ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- isRelPrime_mul_unit_left_left
theorem isRelPrime_mul_unit_left_left : IsRelPrime (x * y) z ↔ IsRelPrime y z 
Arguments: {α : Type u_1} [CommMonoid α] {x y z : α} (hu : IsUnit x)
Goal: IsRelPrime (x * y) z ↔ IsRelPrime y z
-- isRelPrime_mul_unit_left_right
theorem isRelPrime_mul_unit_left_right : IsRelPrime y (x * z) ↔ IsRelPrime y z 
Arguments: {α : Type u_1} [CommMonoid α] {x y z : α} (hu : IsUnit x)
Goal: IsRelPrime y (x * z) ↔ IsRelPrime y z
-- isRelPrime_mul_unit_left
theorem isRelPrime_mul_unit_left : IsRelPrime (x * y) (x * z) ↔ IsRelPrime y z 
Arguments: {α : Type u_1} [CommMonoid α] {x y z : α} (hu : IsUnit x)
Goal: IsRelPrime (x * y) (x * z) ↔ IsRelPrime y z
-- isRelPrime_mul_unit_right_left
theorem isRelPrime_mul_unit_right_left : IsRelPrime (y * x) z ↔ IsRelPrime y z 
Arguments: {α : Type u_1} [CommMonoid α] {x y z : α} (hu : IsUnit x)
Goal: IsRelPrime (y * x) z ↔ IsRelPrime y z
-- isRelPrime_mul_unit_right_right
theorem isRelPrime_mul_unit_right_right : IsRelPrime y (z * x) ↔ IsRelPrime y z 
Arguments: {α : Type u_1} [CommMonoid α] {x y z : α} (hu : IsUnit x)
Goal: IsRelPrime y (z * x) ↔ IsRelPrime y z
-- isRelPrime_mul_unit_right
theorem isRelPrime_mul_unit_right : IsRelPrime (y * x) (z * x) ↔ IsRelPrime y z 
Arguments: {α : Type u_1} [CommMonoid α] {x y z : α} (hu : IsUnit x)
Goal: IsRelPrime (y * x) (z * x) ↔ IsRelPrime y z
-- IsRelPrime.dvd_of_dvd_mul_right_of_isPrimal
theorem IsRelPrime.dvd_of_dvd_mul_right_of_isPrimal (H1 : IsRelPrime x z) (H2 : x ∣ y * z)
    (h : IsPrimal x) : x ∣ y 
Arguments: {α : Type u_1} [CommMonoid α] {x y z : α} (H1 : IsRelPrime x z) (H2 : x ∣ y * z) (h : IsPrimal x)
Goal: x ∣ y
-- IsRelPrime.dvd_of_dvd_mul_left_of_isPrimal
theorem IsRelPrime.dvd_of_dvd_mul_left_of_isPrimal (H1 : IsRelPrime x y) (H2 : x ∣ y * z)
    (h : IsPrimal x) : x ∣ z 
Arguments: {α : Type u_1} [CommMonoid α] {x y z : α} (H1 : IsRelPrime x y) (H2 : x ∣ y * z) (h : IsPrimal x)
Goal: x ∣ z
-- IsRelPrime.mul_dvd_of_right_isPrimal
theorem IsRelPrime.mul_dvd_of_right_isPrimal (H : IsRelPrime x y) (H1 : x ∣ z) (H2 : y ∣ z)
    (hy : IsPrimal y) : x * y ∣ z 
Arguments: {α : Type u_1} [CommMonoid α] {x y z : α} (H : IsRelPrime x y) (H1 : x ∣ z) (H2 : y ∣ z) (hy : IsPrimal y)
Goal: x * y ∣ z
-- IsRelPrime.mul_dvd_of_left_isPrimal
theorem IsRelPrime.mul_dvd_of_left_isPrimal (H : IsRelPrime x y) (H1 : x ∣ z) (H2 : y ∣ z)
    (hx : IsPrimal x) : x * y ∣ z 
Arguments: {α : Type u_1} [CommMonoid α] {x y z : α} (H : IsRelPrime x y) (H1 : x ∣ z) (H2 : y ∣ z) (hx : IsPrimal x)
Goal: x * y ∣ z
-- IsRelPrime.dvd_of_dvd_mul_right
theorem IsRelPrime.dvd_of_dvd_mul_right (H1 : IsRelPrime x z) (H2 : x ∣ y * z) : x ∣ y 
Arguments: {α : Type u_1} [CommMonoid α] {x y z : α} [DecompositionMonoid α] (H1 : IsRelPrime x z) (H2 : x ∣ y * z)
Goal: x ∣ y
-- IsRelPrime.dvd_of_dvd_mul_left
