ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- mul_div_cancel_of_invertible
@[simp]
theorem mul_div_cancel_of_invertible (a b : α) [Invertible b] : a * b / b = a 
Arguments: {α : Type u} [GroupWithZero α] (a b : α) [Invertible b]
Goal: a * b / b = a
-- div_self_of_invertible
@[simp]
theorem div_self_of_invertible (a : α) [Invertible a] : a / a = 1 
Arguments: {α : Type u} [GroupWithZero α] (a : α) [Invertible a]
Goal: a / a = 1
-- invOf_div
theorem invOf_div (a b : α) [Invertible a] [Invertible b] [Invertible (a / b)] :
    ⅟ (a / b) = b / a 
Arguments: {α : Type u} [GroupWithZero α] (a b : α) [Invertible a] [Invertible b] [Invertible (a / b)]
Goal: ⅟(a / b) = b / a
-- LieAlgebra.zero_rootSpace_eq_top_of_nilpotent
theorem zero_rootSpace_eq_top_of_nilpotent [IsNilpotent R L] :
    rootSpace (⊤ : LieSubalgebra R L) 0 = ⊤ 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L]
Goal: LieAlgebra.rootSpace ⊤ 0 = ⊤
-- LieAlgebra.rootSpace_comap_eq_weightSpace
@[simp]
theorem rootSpace_comap_eq_weightSpace (χ : H → R) :
    (rootSpace H χ).comap H.incl' = weightSpace H χ 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] (χ : ↥H → R)
Goal: LieSubmodule.comap H.incl' (LieAlgebra.rootSpace H χ) = LieModule.weightSpace (↥H) χ
-- LieAlgebra.lie_mem_weightSpace_of_mem_weightSpace
theorem lie_mem_weightSpace_of_mem_weightSpace {χ₁ χ₂ : H → R} {x : L} {m : M}
    (hx : x ∈ rootSpace H χ₁) (hm : m ∈ weightSpace M χ₂) : ⁅x, m⁆ ∈ weightSpace M (χ₁ + χ₂) 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {H : LieSubalgebra R L} [LieAlgebra.IsNilpotent R ↥H] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {χ₁ χ₂ : ↥H → R} {x : L} {m : M} (hx : x ∈ LieAlgebra.rootSpace H χ₁) (hm : m ∈ LieModule.weightSpace M χ₂)
Goal: ⁅x, m⁆ ∈ LieModule.weightSpace M (χ₁ + χ₂)
-- LieAlgebra.toEnd_pow_apply_mem
lemma toEnd_pow_apply_mem {χ₁ χ₂ : H → R} {x : L} {m : M}
    (hx : x ∈ rootSpace H χ₁) (hm : m ∈ weightSpace M χ₂) (n) :
    (toEnd R L M x ^ n : Module.End R M) m ∈ weightSpace M (n • χ₁ + χ₂) 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {H : LieSubalgebra R L} [LieAlgebra.IsNilpotent R ↥H] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {χ₁ χ₂ : ↥H → R} {x : L} {m : M} (hx : x ∈ LieAlgebra.rootSpace H χ₁) (hm : m ∈ LieModule.weightSpace M χ₂) (n : ℕ)
Goal: ((LieModule.toEnd R L M) x ^ n) m ∈ LieModule.weightSpace M (n • χ₁ + χ₂)
-- LieAlgebra.coe_rootSpaceWeightSpaceProduct_tmul
@[simp]
theorem coe_rootSpaceWeightSpaceProduct_tmul (χ₁ χ₂ χ₃ : H → R) (hχ : χ₁ + χ₂ = χ₃)
    (x : rootSpace H χ₁) (m : weightSpace M χ₂) :
    (rootSpaceWeightSpaceProduct R L H M χ₁ χ₂ χ₃ hχ (x ⊗ₜ m) : M) = ⁅(x : L), (m : M)⁆ 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] (M : Type u_3) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (χ₁ χ₂ χ₃ : ↥H → R) (hχ : χ₁ + χ₂ = χ₃) (x : ↥(LieAlgebra.rootSpace H χ₁)) (m : ↥(LieModule.weightSpace M χ₂))
