ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- _private.Mathlib.Algebra.Lie.Weights.RootSystem.0.LieAlgebra.IsKilling.chainLength_aux
private lemma chainLength_aux {x} (hx : x ∈ rootSpace H (chainTop α β)) :
    ∃ n : ℕ, n • x = ⁅coroot α, x⁆ 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L) (hα : α.IsNonZero) {x : L} (hx : x ∈ LieAlgebra.rootSpace H ⇑(LieModule.chainTop (⇑α) β))
Goal: ∃ n, n • x = ⁅LieAlgebra.IsKilling.coroot α, x⁆
-- LieAlgebra.IsKilling.chainLength_of_isZero
lemma chainLength_of_isZero (hα : α.IsZero) : chainLength α β = 0 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L) (hα : α.IsZero)
Goal: LieAlgebra.IsKilling.chainLength α β = 0
-- LieAlgebra.IsKilling.chainLength_nsmul
lemma chainLength_nsmul {x} (hx : x ∈ rootSpace H (chainTop α β)) :
    chainLength α β • x = ⁅coroot α, x⁆ 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L) {x : L} (hx : x ∈ LieAlgebra.rootSpace H ⇑(LieModule.chainTop (⇑α) β))
Goal: LieAlgebra.IsKilling.chainLength α β • x = ⁅LieAlgebra.IsKilling.coroot α, x⁆
-- LieAlgebra.IsKilling.chainLength_smul
lemma chainLength_smul {x} (hx : x ∈ rootSpace H (chainTop α β)) :
    (chainLength α β : K) • x = ⁅coroot α, x⁆ 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L) {x : L} (hx : x ∈ LieAlgebra.rootSpace H ⇑(LieModule.chainTop (⇑α) β))
Goal: ↑(LieAlgebra.IsKilling.chainLength α β) • x = ⁅LieAlgebra.IsKilling.coroot α, x⁆
-- LieAlgebra.IsKilling.apply_coroot_eq_cast'
lemma apply_coroot_eq_cast' :
    β (coroot α) = ↑(chainLength α β - 2 * chainTopCoeff α β : ℤ) 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L)
Goal: β (LieAlgebra.IsKilling.coroot α) = ↑(↑(LieAlgebra.IsKilling.chainLength α β) - 2 * ↑(LieModule.chainTopCoeff (⇑α) β))
-- LieAlgebra.IsKilling.rootSpace_neg_nsmul_add_chainTop_of_le
lemma rootSpace_neg_nsmul_add_chainTop_of_le {n : ℕ} (hn : n ≤ chainLength α β) :
    rootSpace H (- (n • α) + chainTop α β) ≠ ⊥ 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L) {n : ℕ} (hn : n ≤ LieAlgebra.IsKilling.chainLength α β)
Goal: LieAlgebra.rootSpace H (-(n • ⇑α) + ⇑(LieModule.chainTop (⇑α) β)) ≠ ⊥
-- LieAlgebra.IsKilling.rootSpace_neg_nsmul_add_chainTop_of_lt
lemma rootSpace_neg_nsmul_add_chainTop_of_lt {n : ℕ} (hn : chainLength α β < n) :
    rootSpace H (- (n • α) + chainTop α β) = ⊥ 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L) (hα : α.IsNonZero) {n : ℕ} (hn : LieAlgebra.IsKilling.chainLength α β < n)
Goal: LieAlgebra.rootSpace H (-(n • ⇑α) + ⇑(LieModule.chainTop (⇑α) β)) = ⊥
-- LieAlgebra.IsKilling.chainTopCoeff_le_chainLength
lemma chainTopCoeff_le_chainLength : chainTopCoeff α β ≤ chainLength α β 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L)
Goal: LieModule.chainTopCoeff (⇑α) β ≤ LieAlgebra.IsKilling.chainLength α β
-- LieAlgebra.IsKilling.chainBotCoeff_add_chainTopCoeff
lemma chainBotCoeff_add_chainTopCoeff :
    chainBotCoeff α β + chainTopCoeff α β = chainLength α β 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L)
Goal: LieModule.chainBotCoeff (⇑α) β + LieModule.chainTopCoeff (⇑α) β = LieAlgebra.IsKilling.chainLength α β