theorem IsRelPrime.dvd_of_dvd_mul_left (H1 : IsRelPrime x y) (H2 : x ∣ y * z) : x ∣ z 
Arguments: {α : Type u_1} [CommMonoid α] {x y z : α} [DecompositionMonoid α] (H1 : IsRelPrime x y) (H2 : x ∣ y * z)
Goal: x ∣ z
-- IsRelPrime.mul_left
theorem IsRelPrime.mul_left (H1 : IsRelPrime x z) (H2 : IsRelPrime y z) : IsRelPrime (x * y) z 
Arguments: {α : Type u_1} [CommMonoid α] {x y z : α} [DecompositionMonoid α] (H1 : IsRelPrime x z) (H2 : IsRelPrime y z)
Goal: IsRelPrime (x * y) z
-- IsRelPrime.mul_right
theorem IsRelPrime.mul_right (H1 : IsRelPrime x y) (H2 : IsRelPrime x z) :
    IsRelPrime x (y * z) 
Arguments: {α : Type u_1} [CommMonoid α] {x y z : α} [DecompositionMonoid α] (H1 : IsRelPrime x y) (H2 : IsRelPrime x z)
Goal: IsRelPrime x (y * z)
-- IsRelPrime.mul_left_iff
theorem IsRelPrime.mul_left_iff : IsRelPrime (x * y) z ↔ IsRelPrime x z ∧ IsRelPrime y z 
Arguments: {α : Type u_1} [CommMonoid α] {x y z : α} [DecompositionMonoid α]
Goal: IsRelPrime (x * y) z ↔ IsRelPrime x z ∧ IsRelPrime y z
-- IsRelPrime.mul_right_iff
theorem IsRelPrime.mul_right_iff : IsRelPrime x (y * z) ↔ IsRelPrime x y ∧ IsRelPrime x z 
Arguments: {α : Type u_1} [CommMonoid α] {x y z : α} [DecompositionMonoid α]
Goal: IsRelPrime x (y * z) ↔ IsRelPrime x y ∧ IsRelPrime x z
-- IsRelPrime.mul_dvd
theorem IsRelPrime.mul_dvd (H : IsRelPrime x y) (H1 : x ∣ z) (H2 : y ∣ z) : x * y ∣ z 
Arguments: {α : Type u_1} [CommMonoid α] {x y z : α} [DecompositionMonoid α] (H : IsRelPrime x y) (H1 : x ∣ z) (H2 : y ∣ z)
Goal: x * y ∣ z
-- Dvd.intro
theorem Dvd.intro (c : α) (h : a * c = b) : a ∣ b 
Arguments: {α : Type u_1} [Semigroup α] {a b : α} (c : α) (h : a * c = b)
Goal: a ∣ b
-- exists_eq_mul_right_of_dvd
theorem exists_eq_mul_right_of_dvd (h : a ∣ b) : ∃ c, b = a * c 
Arguments: {α : Type u_1} [Semigroup α] {a b : α} (h : a ∣ b)
Goal: ∃ c, b = a * c
-- dvd_def
theorem dvd_def : a ∣ b ↔ ∃ c, b = a * c 
Arguments: {α : Type u_1} [Semigroup α] {a b : α}
Goal: a ∣ b ↔ ∃ c, b = a * c
-- Dvd.elim
theorem Dvd.elim {P : Prop} {a b : α} (H₁ : a ∣ b) (H₂ : ∀ c, b = a * c → P) : P 
Arguments: {α : Type u_1} [Semigroup α] {P : Prop} {a b : α} (H₁ : a ∣ b) (H₂ : ∀ (c : α), b = a * c → P)
Goal: P
-- dvd_trans
@[trans]
theorem dvd_trans : a ∣ b → b ∣ c → a ∣ c
  
Arguments: {α : Type u_1} [Semigroup α] {a b c : α}
Goal: a ∣ b → b ∣ c → a ∣ c
-- dvd_mul_right
@[simp]
theorem dvd_mul_right (a b : α) : a ∣ a * b 
Arguments: {α : Type u_1} [Semigroup α] (a b : α)
Goal: a ∣ a * b
-- dvd_mul_of_dvd_left
theorem dvd_mul_of_dvd_left (h : a ∣ b) (c : α) : a ∣ b * c 
Arguments: {α : Type u_1} [Semigroup α] {a b : α} (h : a ∣ b) (c : α)
Goal: a ∣ b * c
-- dvd_of_mul_right_dvd
theorem dvd_of_mul_right_dvd (h : a * b ∣ c) : a ∣ c 
Arguments: {α : Type u_1} [Semigroup α] {a b c : α} (h : a * b ∣ c)
Goal: a ∣ c
-- map_dvd
theorem map_dvd [Semigroup M] [Semigroup N] {F : Type*} [FunLike F M N] [MulHomClass F M N]