Goal: ↑((LieAlgebra.rootSpaceWeightSpaceProduct R L H M χ₁ χ₂ χ₃ hχ) (x ⊗ₜ[R] m)) = ⁅↑x, ↑m⁆
-- LieAlgebra.mapsTo_toEnd_weightSpace_add_of_mem_rootSpace
theorem mapsTo_toEnd_weightSpace_add_of_mem_rootSpace (α χ : H → R)
    {x : L} (hx : x ∈ rootSpace H α) :
    MapsTo (toEnd R L M x) (weightSpace M χ) (weightSpace M (α + χ)) 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] (M : Type u_3) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (α χ : ↥H → R) {x : L} (hx : x ∈ LieAlgebra.rootSpace H α)
Goal: Set.MapsTo ⇑((LieModule.toEnd R L M) x) ↑(LieModule.weightSpace M χ) ↑(LieModule.weightSpace M (α + χ))
-- LieAlgebra.rootSpaceProduct_def
@[simp]
theorem rootSpaceProduct_def : rootSpaceProduct R L H = rootSpaceWeightSpaceProduct R L H L 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H]
Goal: LieAlgebra.rootSpaceProduct R L H = LieAlgebra.rootSpaceWeightSpaceProduct R L H L
-- LieAlgebra.rootSpaceProduct_tmul
theorem rootSpaceProduct_tmul
    (χ₁ χ₂ χ₃ : H → R) (hχ : χ₁ + χ₂ = χ₃) (x : rootSpace H χ₁) (y : rootSpace H χ₂) :
    (rootSpaceProduct R L H χ₁ χ₂ χ₃ hχ (x ⊗ₜ y) : L) = ⁅(x : L), (y : L)⁆ 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] (χ₁ χ₂ χ₃ : ↥H → R) (hχ : χ₁ + χ₂ = χ₃) (x : ↥(LieAlgebra.rootSpace H χ₁)) (y : ↥(LieAlgebra.rootSpace H χ₂))
Goal: ↑((LieAlgebra.rootSpaceProduct R L H χ₁ χ₂ χ₃ hχ) (x ⊗ₜ[R] y)) = ⁅↑x, ↑y⁆
-- LieAlgebra.coe_zeroRootSubalgebra
@[simp]
theorem coe_zeroRootSubalgebra : (zeroRootSubalgebra R L H : Submodule R L) = rootSpace H 0 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H]
Goal: (LieAlgebra.zeroRootSubalgebra R L H).toSubmodule = ↑(LieAlgebra.rootSpace H 0)
-- LieAlgebra.mem_zeroRootSubalgebra
theorem mem_zeroRootSubalgebra (x : L) :
    x ∈ zeroRootSubalgebra R L H ↔ ∀ y : H, ∃ k : ℕ, (toEnd R H L y ^ k) x = 0 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] (x : L)
Goal: x ∈ LieAlgebra.zeroRootSubalgebra R L H ↔ ∀ (y : ↥H), ∃ k, ((LieModule.toEnd R (↥H) L) y ^ k) x = 0
-- LieAlgebra.toLieSubmodule_le_rootSpace_zero
theorem toLieSubmodule_le_rootSpace_zero : H.toLieSubmodule ≤ rootSpace H 0 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H]
Goal: H.toLieSubmodule ≤ LieAlgebra.rootSpace H 0
-- LieAlgebra.le_zeroRootSubalgebra
theorem le_zeroRootSubalgebra : H ≤ zeroRootSubalgebra R L H 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H]
Goal: H ≤ LieAlgebra.zeroRootSubalgebra R L H
-- LieAlgebra.zeroRootSubalgebra_normalizer_eq_self
@[simp]
theorem zeroRootSubalgebra_normalizer_eq_self :
    (zeroRootSubalgebra R L H).normalizer = zeroRootSubalgebra R L H 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H]
