_2} {L : Type u_3} (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (x : L)
Goal: ↑(LieModule.weightSpaceOf M 0 x) = ⨆ k, LinearMap.ker ((LieModule.toEnd R L M) x ^ k)
-- LieModule.mem_weightSpace
theorem mem_weightSpace (χ : L → R) (m : M) :
    m ∈ weightSpace M χ ↔ ∀ x, ∃ k : ℕ, ((toEnd R L M x - χ x • ↑1) ^ k) m = 0 
Arguments: {R : Type u_2} {L : Type u_3} (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (χ : L → R) (m : M)
Goal: m ∈ LieModule.weightSpace M χ ↔ ∀ (x : L), ∃ k, (((LieModule.toEnd R L M) x - χ x • 1) ^ k) m = 0
-- LieModule.weightSpace_le_weightSpaceOf
lemma weightSpace_le_weightSpaceOf (x : L) (χ : L → R) :
    weightSpace M χ ≤ weightSpaceOf M (χ x) x 
Arguments: {R : Type u_2} {L : Type u_3} (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (x : L) (χ : L → R)
Goal: LieModule.weightSpace M χ ≤ LieModule.weightSpaceOf M (χ x) x
-- LieModule.Weight.weightSpace_ne_bot'
structure Weight where
    toFun : L → R
  weightSpace_ne_bot' : weightSpace M toFun ≠ ⊥
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (self : LieModule.Weight R L M)
Goal: LieModule.weightSpace M self.toFun ≠ ⊥
-- LieModule.Weight.coe_weight_mk
@[simp] lemma coe_weight_mk (χ : L → R) (h) :
    (↑(⟨χ, h⟩ : Weight R L M) : L → R) = χ 
Arguments: {R : Type u_2} {L : Type u_3} (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (χ : L → R) (h : LieModule.weightSpace M χ ≠ ⊥)
Goal: ⇑{ toFun := χ, weightSpace_ne_bot' := h } = χ
-- LieModule.Weight.weightSpace_ne_bot
lemma weightSpace_ne_bot (χ : Weight R L M) : weightSpace M χ ≠ ⊥ 
Arguments: {R : Type u_2} {L : Type u_3} (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (χ : LieModule.Weight R L M)
Goal: LieModule.weightSpace M ⇑χ ≠ ⊥
-- LieModule.Weight.ext
@[ext] lemma ext {χ₁ χ₂ : Weight R L M} (h : ∀ x, χ₁ x = χ₂ x) : χ₁ = χ₂ 
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {χ₁ χ₂ : LieModule.Weight R L M} (h : ∀ (x : L), χ₁ x = χ₂ x)
Goal: χ₁ = χ₂
-- LieModule.Weight.ext_iff
protected lemma ext_iff {χ₁ χ₂ : Weight R L M} : χ₁ = χ₂ ↔ (χ₁ : L → R) = χ₂ 
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {χ₁ χ₂ : LieModule.Weight R L M}
Goal: χ₁ = χ₂ ↔ ⇑χ₁ = ⇑χ₂
-- LieModule.Weight.exists_ne_zero
lemma exists_ne_zero (χ : Weight R L M) :
    ∃ x ∈ weightSpace M χ, x ≠ 0 
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (χ : LieModule.Weight R L M)
Goal: ∃ x ∈ LieModule.weightSpace M ⇑χ, x ≠ 0
-- LieModule.Weight.coe_zero
@[simp]
lemma coe_zero [Nontrivial (weightSpace M (0 : L → R))] : ((0 : Weight R L M) : L → R) = 0 
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [Nontrivial ↥↑(LieModule.weightSpace M 0)]
Goal: ⇑0 = 0
-- LieModule.Weight.zero_apply
lemma zero_apply [Nontrivial (weightSpace M (0 : L → R))] (x) : (0 : Weight R L M) x = 0 
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [Nontrivial ↥↑(LieModule.weightSpace M 0)] (x : L)
Goal: 0 x = 0
-- LieModule.Weight.IsZero.eq
@[simp] lemma IsZero.eq {χ : Weight R L M} (hχ : χ.IsZero) : (χ : L → R) = 0 
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {χ : LieModule.Weight R L M} (hχ : χ.IsZero)
Goal: ⇑χ = 0
-- LieModule.Weight.coe_eq_zero_iff
@[simp] lemma coe_eq_zero_iff (χ : Weight R L M) : (χ : L → R) = 0 ↔ χ.IsZero 
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (χ : LieModule.Weight R L M)
Goal: ⇑χ = 0 ↔ χ.IsZero
-- LieModule.Weight.isZero_iff_eq_zero
lemma isZero_iff_eq_zero [Nontrivial (weightSpace M (0 : L → R))] {χ : Weight R L M} :
    χ.IsZero ↔ χ = 0 
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [Nontrivial ↥↑(LieModule.weightSpace M 0)] {χ : LieModule.Weight R L M}