    (f : F) {a b} : a ∣ b → f a ∣ f b
  
Arguments: {M : Type u_2} {N : Type u_3} [Semigroup M] [Semigroup N] {F : Type u_4} [FunLike F M N] [MulHomClass F M N] (f : F) {a b : M}
Goal: a ∣ b → f a ∣ f b
-- MulHom.map_dvd
theorem MulHom.map_dvd [Semigroup M] [Semigroup N] (f : M →ₙ* N) {a b} : a ∣ b → f a ∣ f b 
Arguments: {M : Type u_2} {N : Type u_3} [Semigroup M] [Semigroup N] (f : M →ₙ* N) {a b : M}
Goal: a ∣ b → f a ∣ f b
-- MonoidHom.map_dvd
theorem MonoidHom.map_dvd [Monoid M] [Monoid N] (f : M →* N) {a b} : a ∣ b → f a ∣ f b 
Arguments: {M : Type u_2} {N : Type u_3} [Monoid M] [Monoid N] (f : M →* N) {a b : M}
Goal: a ∣ b → f a ∣ f b
-- decompositionMonoid_iff
@[mk_iff] class DecompositionMonoid : Prop where
  primal (a : α) : IsPrimal a
Arguments: (α : Type u_1) [Semigroup α]
Goal: DecompositionMonoid α ↔ ∀ (a : α), IsPrimal a
-- DecompositionMonoid.primal
@[mk_iff] class DecompositionMonoid : Prop where
  primal (a : α) : IsPrimal a
Arguments: {α : Type u_1} [Semigroup α] [self : DecompositionMonoid α] (a : α)
Goal: IsPrimal a
-- exists_dvd_and_dvd_of_dvd_mul
theorem exists_dvd_and_dvd_of_dvd_mul [DecompositionMonoid α] {b c a : α} (H : a ∣ b * c) :
    ∃ a₁ a₂, a₁ ∣ b ∧ a₂ ∣ c ∧ a = a₁ * a₂ 
Arguments: {α : Type u_1} [Semigroup α] [DecompositionMonoid α] {b c a : α} (H : a ∣ b * c)
Goal: ∃ a₁ a₂, a₁ ∣ b ∧ a₂ ∣ c ∧ a = a₁ * a₂
-- dvd_refl
@[refl, simp]
theorem dvd_refl (a : α) : a ∣ a 
Arguments: {α : Type u_1} [Monoid α] (a : α)
Goal: a ∣ a
-- dvd_rfl
theorem dvd_rfl : ∀ {a : α}, a ∣ a 
Arguments: {α : Type u_1} [Monoid α] {a : α}
Goal: a ∣ a
-- one_dvd
theorem one_dvd (a : α) : 1 ∣ a 
Arguments: {α : Type u_1} [Monoid α] (a : α)
Goal: 1 ∣ a
-- dvd_of_eq
theorem dvd_of_eq (h : a = b) : a ∣ b 
Arguments: {α : Type u_1} [Monoid α] {a b : α} (h : a = b)
Goal: a ∣ b
-- pow_dvd_pow
lemma pow_dvd_pow (a : α) (h : m ≤ n) : a ^ m ∣ a ^ n 
Arguments: {α : Type u_1} [Monoid α] {m n : ℕ} (a : α) (h : m ≤ n)
Goal: a ^ m ∣ a ^ n
-- dvd_pow
lemma dvd_pow (hab : a ∣ b) : ∀ {n : ℕ} (_ : n ≠ 0), a ∣ b ^ n
  
Arguments: {α : Type u_1} [Monoid α] {a b : α} (hab : a ∣ b) {n : ℕ}
Goal: n ≠ 0 → a ∣ b ^ n
-- dvd_pow_self
lemma dvd_pow_self (a : α) {n : ℕ} (hn : n ≠ 0) : a ∣ a ^ n 
Arguments: {α : Type u_1} [Monoid α] (a : α) {n : ℕ} (hn : n ≠ 0)
Goal: a ∣ a ^ n
-- mul_dvd_mul_left
theorem mul_dvd_mul_left (a : α) (h : b ∣ c) : a * b ∣ a * c 
Arguments: {α : Type u_1} [Monoid α] {b c : α} (a : α) (h : b ∣ c)
Goal: a * b ∣ a * c
-- Dvd.intro_left
theorem Dvd.intro_left (c : α) (h : c * a = b) : a ∣ b 
Arguments: {α : Type u_1} [CommSemigroup α] {a b : α} (c : α) (h : c * a = b)
Goal: a ∣ b
-- exists_eq_mul_left_of_dvd
theorem exists_eq_mul_left_of_dvd (h : a ∣ b) : ∃ c, b = c * a 
Arguments: {α : Type u_1} [CommSemigroup α] {a b : α} (h : a ∣ b)
Goal: ∃ c, b = c * a
-- dvd_iff_exists_eq_mul_left
theorem dvd_iff_exists_eq_mul_left : a ∣ b ↔ ∃ c, b = c * a 