Goal: (LieAlgebra.zeroRootSubalgebra R L H).normalizer = LieAlgebra.zeroRootSubalgebra R L H
-- LieAlgebra.is_cartan_of_zeroRootSubalgebra_eq
theorem is_cartan_of_zeroRootSubalgebra_eq (h : zeroRootSubalgebra R L H = H) :
    H.IsCartanSubalgebra 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] (h : LieAlgebra.zeroRootSubalgebra R L H = H)
Goal: H.IsCartanSubalgebra
-- LieAlgebra.zeroRootSubalgebra_eq_of_is_cartan
@[simp]
theorem zeroRootSubalgebra_eq_of_is_cartan (H : LieSubalgebra R L) [H.IsCartanSubalgebra]
    [IsNoetherian R L] : zeroRootSubalgebra R L H = H 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [H.IsCartanSubalgebra] [IsNoetherian R L]
Goal: LieAlgebra.zeroRootSubalgebra R L H = H
-- LieAlgebra.zeroRootSubalgebra_eq_iff_is_cartan
theorem zeroRootSubalgebra_eq_iff_is_cartan [IsNoetherian R L] :
    zeroRootSubalgebra R L H = H ↔ H.IsCartanSubalgebra 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] [IsNoetherian R L]
Goal: LieAlgebra.zeroRootSubalgebra R L H = H ↔ H.IsCartanSubalgebra
-- LieAlgebra.rootSpace_zero_eq
@[simp]
theorem rootSpace_zero_eq (H : LieSubalgebra R L) [H.IsCartanSubalgebra] [IsNoetherian R L] :
    rootSpace H 0 = H.toLieSubmodule 
Arguments: (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [H.IsCartanSubalgebra] [IsNoetherian R L]
Goal: LieAlgebra.rootSpace H 0 = H.toLieSubmodule
-- LieAlgebra.mem_corootSpace
lemma mem_corootSpace {x : H} :
    x ∈ corootSpace α ↔
    (x : L) ∈ Submodule.span R {⁅y, z⁆ | (y ∈ rootSpace H α) (z ∈ rootSpace H (-α))} 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {H : LieSubalgebra R L} [LieAlgebra.IsNilpotent R ↥H] [H.IsCartanSubalgebra] [IsNoetherian R L] (α : ↥H → R) {x : ↥H}
Goal: x ∈ LieAlgebra.corootSpace α ↔
  ↑x ∈ Submodule.span R {x | ∃ y ∈ LieAlgebra.rootSpace H α, ∃ z ∈ LieAlgebra.rootSpace H (-α), ⁅y, z⁆ = x}
-- LieAlgebra.mem_corootSpace'
lemma mem_corootSpace' {x : H} :
    x ∈ corootSpace α ↔
    x ∈ Submodule.span R ({⁅y, z⁆ | (y ∈ rootSpace H α) (z ∈ rootSpace H (-α))} : Set H) 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {H : LieSubalgebra R L} [LieAlgebra.IsNilpotent R ↥H] [H.IsCartanSubalgebra] [IsNoetherian R L] (α : ↥H → R) {x : ↥H}
Goal: x ∈ LieAlgebra.corootSpace α ↔
  x ∈ Submodule.span R {x | ∃ y ∈ LieAlgebra.rootSpace H α, ∃ z ∈ LieAlgebra.rootSpace H (-α), ⁅y, z⁆ = ↑x}
-- eq_zero_of_zero_dvd
theorem eq_zero_of_zero_dvd (h : 0 ∣ a) : a = 0 
Arguments: {α : Type u_1} [SemigroupWithZero α] {a : α} (h : 0 ∣ a)
Goal: a = 0
-- zero_dvd_iff
@[simp]
theorem zero_dvd_iff : 0 ∣ a ↔ a = 0 
Arguments: {α : Type u_1} [SemigroupWithZero α] {a : α}
Goal: 0 ∣ a ↔ a = 0
-- dvd_zero
@[simp]
theorem dvd_zero (a : α) : a ∣ 0 
Arguments: {α : Type u_1} [SemigroupWithZero α] (a : α)