Goal: χ.IsZero ↔ χ = 0
-- LieModule.Weight.isZero_zero
lemma isZero_zero [Nontrivial (weightSpace M (0 : L → R))] : IsZero (0 : Weight R L M) 
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [Nontrivial ↥↑(LieModule.weightSpace M 0)]
Goal: LieModule.Weight.IsZero 0
-- LieModule.Weight.isNonZero_iff_ne_zero
lemma isNonZero_iff_ne_zero [Nontrivial (weightSpace M (0 : L → R))] {χ : Weight R L M} :
    χ.IsNonZero ↔ χ ≠ 0 
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [Nontrivial ↥↑(LieModule.weightSpace M 0)] {χ : LieModule.Weight R L M}
Goal: χ.IsNonZero ↔ χ ≠ 0
-- LieModule.Weight.weightSpaceOf_ne_bot
lemma weightSpaceOf_ne_bot (χ : Weight R L M) (x : L) :
    weightSpaceOf M (χ x) x ≠ ⊥ 
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (χ : LieModule.Weight R L M) (x : L)
Goal: LieModule.weightSpaceOf M (χ x) x ≠ ⊥
-- LieModule.Weight.hasEigenvalueAt
lemma hasEigenvalueAt (χ : Weight R L M) (x : L) :
    (toEnd R L M x).HasEigenvalue (χ x) 
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (χ : LieModule.Weight R L M) (x : L)
Goal: ((LieModule.toEnd R L M) x).HasEigenvalue (χ x)
-- LieModule.Weight.apply_eq_zero_of_isNilpotent
lemma apply_eq_zero_of_isNilpotent [NoZeroSMulDivisors R M] [IsReduced R]
    (x : L) (h : _root_.IsNilpotent (toEnd R L M x)) (χ : Weight R L M) :
    χ x = 0 
Arguments: {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [NoZeroSMulDivisors R M] [IsReduced R] (x : L) (h : IsNilpotent ((LieModule.toEnd R L M) x)) (χ : LieModule.Weight R L M)
Goal: χ x = 0
-- LieModule.zero_weightSpace_eq_top_of_nilpotent'
@[simp]
theorem zero_weightSpace_eq_top_of_nilpotent' [IsNilpotent R L M] :
    weightSpace M (0 : L → R) = ⊤ 
Arguments: {R : Type u_2} {L : Type u_3} (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieModule.IsNilpotent R L M]
Goal: LieModule.weightSpace M 0 = ⊤
-- LieModule.coe_weightSpace_of_top
theorem coe_weightSpace_of_top (χ : L → R) :
    (weightSpace M (χ ∘ (⊤ : LieSubalgebra R L).incl) : Submodule R M) = weightSpace M χ 
Arguments: {R : Type u_2} {L : Type u_3} (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (χ : L → R)
Goal: ↑(LieModule.weightSpace M (χ ∘ ⇑⊤.incl)) = ↑(LieModule.weightSpace M χ)
-- LieModule.zero_weightSpace_eq_top_of_nilpotent
@[simp]
theorem zero_weightSpace_eq_top_of_nilpotent [IsNilpotent R L M] :
    weightSpace M (0 : (⊤ : LieSubalgebra R L) → R) = ⊤ 
Arguments: {R : Type u_2} {L : Type u_3} (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieModule.IsNilpotent R L M]
Goal: LieModule.weightSpace M 0 = ⊤
-- LieModule.exists_weightSpace_le_ker_of_isNoetherian
theorem exists_weightSpace_le_ker_of_isNoetherian [IsNoetherian R M] (χ : L → R) (x : L) :
    ∃ k : ℕ,
      weightSpace M χ ≤ LinearMap.ker ((toEnd R L M x - algebraMap R _ (χ x)) ^ k) 
Arguments: {R : Type u_2} {L : Type u_3} (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [IsNoetherian R M] (χ : L → R) (x : L)
Goal: ∃ k,
  ↑(LieModule.weightSpace M χ) ≤ LinearMap.ker (((LieModule.toEnd R L M) x - (algebraMap R (Module.End R M)) (χ x)) ^ k)
-- LieModule.exists_weightSpace_zero_le_ker_of_isNoetherian
theorem exists_weightSpace_zero_le_ker_of_isNoetherian
    [IsNoetherian R M] (x : L) :
    ∃ k : ℕ, weightSpace M (0 : L → R) ≤ LinearMap.ker (toEnd R L M x ^ k) 
Arguments: (R : Type u_2) {L : Type u_3} (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [IsNoetherian R M] (x : L)
Goal: ∃ k, ↑(LieModule.weightSpace M 0) ≤ LinearMap.ker ((LieModule.toEnd R L M) x ^ k)