Arguments: {α : Type u_1} [CommSemigroup α] {a b : α}
Goal: a ∣ b ↔ ∃ c, b = c * a
-- Dvd.elim_left
theorem Dvd.elim_left {P : Prop} (h₁ : a ∣ b) (h₂ : ∀ c, b = c * a → P) : P 
Arguments: {α : Type u_1} [CommSemigroup α] {a b : α} {P : Prop} (h₁ : a ∣ b) (h₂ : ∀ (c : α), b = c * a → P)
Goal: P
-- dvd_mul_left
@[simp]
theorem dvd_mul_left (a b : α) : a ∣ b * a 
Arguments: {α : Type u_1} [CommSemigroup α] (a b : α)
Goal: a ∣ b * a
-- dvd_mul_of_dvd_right
theorem dvd_mul_of_dvd_right (h : a ∣ b) (c : α) : a ∣ c * b 
Arguments: {α : Type u_1} [CommSemigroup α] {a b : α} (h : a ∣ b) (c : α)
Goal: a ∣ c * b
-- mul_dvd_mul
theorem mul_dvd_mul : ∀ {a b c d : α}, a ∣ b → c ∣ d → a * c ∣ b * d
  
Arguments: {α : Type u_1} [CommSemigroup α] {a b c d : α}
Goal: a ∣ b → c ∣ d → a * c ∣ b * d
-- dvd_of_mul_left_dvd
theorem dvd_of_mul_left_dvd (h : a * b ∣ c) : b ∣ c 
Arguments: {α : Type u_1} [CommSemigroup α] {a b c : α} (h : a * b ∣ c)
Goal: b ∣ c
-- dvd_mul
theorem dvd_mul [DecompositionMonoid α] {k m n : α} :
    k ∣ m * n ↔ ∃ d₁ d₂, d₁ ∣ m ∧ d₂ ∣ n ∧ k = d₁ * d₂ 
Arguments: {α : Type u_1} [CommSemigroup α] [DecompositionMonoid α] {k m n : α}
Goal: k ∣ m * n ↔ ∃ d₁ d₂, d₁ ∣ m ∧ d₂ ∣ n ∧ k = d₁ * d₂
-- mul_dvd_mul_right
theorem mul_dvd_mul_right (h : a ∣ b) (c : α) : a * c ∣ b * c 
Arguments: {α : Type u_1} [CommMonoid α] {a b : α} (h : a ∣ b) (c : α)
Goal: a * c ∣ b * c
-- pow_dvd_pow_of_dvd
theorem pow_dvd_pow_of_dvd (h : a ∣ b) : ∀ n : ℕ, a ^ n ∣ b ^ n
  
Arguments: {α : Type u_1} [CommMonoid α] {a b : α} (h : a ∣ b) (n : ℕ)
Goal: a ^ n ∣ b ^ n
-- map_inv_natCast_smul
theorem map_inv_natCast_smul [AddCommMonoid M] [AddCommMonoid M₂] {F : Type*} [FunLike F M M₂]
    [AddMonoidHomClass F M M₂] (f : F) (R S : Type*)
    [DivisionSemiring R] [DivisionSemiring S] [Module R M]
    [Module S M₂] (n : ℕ) (x : M) : f ((n⁻¹ : R) • x) = (n⁻¹ : S) • f x 
Arguments: {M : Type u_3} {M₂ : Type u_4} [AddCommMonoid M] [AddCommMonoid M₂] {F : Type u_5} [FunLike F M M₂] [AddMonoidHomClass F M M₂] (f : F) (R : Type u_6) (S : Type u_7) [DivisionSemiring R] [DivisionSemiring S] [Module R M] [Module S M₂] (n : ℕ) (x : M)
Goal: f ((↑n)⁻¹ • x) = (↑n)⁻¹ • f x
-- map_inv_intCast_smul
theorem map_inv_intCast_smul [AddCommGroup M] [AddCommGroup M₂] {F : Type*} [FunLike F M M₂]
    [AddMonoidHomClass F M M₂] (f : F) (R S : Type*) [DivisionRing R] [DivisionRing S] [Module R M]
    [Module S M₂] (z : ℤ) (x : M) : f ((z⁻¹ : R) • x) = (z⁻¹ : S) • f x 
Arguments: {M : Type u_3} {M₂ : Type u_4} [AddCommGroup M] [AddCommGroup M₂] {F : Type u_5} [FunLike F M M₂] [AddMonoidHomClass F M M₂] (f : F) (R : Type u_6) (S : Type u_7) [DivisionRing R] [DivisionRing S] [Module R M] [Module S M₂] (z : ℤ) (x : M)
Goal: f ((↑z)⁻¹ • x) = (↑z)⁻¹ • f x
-- inv_natCast_smul_eq
theorem inv_natCast_smul_eq {E : Type*} (R S : Type*) [AddCommMonoid E] [DivisionSemiring R]