Goal: a ∣ 0
-- mul_dvd_mul_iff_left
theorem mul_dvd_mul_iff_left [CancelMonoidWithZero α] {a b c : α} (ha : a ≠ 0) :
    a * b ∣ a * c ↔ b ∣ c 
Arguments: {α : Type u_1} [CancelMonoidWithZero α] {a b c : α} (ha : a ≠ 0)
Goal: a * b ∣ a * c ↔ b ∣ c
-- mul_dvd_mul_iff_right
theorem mul_dvd_mul_iff_right [CancelCommMonoidWithZero α] {a b c : α} (hc : c ≠ 0) :
    a * c ∣ b * c ↔ a ∣ b 
Arguments: {α : Type u_1} [CancelCommMonoidWithZero α] {a b c : α} (hc : c ≠ 0)
Goal: a * c ∣ b * c ↔ a ∣ b
-- dvdNotUnit_of_dvd_of_not_dvd
theorem dvdNotUnit_of_dvd_of_not_dvd {a b : α} (hd : a ∣ b) (hnd : ¬b ∣ a) : DvdNotUnit a b 
Arguments: {α : Type u_1} [CommMonoidWithZero α] {a b : α} (hd : a ∣ b) (hnd : ¬b ∣ a)
Goal: DvdNotUnit a b
-- isRelPrime_zero_left
theorem isRelPrime_zero_left : IsRelPrime 0 x ↔ IsUnit x 
Arguments: {α : Type u_1} [CommMonoidWithZero α] {x : α}
Goal: IsRelPrime 0 x ↔ IsUnit x
-- isRelPrime_zero_right
theorem isRelPrime_zero_right : IsRelPrime x 0 ↔ IsUnit x 
Arguments: {α : Type u_1} [CommMonoidWithZero α] {x : α}
Goal: IsRelPrime x 0 ↔ IsUnit x
-- not_isRelPrime_zero_zero
theorem not_isRelPrime_zero_zero [Nontrivial α] : ¬IsRelPrime (0 : α) 0 
Arguments: {α : Type u_1} [CommMonoidWithZero α] [Nontrivial α]
Goal: ¬IsRelPrime 0 0
-- IsRelPrime.ne_zero_or_ne_zero
theorem IsRelPrime.ne_zero_or_ne_zero [Nontrivial α] (h : IsRelPrime x y) : x ≠ 0 ∨ y ≠ 0 
Arguments: {α : Type u_1} [CommMonoidWithZero α] {x y : α} [Nontrivial α] (h : IsRelPrime x y)
Goal: x ≠ 0 ∨ y ≠ 0
-- isRelPrime_of_no_nonunits_factors
theorem isRelPrime_of_no_nonunits_factors [MonoidWithZero α] {x y : α} (nonzero : ¬(x = 0 ∧ y = 0))
    (H : ∀ z, ¬ IsUnit z → z ≠ 0 → z ∣ x → ¬z ∣ y) : IsRelPrime x y 
Arguments: {α : Type u_1} [MonoidWithZero α] {x y : α} (nonzero : ¬(x = 0 ∧ y = 0)) (H : ∀ (z : α), ¬IsUnit z → z ≠ 0 → z ∣ x → ¬z ∣ y)
Goal: IsRelPrime x y
-- dvd_and_not_dvd_iff
theorem dvd_and_not_dvd_iff [CancelCommMonoidWithZero α] {x y : α} :
    x ∣ y ∧ ¬y ∣ x ↔ DvdNotUnit x y 
Arguments: {α : Type u_1} [CancelCommMonoidWithZero α] {x y : α}
Goal: x ∣ y ∧ ¬y ∣ x ↔ DvdNotUnit x y
-- ne_zero_of_dvd_ne_zero
theorem ne_zero_of_dvd_ne_zero {p q : α} (h₁ : q ≠ 0) (h₂ : p ∣ q) : p ≠ 0 
Arguments: {α : Type u_1} [MonoidWithZero α] {p q : α} (h₁ : q ≠ 0) (h₂ : p ∣ q)
Goal: p ≠ 0
-- isPrimal_zero
theorem isPrimal_zero : IsPrimal (0 : α) 
Arguments: {α : Type u_1} [MonoidWithZero α]
Goal: IsPrimal 0
-- IsPrimal.mul
theorem IsPrimal.mul {α} [CancelCommMonoidWithZero α] {m n : α}
    (hm : IsPrimal m) (hn : IsPrimal n) : IsPrimal (m * n) 
Arguments: {α : Type u_2} [CancelCommMonoidWithZero α] {m n : α} (hm : IsPrimal m) (hn : IsPrimal n)
Goal: IsPrimal (m * n)
-- dvd_antisymm
theorem dvd_antisymm : a ∣ b → b ∣ a → a = b 