-- LieAlgebra.IsKilling.chainTopCoeff_add_chainBotCoeff
lemma chainTopCoeff_add_chainBotCoeff :
    chainTopCoeff α β + chainBotCoeff α β = chainLength α β 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L)
Goal: LieModule.chainTopCoeff (⇑α) β + LieModule.chainBotCoeff (⇑α) β = LieAlgebra.IsKilling.chainLength α β
-- LieAlgebra.IsKilling.chainBotCoeff_le_chainLength
lemma chainBotCoeff_le_chainLength : chainBotCoeff α β ≤ chainLength α β 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L)
Goal: LieModule.chainBotCoeff (⇑α) β ≤ LieAlgebra.IsKilling.chainLength α β
-- LieAlgebra.IsKilling.chainLength_neg
@[simp]
lemma chainLength_neg :
    chainLength (-α) β = chainLength α β 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L)
Goal: LieAlgebra.IsKilling.chainLength (-α) β = LieAlgebra.IsKilling.chainLength α β
-- LieAlgebra.IsKilling.chainLength_zero
@[simp]
lemma chainLength_zero [Nontrivial L] : chainLength 0 β = 0 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (β : LieModule.Weight K (↥H) L) [Nontrivial L]
Goal: LieAlgebra.IsKilling.chainLength 0 β = 0
-- LieAlgebra.IsKilling.apply_coroot_eq_cast
lemma apply_coroot_eq_cast :
    β (coroot α) = (chainBotCoeff α β - chainTopCoeff α β : ℤ) 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L)
Goal: β (LieAlgebra.IsKilling.coroot α) = ↑(↑(LieModule.chainBotCoeff (⇑α) β) - ↑(LieModule.chainTopCoeff (⇑α) β))
-- LieAlgebra.IsKilling.le_chainBotCoeff_of_rootSpace_ne_top
lemma le_chainBotCoeff_of_rootSpace_ne_top (n : ℤ) (hn : rootSpace H (-n • α + β) ≠ ⊥) :
    n ≤ chainBotCoeff α β 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L) (hα : α.IsNonZero) (n : ℤ) (hn : LieAlgebra.rootSpace H (-n • ⇑α + ⇑β) ≠ ⊥)
Goal: n ≤ ↑(LieModule.chainBotCoeff (⇑α) β)
-- LieAlgebra.IsKilling.rootSpace_zsmul_add_ne_bot_iff
lemma rootSpace_zsmul_add_ne_bot_iff (n : ℤ) :
    rootSpace H (n • α + β) ≠ ⊥ ↔ n ≤ chainTopCoeff α β ∧ -n ≤ chainBotCoeff α β 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L) (hα : α.IsNonZero) (n : ℤ)
Goal: LieAlgebra.rootSpace H (n • ⇑α + ⇑β) ≠ ⊥ ↔
  n ≤ ↑(LieModule.chainTopCoeff (⇑α) β) ∧ -n ≤ ↑(LieModule.chainBotCoeff (⇑α) β)
-- LieAlgebra.IsKilling.rootSpace_zsmul_add_ne_bot_iff_mem
lemma rootSpace_zsmul_add_ne_bot_iff_mem (n : ℤ) :
    rootSpace H (n • α + β) ≠ ⊥ ↔ n ∈ Finset.Icc (-chainBotCoeff α β : ℤ) (chainTopCoeff α β) 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L) (hα : α.IsNonZero) (n : ℤ)
Goal: LieAlgebra.rootSpace H (n • ⇑α + ⇑β) ≠ ⊥ ↔
  n ∈ Finset.Icc (-↑(LieModule.chainBotCoeff (⇑α) β)) ↑(LieModule.chainTopCoeff (⇑α) β)
-- LieAlgebra.IsKilling.chainTopCoeff_of_eq_zsmul_add
lemma chainTopCoeff_of_eq_zsmul_add (β' : Weight K H L) (n : ℤ) (hβ' : (β' : H → K) = n • α + β) :
    chainTopCoeff α β' = chainTopCoeff α β - n 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L) (hα : α.IsNonZero) (β' : LieModule.Weight K (↥H) L) (n : ℤ) (hβ' : ⇑β' = n • ⇑α + ⇑β)
Goal: ↑(LieModule.chainTopCoeff (⇑α) β') = ↑(LieModule.chainTopCoeff (⇑α) β) - n
-- LieAlgebra.IsKilling.chainBotCoeff_of_eq_zsmul_add
lemma chainBotCoeff_of_eq_zsmul_add (β' : Weight K H L) (n : ℤ) (hβ' : (β' : H → K) = n • α + β) :