    [DivisionSemiring S] [Module R E] [Module S E] (n : ℕ) (x : E) :
    (n⁻¹ : R) • x = (n⁻¹ : S) • x 
Arguments: {E : Type u_5} (R : Type u_6) (S : Type u_7) [AddCommMonoid E] [DivisionSemiring R] [DivisionSemiring S] [Module R E] [Module S E] (n : ℕ) (x : E)
Goal: (↑n)⁻¹ • x = (↑n)⁻¹ • x
-- inv_intCast_smul_eq
theorem inv_intCast_smul_eq {E : Type*} (R S : Type*) [AddCommGroup E] [DivisionRing R]
    [DivisionRing S] [Module R E] [Module S E] (n : ℤ) (x : E) : (n⁻¹ : R) • x = (n⁻¹ : S) • x 
Arguments: {E : Type u_5} (R : Type u_6) (S : Type u_7) [AddCommGroup E] [DivisionRing R] [DivisionRing S] [Module R E] [Module S E] (n : ℤ) (x : E)
Goal: (↑n)⁻¹ • x = (↑n)⁻¹ • x
-- inv_natCast_smul_comm
theorem inv_natCast_smul_comm {α E : Type*} (R : Type*) [AddCommMonoid E] [DivisionSemiring R]
    [Monoid α] [Module R E] [DistribMulAction α E] (n : ℕ) (s : α) (x : E) :
    (n⁻¹ : R) • s • x = s • (n⁻¹ : R) • x 
Arguments: {α : Type u_5} {E : Type u_6} (R : Type u_7) [AddCommMonoid E] [DivisionSemiring R] [Monoid α] [Module R E] [DistribMulAction α E] (n : ℕ) (s : α) (x : E)
Goal: (↑n)⁻¹ • s • x = s • (↑n)⁻¹ • x
-- inv_intCast_smul_comm
theorem inv_intCast_smul_comm {α E : Type*} (R : Type*) [AddCommGroup E] [DivisionRing R]
    [Monoid α] [Module R E] [DistribMulAction α E] (n : ℤ) (s : α) (x : E) :
    (n⁻¹ : R) • s • x = s • (n⁻¹ : R) • x 
Arguments: {α : Type u_5} {E : Type u_6} (R : Type u_7) [AddCommGroup E] [DivisionRing R] [Monoid α] [Module R E] [DistribMulAction α E] (n : ℤ) (s : α) (x : E)
Goal: (↑n)⁻¹ • s • x = s • (↑n)⁻¹ • x
-- Function.support_smul_subset_left
lemma support_smul_subset_left [Zero R] [Zero M] [SMulWithZero R M] (f : α → R) (g : α → M) :
    support (f • g) ⊆ support f 
Arguments: {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero R] [Zero M] [SMulWithZero R M] (f : α → R) (g : α → M)
Goal: Function.support (f • g) ⊆ Function.support f
-- Function.support_smul_subset_right
lemma support_smul_subset_right [Zero M] [SMulZeroClass R M] (f : α → R) (g : α → M) :
    support (f • g) ⊆ support g 
Arguments: {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero M] [SMulZeroClass R M] (f : α → R) (g : α → M)
Goal: Function.support (f • g) ⊆ Function.support g
-- Function.support_const_smul_of_ne_zero
lemma support_const_smul_of_ne_zero [Zero R] [Zero M] [SMulWithZero R M] [NoZeroSMulDivisors R M]
    (c : R) (g : α → M) (hc : c ≠ 0) : support (c • g) = support g 
Arguments: {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero R] [Zero M] [SMulWithZero R M] [NoZeroSMulDivisors R M] (c : R) (g : α → M) (hc : c ≠ 0)
Goal: Function.support (c • g) = Function.support g
-- Function.support_smul
lemma support_smul [Zero R] [Zero M] [SMulWithZero R M] [NoZeroSMulDivisors R M] (f : α → R)
    (g : α → M) : support (f • g) = support f ∩ support g 
Arguments: {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero R] [Zero M] [SMulWithZero R M] [NoZeroSMulDivisors R M] (f : α → R) (g : α → M)