Arguments: {α : Type u_1} [CancelCommMonoidWithZero α] {a b : α} [Subsingleton αˣ]
Goal: a ∣ b → b ∣ a → a = b
-- dvd_antisymm'
theorem dvd_antisymm' : a ∣ b → b ∣ a → b = a 
Arguments: {α : Type u_1} [CancelCommMonoidWithZero α] {a b : α} [Subsingleton αˣ]
Goal: a ∣ b → b ∣ a → b = a
-- eq_of_forall_dvd
theorem eq_of_forall_dvd (h : ∀ c, a ∣ c ↔ b ∣ c) : a = b 
Arguments: {α : Type u_1} [CancelCommMonoidWithZero α] {a b : α} [Subsingleton αˣ] (h : ∀ (c : α), a ∣ c ↔ b ∣ c)
Goal: a = b
-- eq_of_forall_dvd'
theorem eq_of_forall_dvd' (h : ∀ c, c ∣ a ↔ c ∣ b) : a = b 
Arguments: {α : Type u_1} [CancelCommMonoidWithZero α] {a b : α} [Subsingleton αˣ] (h : ∀ (c : α), c ∣ a ↔ c ∣ b)
Goal: a = b
-- pow_dvd_pow_iff
lemma pow_dvd_pow_iff (ha₀ : a ≠ 0) (ha : ¬IsUnit a) : a ^ n ∣ a ^ m ↔ n ≤ m 
Arguments: {α : Type u_1} [CancelCommMonoidWithZero α] {a : α} {m n : ℕ} (ha₀ : a ≠ 0) (ha : ¬IsUnit a)
Goal: a ^ n ∣ a ^ m ↔ n ≤ m
-- Set.indicator_mul
lemma indicator_mul (s : Set ι) (f g : ι → M₀) :
    indicator s (fun i ↦ f i * g i) = fun i ↦ indicator s f i * indicator s g i 
Arguments: {ι : Type u_1} {M₀ : Type u_4} [MulZeroClass M₀] (s : Set ι) (f g : ι → M₀)
Goal: (s.indicator fun i => f i * g i) = fun i => s.indicator f i * s.indicator g i
-- Set.indicator_mul_left
lemma indicator_mul_left (s : Set ι) (f g : ι → M₀) :
    indicator s (fun j ↦ f j * g j) i = indicator s f i * g i 
Arguments: {ι : Type u_1} {M₀ : Type u_4} [MulZeroClass M₀] {i : ι} (s : Set ι) (f g : ι → M₀)
Goal: s.indicator (fun j => f j * g j) i = s.indicator f i * g i
-- Set.indicator_mul_right
lemma indicator_mul_right (s : Set ι) (f g : ι → M₀) :
    indicator s (fun j ↦ f j * g j) i = f i * indicator s g i 
Arguments: {ι : Type u_1} {M₀ : Type u_4} [MulZeroClass M₀] {i : ι} (s : Set ι) (f g : ι → M₀)
Goal: s.indicator (fun j => f j * g j) i = f i * s.indicator g i
-- Set.inter_indicator_mul
lemma inter_indicator_mul (f g : ι → M₀) (i : ι) :
    (s ∩ t).indicator (fun j ↦ f j * g j) i = s.indicator f i * t.indicator g i 
Arguments: {ι : Type u_1} {M₀ : Type u_4} [MulZeroClass M₀] {s t : Set ι} (f g : ι → M₀) (i : ι)
Goal: (s ∩ t).indicator (fun j => f j * g j) i = s.indicator f i * t.indicator g i
-- Set.inter_indicator_one
lemma inter_indicator_one : (s ∩ t).indicator (1 : ι → M₀) = s.indicator 1 * t.indicator 1 
Arguments: {ι : Type u_1} {M₀ : Type u_4} [MulZeroOneClass M₀] {s t : Set ι}
Goal: (s ∩ t).indicator 1 = s.indicator 1 * t.indicator 1
-- Set.indicator_prod_one
lemma indicator_prod_one {t : Set κ} {j : κ} :
    (s ×ˢ t).indicator (1 : ι × κ → M₀) (i, j) = s.indicator 1 i * t.indicator 1 j 
Arguments: {ι : Type u_1} {κ : Type u_2} {M₀ : Type u_4} [MulZeroOneClass M₀] {s : Set ι} {i : ι} {t : Set κ} {j : κ}