    chainBotCoeff α β' = chainBotCoeff α β + n 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L) (hα : α.IsNonZero) (β' : LieModule.Weight K (↥H) L) (n : ℤ) (hβ' : ⇑β' = n • ⇑α + ⇑β)
Goal: ↑(LieModule.chainBotCoeff (⇑α) β') = ↑(LieModule.chainBotCoeff (⇑α) β) + n
-- LieAlgebra.IsKilling.chainLength_of_eq_zsmul_add
lemma chainLength_of_eq_zsmul_add (β' : Weight K H L) (n : ℤ) (hβ' : (β' : H → K) = n • α + β) :
    chainLength α β' = chainLength α β 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β β' : LieModule.Weight K (↥H) L) (n : ℤ) (hβ' : ⇑β' = n • ⇑α + ⇑β)
Goal: LieAlgebra.IsKilling.chainLength α β' = LieAlgebra.IsKilling.chainLength α β
-- LieAlgebra.IsKilling.chainTopCoeff_zero_right
lemma chainTopCoeff_zero_right [Nontrivial L] :
    chainTopCoeff α (0 : Weight K H L) = 1 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α : LieModule.Weight K (↥H) L) (hα : α.IsNonZero) [Nontrivial L]
Goal: LieModule.chainTopCoeff (⇑α) 0 = 1
-- LieAlgebra.IsKilling.chainBotCoeff_zero_right
lemma chainBotCoeff_zero_right [Nontrivial L] : chainBotCoeff α (0 : Weight K H L) = 1 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α : LieModule.Weight K (↥H) L) (hα : α.IsNonZero) [Nontrivial L]
Goal: LieModule.chainBotCoeff (⇑α) 0 = 1
-- LieAlgebra.IsKilling.chainLength_zero_right
lemma chainLength_zero_right [Nontrivial L] : chainLength α 0 = 2 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α : LieModule.Weight K (↥H) L) (hα : α.IsNonZero) [Nontrivial L]
Goal: LieAlgebra.IsKilling.chainLength α 0 = 2
-- LieAlgebra.IsKilling.rootSpace_two_smul
lemma rootSpace_two_smul : rootSpace H (2 • α) = ⊥ 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α : LieModule.Weight K (↥H) L) (hα : α.IsNonZero)
Goal: LieAlgebra.rootSpace H (2 • ⇑α) = ⊥
-- LieAlgebra.IsKilling.rootSpace_one_div_two_smul
lemma rootSpace_one_div_two_smul : rootSpace H ((2⁻¹ : K) • α) = ⊥ 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α : LieModule.Weight K (↥H) L) (hα : α.IsNonZero)
Goal: LieAlgebra.rootSpace H (2⁻¹ • ⇑α) = ⊥
-- LieAlgebra.IsKilling.eq_neg_one_or_eq_zero_or_eq_one_of_eq_smul
lemma eq_neg_one_or_eq_zero_or_eq_one_of_eq_smul (k : K) (h : (β : H → K) = k • α) :
    k = -1 ∨ k = 0 ∨ k = 1 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L) (hα : α.IsNonZero) (k : K) (h : ⇑β = k • ⇑α)
Goal: k = -1 ∨ k = 0 ∨ k = 1
-- LieAlgebra.IsKilling.eq_neg_or_eq_of_eq_smul
lemma eq_neg_or_eq_of_eq_smul (hβ : β.IsNonZero) (k : K) (h : (β : H → K) = k • α) :
    β = -α ∨ β = α 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L) (hβ : β.IsNonZero) (k : K) (h : ⇑β = k • ⇑α)
Goal: β = -α ∨ β = α
-- LieAlgebra.IsKilling.reflectRoot_isNonZero
lemma reflectRoot_isNonZero (α β : Weight K H L) (hβ : β.IsNonZero) :
    (reflectRoot α β).IsNonZero 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L) (hβ : β.IsNonZero)
Goal: (LieAlgebra.IsKilling.reflectRoot α β).IsNonZero
-- LieAlgebra.IsKilling.rootSystem_toLin_apply
@[simp] lemma rootSystem_toLin_apply (f x) : (rootSystem H).toLin f x = f x 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] (H : LieSubalgebra K L) [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (f : Module.Dual K ↥H) (x : ↥H)