Goal: Function.support (f • g) = Function.support f ∩ Function.support g
-- Function.support_const_smul_subset
lemma support_const_smul_subset [Zero M] [SMulZeroClass R M] (a : R) (f : α → M) :
    support (a • f) ⊆ support f 
Arguments: {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero M] [SMulZeroClass R M] (a : R) (f : α → M)
Goal: Function.support (a • f) ⊆ Function.support f
-- Set.indicator_smul_apply
lemma indicator_smul_apply (s : Set α) (r : α → R) (f : α → M) (a : α) :
    indicator s (fun a ↦ r a • f a) a = r a • indicator s f a 
Arguments: {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero M] [SMulZeroClass R M] (s : Set α) (r : α → R) (f : α → M) (a : α)
Goal: s.indicator (fun a => r a • f a) a = r a • s.indicator f a
-- Set.indicator_smul
lemma indicator_smul (s : Set α) (r : α → R) (f : α → M) :
    indicator s (fun a ↦ r a • f a) = fun a ↦ r a • indicator s f a 
Arguments: {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero M] [SMulZeroClass R M] (s : Set α) (r : α → R) (f : α → M)
Goal: (s.indicator fun a => r a • f a) = fun a => r a • s.indicator f a
-- Set.indicator_const_smul_apply
lemma indicator_const_smul_apply (s : Set α) (r : R) (f : α → M) (a : α) :
    indicator s (r • f ·) a = r • indicator s f a 
Arguments: {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero M] [SMulZeroClass R M] (s : Set α) (r : R) (f : α → M) (a : α)
Goal: s.indicator (fun x => r • f x) a = r • s.indicator f a
-- Set.indicator_const_smul
lemma indicator_const_smul (s : Set α) (r : R) (f : α → M) :
    indicator s (r • f ·) = (r • indicator s f ·) 
Arguments: {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero M] [SMulZeroClass R M] (s : Set α) (r : R) (f : α → M)
Goal: (s.indicator fun x => r • f x) = fun x => r • s.indicator f x
-- Set.indicator_smul_apply_left
lemma indicator_smul_apply_left (s : Set α) (r : α → R) (f : α → M) (a : α) :
    indicator s (fun a ↦ r a • f a) a = indicator s r a • f a 
Arguments: {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero R] [Zero M] [SMulWithZero R M] (s : Set α) (r : α → R) (f : α → M) (a : α)
Goal: s.indicator (fun a => r a • f a) a = s.indicator r a • f a
-- Set.indicator_smul_left
lemma indicator_smul_left (s : Set α) (r : α → R) (f : α → M) :
    indicator s (fun a ↦ r a • f a) = fun a ↦ indicator s r a • f a 
Arguments: {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero R] [Zero M] [SMulWithZero R M] (s : Set α) (r : α → R) (f : α → M)
Goal: (s.indicator fun a => r a • f a) = fun a => s.indicator r a • f a
-- Set.indicator_smul_const_apply
lemma indicator_smul_const_apply (s : Set α) (r : α → R) (m : M) (a : α) :
    indicator s (r · • m) a = indicator s r a • m 
Arguments: {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero R] [Zero M] [SMulWithZero R M] (s : Set α) (r : α → R) (m : M) (a : α)