Goal: (s ×ˢ t).indicator 1 (i, j) = s.indicator 1 i * t.indicator 1 j
-- Set.indicator_eq_zero_iff_not_mem
lemma indicator_eq_zero_iff_not_mem : indicator s 1 i = (0 : M₀) ↔ i ∉ s 
Arguments: {ι : Type u_1} (M₀ : Type u_4) [MulZeroOneClass M₀] {s : Set ι} {i : ι} [Nontrivial M₀]
Goal: s.indicator 1 i = 0 ↔ i ∉ s
-- Set.indicator_eq_one_iff_mem
lemma indicator_eq_one_iff_mem : indicator s 1 i = (1 : M₀) ↔ i ∈ s 
Arguments: {ι : Type u_1} (M₀ : Type u_4) [MulZeroOneClass M₀] {s : Set ι} {i : ι} [Nontrivial M₀]
Goal: s.indicator 1 i = 1 ↔ i ∈ s
-- Set.indicator_one_inj
lemma indicator_one_inj (h : indicator s (1 : ι → M₀) = indicator t 1) : s = t 
Arguments: {ι : Type u_1} (M₀ : Type u_4) [MulZeroOneClass M₀] {s t : Set ι} [Nontrivial M₀] (h : s.indicator 1 = t.indicator 1)
Goal: s = t
-- Function.support_one
@[simp] lemma support_one : support (1 : ι → R) = univ 
Arguments: {ι : Type u_1} (R : Type u_5) [Zero R] [One R] [NeZero 1]
Goal: Function.support 1 = Set.univ
-- Function.mulSupport_zero
@[simp] lemma mulSupport_zero : mulSupport (0 : ι → R) = univ 
Arguments: {ι : Type u_1} (R : Type u_5) [Zero R] [One R] [NeZero 1]
Goal: Function.mulSupport 0 = Set.univ
-- Function.support_mul_subset_left
lemma support_mul_subset_left (f g : ι → M₀) : support (fun x ↦ f x * g x) ⊆ support f 
Arguments: {ι : Type u_1} {M₀ : Type u_4} [MulZeroClass M₀] (f g : ι → M₀)
Goal: (Function.support fun x => f x * g x) ⊆ Function.support f
-- Function.support_mul_subset_right
lemma support_mul_subset_right (f g : ι → M₀) : support (fun x ↦ f x * g x) ⊆ support g 
Arguments: {ι : Type u_1} {M₀ : Type u_4} [MulZeroClass M₀] (f g : ι → M₀)
Goal: (Function.support fun x => f x * g x) ⊆ Function.support g
-- Function.support_mul
@[simp] lemma support_mul (f g : ι → M₀) : support (fun x ↦ f x * g x) = support f ∩ support g 
Arguments: {ι : Type u_1} {M₀ : Type u_4} [MulZeroClass M₀] [NoZeroDivisors M₀] (f g : ι → M₀)
Goal: (Function.support fun x => f x * g x) = Function.support f ∩ Function.support g
-- Function.support_mul'
@[simp] lemma support_mul' (f g : ι → M₀) : support (f * g) = support f ∩ support g 
Arguments: {ι : Type u_1} {M₀ : Type u_4} [MulZeroClass M₀] [NoZeroDivisors M₀] (f g : ι → M₀)
Goal: Function.support (f * g) = Function.support f ∩ Function.support g
-- Function.support_pow
@[simp] lemma support_pow (f : ι → M₀) (hn : n ≠ 0) : support (fun a ↦ f a ^ n) = support f 
Arguments: {ι : Type u_1} {M₀ : Type u_4} [MonoidWithZero M₀] [NoZeroDivisors M₀] {n : ℕ} (f : ι → M₀) (hn : n ≠ 0)
Goal: (Function.support fun a => f a ^ n) = Function.support f
-- Function.support_pow'
@[simp] lemma support_pow' (f : ι → M₀) (hn : n ≠ 0) : support (f ^ n) = support f 
Arguments: {ι : Type u_1} {M₀ : Type u_4} [MonoidWithZero M₀] [NoZeroDivisors M₀] {n : ℕ} (f : ι → M₀) (hn : n ≠ 0)