Goal: ((LieAlgebra.IsKilling.rootSystem H).toLin f) x = f x
-- LieAlgebra.IsKilling.rootSystem_pairing_apply
@[simp] lemma rootSystem_pairing_apply (α β) : (rootSystem H).pairing β α = β.1 (coroot α.1) 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] (H : LieSubalgebra K L) [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : { α // α.IsNonZero })
Goal: (LieAlgebra.IsKilling.rootSystem H).pairing β α = ↑β (LieAlgebra.IsKilling.coroot ↑α)
-- LieAlgebra.IsKilling.rootSystem_root_apply
@[simp] lemma rootSystem_root_apply (α) : (rootSystem H).root α = α 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] (H : LieSubalgebra K L) [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α : { α // α.IsNonZero })
Goal: (LieAlgebra.IsKilling.rootSystem H).root α = LieModule.Weight.toLinear K (↥H) L ↑α
-- LieAlgebra.IsKilling.rootSystem_coroot_apply
@[simp] lemma rootSystem_coroot_apply (α) : (rootSystem H).coroot α = coroot α 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] (H : LieSubalgebra K L) [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α : { α // α.IsNonZero })
Goal: (LieAlgebra.IsKilling.rootSystem H).coroot α = LieAlgebra.IsKilling.coroot ↑α
-- LieAlgebra.IsKilling.isCrystallographic_rootSystem
theorem isCrystallographic_rootSystem : (rootSystem H).IsCrystallographic 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] (H : LieSubalgebra K L) [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L]
Goal: (LieAlgebra.IsKilling.rootSystem H).IsCrystallographic
-- LieAlgebra.IsKilling.isReduced_rootSystem
theorem isReduced_rootSystem : (rootSystem H).IsReduced 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [LieAlgebra.IsKilling K L] [FiniteDimensional K L] (H : LieSubalgebra K L) [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L]
Goal: (LieAlgebra.IsKilling.rootSystem H).IsReduced
-- SemilinearEquivClass.map_smulₛₗ
class SemilinearEquivClass (F : Type*) {R S : outParam Type*} [Semiring R] [Semiring S]
  (σ : outParam <| R →+* S) {σ' : outParam <| S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ]
  (M M₂ : outParam Type*) [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂]
  [EquivLike F M M₂]
  extends AddEquivClass F M M₂ : Prop where
    map_smulₛₗ : ∀ (f : F) (r : R) (x : M), f (r • x) = σ r • f x
Arguments: {F : Type u_17} {R : outParam (Type u_18)} {S : outParam (Type u_19)} [Semiring R] [Semiring S] {σ : outParam (R →+* S)} {σ' : outParam (S →+* R)} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {M : outParam (Type u_20)} {M₂ : outParam (Type u_21)} [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] [EquivLike F M M₂] [self : SemilinearEquivClass F σ M M₂] (f : F) (r : R) (x : M)
Goal: f (r • x) = σ r • f x
-- LinearEquiv.toEquiv_injective
theorem toEquiv_injective : Function.Injective (toEquiv : (M ≃ₛₗ[σ] M₂) → M ≃ M₂) 
Arguments: {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ]
Goal: Function.Injective LinearEquiv.toEquiv
-- LinearEquiv.toEquiv_inj
@[simp]