Goal: s.indicator (fun x => r x • m) a = s.indicator r a • m
-- Set.indicator_smul_const
lemma indicator_smul_const (s : Set α) (r : α → R) (m : M) :
    indicator s (r · • m) = (indicator s r · • m) 
Arguments: {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero R] [Zero M] [SMulWithZero R M] (s : Set α) (r : α → R) (m : M)
Goal: (s.indicator fun x => r x • m) = fun x => s.indicator r x • m
-- IsLinearMap.isLinearMap_add
theorem isLinearMap_add [AddCommMonoid M] [Module R M] :
    IsLinearMap R fun x : M × M => x.1 + x.2 
Arguments: {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M]
Goal: IsLinearMap R fun x => x.1 + x.2
-- IsLinearMap.isLinearMap_sub
theorem isLinearMap_sub [AddCommGroup M] [Module R M] :
    IsLinearMap R fun x : M × M => x.1 - x.2 
Arguments: {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommGroup M] [Module R M]
Goal: IsLinearMap R fun x => x.1 - x.2
-- DomMulAct.smul_linearMap_apply
theorem _root_.DomMulAct.smul_linearMap_apply (a : S'ᵈᵐᵃ) (f : M →ₛₗ[σ₁₂] M₂) (x : M) :
    (a • f) x = f (DomMulAct.mk.symm a • x) 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {S' : Type u_17} [Monoid S'] [DistribMulAction S' M] [SMulCommClass R S' M] (a : S'ᵈᵐᵃ) (f : M →ₛₗ[σ₁₂] M₂) (x : M)
Goal: (a • f) x = f (DomMulAct.mk.symm a • x)
-- DomMulAct.mk_smul_linearMap_apply
@[simp]
theorem _root_.DomMulAct.mk_smul_linearMap_apply (a : S') (f : M →ₛₗ[σ₁₂] M₂) (x : M) :
    (DomMulAct.mk a • f) x = f (a • x) 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {S' : Type u_17} [Monoid S'] [DistribMulAction S' M] [SMulCommClass R S' M] (a : S') (f : M →ₛₗ[σ₁₂] M₂) (x : M)
Goal: (DomMulAct.mk a • f) x = f (a • x)
-- DomMulAct.coe_smul_linearMap
theorem  _root_.DomMulAct.coe_smul_linearMap (a : S'ᵈᵐᵃ) (f : M →ₛₗ[σ₁₂] M₂) :
    (a • f : M →ₛₗ[σ₁₂] M₂) = a • (f : M → M₂) 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {S' : Type u_17} [Monoid S'] [DistribMulAction S' M] [SMulCommClass R S' M] (a : S'ᵈᵐᵃ) (f : M →ₛₗ[σ₁₂] M₂)
Goal: ⇑(a • f) = a • ⇑f
-- LieModule.eventually_weightSpace_smul_add_eq_bot
lemma eventually_weightSpace_smul_add_eq_bot :
    ∀ᶠ (k : ℕ) in Filter.atTop, weightSpace M (k • χ₁ + χ₂) = ⊥ 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (M : Type u_3) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] (χ₁ χ₂ : L → R) [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (hχ₁ : χ₁ ≠ 0)
Goal: ∀ᶠ (k : ℕ) in Filter.atTop, LieModule.weightSpace M (k • χ₁ + χ₂) = ⊥
-- LieModule.exists_weightSpace_smul_add_eq_bot
lemma exists_weightSpace_smul_add_eq_bot :
    ∃ k > 0, weightSpace M (k • χ₁ + χ₂) = ⊥ 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (M : Type u_3) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] (χ₁ χ₂ : L → R) [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (hχ₁ : χ₁ ≠ 0)
Goal: ∃ k > 0, LieModule.weightSpace M (k • χ₁ + χ₂) = ⊥