Goal: Function.support (f ^ n) = Function.support f
-- Function.support_inv
@[simp] lemma support_inv (f : ι → G₀) : support (fun a ↦ (f a)⁻¹) = support f 
Arguments: {ι : Type u_1} {G₀ : Type u_3} [GroupWithZero G₀] (f : ι → G₀)
Goal: (Function.support fun a => (f a)⁻¹) = Function.support f
-- Function.support_inv'
@[simp] lemma support_inv' (f : ι → G₀) : support f⁻¹ = support f 
Arguments: {ι : Type u_1} {G₀ : Type u_3} [GroupWithZero G₀] (f : ι → G₀)
Goal: Function.support f⁻¹ = Function.support f
-- Function.support_div
@[simp] lemma support_div (f g : ι → G₀) : support (fun a ↦ f a / g a) = support f ∩ support g 
Arguments: {ι : Type u_1} {G₀ : Type u_3} [GroupWithZero G₀] (f g : ι → G₀)
Goal: (Function.support fun a => f a / g a) = Function.support f ∩ Function.support g
-- Function.support_div'
@[simp] lemma support_div' (f g : ι → G₀) : support (f / g) = support f ∩ support g 
Arguments: {ι : Type u_1} {G₀ : Type u_3} [GroupWithZero G₀] (f g : ι → G₀)
Goal: Function.support (f / g) = Function.support f ∩ Function.support g
-- Function.mulSupport_one_add
lemma mulSupport_one_add [AddLeftCancelMonoid R] (f : ι → R) :
    mulSupport (fun x ↦ 1 + f x) = support f 
Arguments: {ι : Type u_1} {R : Type u_5} [One R] [AddLeftCancelMonoid R] (f : ι → R)
Goal: (Function.mulSupport fun x => 1 + f x) = Function.support f
-- Function.mulSupport_one_add'
lemma mulSupport_one_add' [AddLeftCancelMonoid R] (f : ι → R) : mulSupport (1 + f) = support f 
Arguments: {ι : Type u_1} {R : Type u_5} [One R] [AddLeftCancelMonoid R] (f : ι → R)
Goal: Function.mulSupport (1 + f) = Function.support f
-- Function.mulSupport_add_one
lemma mulSupport_add_one [AddRightCancelMonoid R] (f : ι → R) :
    mulSupport (fun x ↦ f x + 1) = support f 
Arguments: {ι : Type u_1} {R : Type u_5} [One R] [AddRightCancelMonoid R] (f : ι → R)
Goal: (Function.mulSupport fun x => f x + 1) = Function.support f
-- Function.mulSupport_add_one'
lemma mulSupport_add_one' [AddRightCancelMonoid R] (f : ι → R) : mulSupport (f + 1) = support f 
Arguments: {ι : Type u_1} {R : Type u_5} [One R] [AddRightCancelMonoid R] (f : ι → R)
Goal: Function.mulSupport (f + 1) = Function.support f
-- Function.mulSupport_one_sub'
lemma mulSupport_one_sub' [AddGroup R] (f : ι → R) : mulSupport (1 - f) = support f 
Arguments: {ι : Type u_1} {R : Type u_5} [One R] [AddGroup R] (f : ι → R)
Goal: Function.mulSupport (1 - f) = Function.support f
-- Function.mulSupport_one_sub
lemma mulSupport_one_sub [AddGroup R] (f : ι → R) :
    mulSupport (fun x ↦ 1 - f x) = support f 
Arguments: {ι : Type u_1} {R : Type u_5} [One R] [AddGroup R] (f : ι → R)
Goal: (Function.mulSupport fun x => 1 - f x) = Function.support f
-- LieSubmodule.lcs_zero
@[simp]