theorem toEquiv_inj {e₁ e₂ : M ≃ₛₗ[σ] M₂} : e₁.toEquiv = e₂.toEquiv ↔ e₁ = e₂ 
Arguments: {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {e₁ e₂ : M ≃ₛₗ[σ] M₂}
Goal: e₁.toEquiv = e₂.toEquiv ↔ e₁ = e₂
-- LinearEquiv.toLinearMap_injective
theorem toLinearMap_injective : Injective (toLinearMap : (M ≃ₛₗ[σ] M₂) → M →ₛₗ[σ] M₂) 
Arguments: {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ]
Goal: Function.Injective LinearEquiv.toLinearMap
-- LinearEquiv.toLinearMap_inj
@[simp, norm_cast]
theorem toLinearMap_inj {e₁ e₂ : M ≃ₛₗ[σ] M₂} : (↑e₁ : M →ₛₗ[σ] M₂) = e₂ ↔ e₁ = e₂ 
Arguments: {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {e₁ e₂ : M ≃ₛₗ[σ] M₂}
Goal: ↑e₁ = ↑e₂ ↔ e₁ = e₂
-- LinearEquiv.coe_mk
@[simp]
theorem coe_mk {to_fun inv_fun map_add map_smul left_inv right_inv} :
    (⟨⟨⟨to_fun, map_add⟩, map_smul⟩, inv_fun, left_inv, right_inv⟩ : M ≃ₛₗ[σ] M₂) = to_fun 
Arguments: {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {to_fun : M → M₂} {inv_fun : M₂ → M} {map_add : ∀ (x y : M), to_fun (x + y) = to_fun x + to_fun y} {map_smul :
  ∀ (m : R) (x : M),
    { toFun := to_fun, map_add' := map_add }.toFun (m • x) = σ m • { toFun := to_fun, map_add' := map_add }.toFun x} {left_inv : Function.LeftInverse inv_fun { toFun := to_fun, map_add' := map_add, map_smul' := map_smul }.toFun} {right_inv : Function.RightInverse inv_fun { toFun := to_fun, map_add' := map_add, map_smul' := map_smul }.toFun}
Goal: ⇑{ toFun := to_fun, map_add' := map_add, map_smul' := map_smul, invFun := inv_fun, left_inv := left_inv,
      right_inv := right_inv } =
  to_fun
-- LinearEquiv.coe_injective
theorem coe_injective : @Injective (M ≃ₛₗ[σ] M₂) (M → M₂) CoeFun.coe 
Arguments: {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ]
Goal: Function.Injective CoeFun.coe
-- LinearEquiv.coe_coe
@[simp, norm_cast]
theorem coe_coe : ⇑(e : M →ₛₗ[σ] M₂) = e 
Arguments: {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂)
Goal: ⇑↑e = ⇑e
-- LinearEquiv.coe_toEquiv
@[simp]
theorem coe_toEquiv : ⇑(e.toEquiv) = e 
Arguments: {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂)
Goal: ⇑e.toEquiv = ⇑e
-- LinearEquiv.coe_toLinearMap
@[simp]
theorem coe_toLinearMap : ⇑e.toLinearMap = e 
Arguments: {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂)
Goal: ⇑↑e = ⇑e
-- LinearEquiv.toFun_eq_coe
theorem toFun_eq_coe : e.toFun = e 
Arguments: {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂)
Goal: (↑e).toFun = ⇑e
-- LinearEquiv.ext
@[ext]
theorem ext (h : ∀ x, e x = e' x) : e = e' 
Arguments: {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} {e e' : M ≃ₛₗ[σ] M₂} (h : ∀ (x : M), e x = e' x)
Goal: e = e'
-- LinearEquiv.ext_iff
theorem ext_iff : e = e' ↔ ∀ x, e x = e' x 
Arguments: {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} {e e' : M ≃ₛₗ[σ] M₂}
Goal: e = e' ↔ ∀ (x : M), e x = e' x
-- LinearEquiv.congr_arg
protected theorem congr_arg {x x'} : x = x' → e x = e x' 
Arguments: {R : Type u_1} {S : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂} {σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} {e : M ≃ₛₗ[σ] M₂} {x x' : M}
Goal: x = x' → e x = e x'