-- LieModule.exists₂_weightSpace_smul_add_eq_bot
lemma exists₂_weightSpace_smul_add_eq_bot :
    ∃ᵉ (p < (0 : ℤ)) (q > (0 : ℤ)),
      weightSpace M (p • χ₁ + χ₂) = ⊥ ∧
      weightSpace M (q • χ₁ + χ₂) = ⊥ 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (M : Type u_3) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] (χ₁ χ₂ : L → R) [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (hχ₁ : χ₁ ≠ 0)
Goal: ∃ p < 0, ∃ q > 0, LieModule.weightSpace M (p • χ₁ + χ₂) = ⊥ ∧ LieModule.weightSpace M (q • χ₁ + χ₂) = ⊥
-- LieModule.weightSpaceChain_def
lemma weightSpaceChain_def :
    weightSpaceChain M χ₁ χ₂ p q = ⨆ k ∈ Ioo p q, weightSpace M (k • χ₁ + χ₂) 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (M : Type u_3) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] (χ₁ χ₂ : L → R) (p q : ℤ)
Goal: LieModule.weightSpaceChain M χ₁ χ₂ p q = ⨆ k ∈ Set.Ioo p q, LieModule.weightSpace M (k • χ₁ + χ₂)
-- LieModule.weightSpaceChain_def'
lemma weightSpaceChain_def' :
    weightSpaceChain M χ₁ χ₂ p q = ⨆ k ∈ Finset.Ioo p q, weightSpace M (k • χ₁ + χ₂) 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (M : Type u_3) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] (χ₁ χ₂ : L → R) (p q : ℤ)
Goal: LieModule.weightSpaceChain M χ₁ χ₂ p q = ⨆ k ∈ Finset.Ioo p q, LieModule.weightSpace M (k • χ₁ + χ₂)
-- LieModule.weightSpaceChain_neg
@[simp]
lemma weightSpaceChain_neg :
    weightSpaceChain M (-χ₁) χ₂ (-q) (-p) = weightSpaceChain M χ₁ χ₂ p q 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (M : Type u_3) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] (χ₁ χ₂ : L → R) (p q : ℤ)
Goal: LieModule.weightSpaceChain M (-χ₁) χ₂ (-q) (-p) = LieModule.weightSpaceChain M χ₁ χ₂ p q
-- LieModule.weightSpace_le_weightSpaceChain
lemma weightSpace_le_weightSpaceChain {k : ℤ} (hk : k ∈ Ioo p q) :
    weightSpace M (k • χ₁ + χ₂) ≤ weightSpaceChain M χ₁ χ₂ p q 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (M : Type u_3) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieAlgebra.IsNilpotent R L] (χ₁ χ₂ : L → R) (p q : ℤ) {k : ℤ} (hk : k ∈ Set.Ioo p q)
Goal: LieModule.weightSpace M (k • χ₁ + χ₂) ≤ LieModule.weightSpaceChain M χ₁ χ₂ p q
-- LieModule.lie_mem_weightSpaceChain_of_weightSpace_eq_bot_right
lemma lie_mem_weightSpaceChain_of_weightSpace_eq_bot_right [LieAlgebra.IsNilpotent R H]
    (hq : weightSpace M (q • α + χ) = ⊥)
    {x : L} (hx : x ∈ rootSpace H α)
    {y : M} (hy : y ∈ weightSpaceChain M α χ p q) :
    ⁅x, y⁆ ∈ weightSpaceChain M α χ p q 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (M : Type u_3) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {H : LieSubalgebra R L} (α χ : ↥H → R) (p q : ℤ) [LieAlgebra.IsNilpotent R ↥H] (hq : LieModule.weightSpace M (q • α + χ) = ⊥) {x : L} (hx : x ∈ LieAlgebra.rootSpace H α) {y : M} (hy : y ∈ LieModule.weightSpaceChain M α χ p q)
Goal: ⁅x, y⁆ ∈ LieModule.weightSpaceChain M α χ p q