theorem lcs_zero (N : LieSubmodule R L M) : N.lcs 0 = N 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] (N : LieSubmodule R L M)
Goal: LieSubmodule.lcs 0 N = N
-- LieSubmodule.lcs_succ
@[simp]
theorem lcs_succ : N.lcs (k + 1) = ⁅(⊤ : LieIdeal R L), N.lcs k⁆ 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] (k : ℕ) (N : LieSubmodule R L M)
Goal: LieSubmodule.lcs (k + 1) N = ⁅⊤, LieSubmodule.lcs k N⁆
-- LieSubmodule.lcs_sup
@[simp]
lemma lcs_sup {N₁ N₂ : LieSubmodule R L M} {k : ℕ} :
    (N₁ ⊔ N₂).lcs k = N₁.lcs k ⊔ N₂.lcs k 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] {N₁ N₂ : LieSubmodule R L M} {k : ℕ}
Goal: LieSubmodule.lcs k (N₁ ⊔ N₂) = LieSubmodule.lcs k N₁ ⊔ LieSubmodule.lcs k N₂
-- LieModule.lowerCentralSeries_zero
@[simp]
theorem lowerCentralSeries_zero : lowerCentralSeries R L M 0 = ⊤ 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M]
Goal: LieModule.lowerCentralSeries R L M 0 = ⊤
-- LieModule.lowerCentralSeries_succ
@[simp]
theorem lowerCentralSeries_succ :
    lowerCentralSeries R L M (k + 1) = ⁅(⊤ : LieIdeal R L), lowerCentralSeries R L M k⁆ 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] (k : ℕ)
Goal: LieModule.lowerCentralSeries R L M (k + 1) = ⁅⊤, LieModule.lowerCentralSeries R L M k⁆
-- LieSubmodule.lcs_le_self
theorem lcs_le_self : N.lcs k ≤ N 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] (k : ℕ) (N : LieSubmodule R L M)
Goal: LieSubmodule.lcs k N ≤ N
-- LieSubmodule.lowerCentralSeries_eq_lcs_comap
theorem lowerCentralSeries_eq_lcs_comap : lowerCentralSeries R L N k = (N.lcs k).comap N.incl 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (k : ℕ) (N : LieSubmodule R L M)
Goal: LieModule.lowerCentralSeries R L (↥↑N) k = LieSubmodule.comap N.incl (LieSubmodule.lcs k N)
-- LieSubmodule.lowerCentralSeries_map_eq_lcs
theorem lowerCentralSeries_map_eq_lcs : (lowerCentralSeries R L N k).map N.incl = N.lcs k 
Arguments: {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (k : ℕ) (N : LieSubmodule R L M)
Goal: LieSubmodule.map N.incl (LieModule.lowerCentralSeries R L (↥↑N) k) = LieSubmodule.lcs k N
-- LieModule.antitone_lowerCentralSeries
theorem antitone_lowerCentralSeries : Antitone <| lowerCentralSeries R L M 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M]
Goal: Antitone (LieModule.lowerCentralSeries R L M)
-- LieModule.eventually_iInf_lowerCentralSeries_eq
theorem eventually_iInf_lowerCentralSeries_eq [IsArtinian R M] :
    ∀ᶠ l in Filter.atTop, ⨅ k, lowerCentralSeries R L M k = lowerCentralSeries R L M l 
Arguments: (R : Type u) (L : Type v) (M : Type w) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [IsArtinian R M]
Goal: ∀ᶠ (l : ℕ) in Filter.atTop, ⨅ k, LieModule.lowerCentralSeries R L M k = LieModule.lowerCentralSeries R L M l