ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- PerfectionMap.lift_symm_apply
@[simps]
noncomputable def lift [PerfectRing R p] (S : Type u₂) [CommSemiring S] [CharP S p] (P : Type u₃)
    [CommSemiring P] [CharP P p] [PerfectRing P p] (π : P →+* S) (m : PerfectionMap p π) :
    (R →+* S) ≃ (R →+* P) 
Arguments: (p : ℕ) [Fact (Nat.Prime p)] (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing R p] (S : Type u₂) [CommSemiring S] [CharP S p] (P : Type u₃) [CommSemiring P] [CharP P p] [PerfectRing P p] (π : P →+* S) (m : PerfectionMap p π) (f : R →+* P)
Goal: (PerfectionMap.lift p R S P π m).symm f = π.comp f
-- PerfectionMap.hom_ext
theorem hom_ext [PerfectRing R p] {S : Type u₂} [CommSemiring S] [CharP S p] {P : Type u₃}
    [CommSemiring P] [CharP P p] [PerfectRing P p] (π : P →+* S) (m : PerfectionMap p π)
    {f g : R →+* P} (hfg : ∀ x, π (f x) = π (g x)) : f = g 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] [PerfectRing R p] {S : Type u₂} [CommSemiring S] [CharP S p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] (π : P →+* S) (m : PerfectionMap p π) {f g : R →+* P} (hfg : ∀ (x : R), π (f x) = π (g x))
Goal: f = g
-- PerfectionMap.comp_map
theorem comp_map {π : P →+* R} (m : PerfectionMap p π) {σ : Q →+* S} (n : PerfectionMap p σ)
    (φ : R →+* S) : σ.comp (map p m n φ) = φ.comp π 
Arguments: (p : ℕ) [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] {S : Type u₂} [CommSemiring S] [CharP S p] {Q : Type u₄} [CommSemiring Q] [CharP Q p] [PerfectRing Q p] {π : P →+* R} (m : PerfectionMap p π) {σ : Q →+* S} (n : PerfectionMap p σ) (φ : R →+* S)
Goal: σ.comp (PerfectionMap.map p m n φ) = φ.comp π
-- PerfectionMap.map_map
theorem map_map {π : P →+* R} (m : PerfectionMap p π) {σ : Q →+* S} (n : PerfectionMap p σ)
    (φ : R →+* S) (x : P) : σ (map p m n φ x) = φ (π x) 
Arguments: (p : ℕ) [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] {S : Type u₂} [CommSemiring S] [CharP S p] {Q : Type u₄} [CommSemiring Q] [CharP Q p] [PerfectRing Q p] {π : P →+* R} (m : PerfectionMap p π) {σ : Q →+* S} (n : PerfectionMap p σ) (φ : R →+* S) (x : P)
Goal: σ ((PerfectionMap.map p m n φ) x) = φ (π x)
-- PerfectionMap.map_eq_map
theorem map_eq_map (φ : R →+* S) : map p (of p R) (of p S) φ = Perfection.map p φ 
Arguments: (p : ℕ) [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {S : Type u₂} [CommSemiring S] [CharP S p] (φ : R →+* S)
Goal: PerfectionMap.map p ⋯ ⋯ φ = Perfection.map p φ
-- ModP.preVal_mk
theorem preVal_mk {x : O} (hx : (Ideal.Quotient.mk _ x : ModP K v O hv p) ≠ 0) :
    preVal K v O hv p (Ideal.Quotient.mk _ x) = v (algebraMap O K x) 
Arguments: {K : Type u₁} [Field K] {v : Valuation K ℝ≥0} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} {x : O} (hx : (Ideal.Quotient.mk (Ideal.span {↑p})) x ≠ 0)
Goal: ModP.preVal K v O hv p ((Ideal.Quotient.mk (Ideal.span {↑p})) x) = v ((algebraMap O K) x)
-- ModP.preVal_zero
theorem preVal_zero : preVal K v O hv p 0 = 0 
Arguments: {K : Type u₁} [Field K] {v : Valuation K ℝ≥0} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ}
Goal: ModP.preVal K v O hv p 0 = 0
-- ModP.preVal_mul
theorem preVal_mul {x y : ModP K v O hv p} (hxy0 : x * y ≠ 0) :
    preVal K v O hv p (x * y) = preVal K v O hv p x * preVal K v O hv p y 
Arguments: {K : Type u₁} [Field K] {v : Valuation K ℝ≥0} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} {x y : ModP K v O hv p} (hxy0 : x * y ≠ 0)
Goal: ModP.preVal K v O hv p (x * y) = ModP.preVal K v O hv p x * ModP.preVal K v O hv p y
-- ModP.preVal_add
theorem preVal_add (x y : ModP K v O hv p) :
    preVal K v O hv p (x + y) ≤ max (preVal K v O hv p x) (preVal K v O hv p y) 
Arguments: {K : Type u₁} [Field K] {v : Valuation K ℝ≥0} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} (x y : ModP K v O hv p)
Goal: ModP.preVal K v O hv p (x + y) ≤ max (ModP.preVal K v O hv p x) (ModP.preVal K v O hv p y)
-- ModP.v_p_lt_preVal
theorem v_p_lt_preVal {x : ModP K v O hv p} : v p < preVal K v O hv p x ↔ x ≠ 0 
Arguments: {K : Type u₁} [Field K] {v : Valuation K ℝ≥0} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} {x : ModP K v O hv p}
Goal: v ↑p < ModP.preVal K v O hv p x ↔ x ≠ 0
-- ModP.preVal_eq_zero
theorem preVal_eq_zero {x : ModP K v O hv p} : preVal K v O hv p x = 0 ↔ x = 0 
Arguments: {K : Type u₁} [Field K] {v : Valuation K ℝ≥0} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} {x : ModP K v O hv p}
Goal: ModP.preVal K v O hv p x = 0 ↔ x = 0
-- ModP.v_p_lt_val
theorem v_p_lt_val {x : O} :
    v p < v (algebraMap O K x) ↔ (Ideal.Quotient.mk _ x : ModP K v O hv p) ≠ 0 
Arguments: {K : Type u₁} [Field K] {v : Valuation K ℝ≥0} {O : Type u₂} [CommRing O] [Algebra O K] (hv : v.Integers O) {p : ℕ} {x : O}
Goal: v ↑p < v ((algebraMap O K) x) ↔ (Ideal.Quotient.mk (Ideal.span {↑p})) x ≠ 0
-- ModP.mul_ne_zero_of_pow_p_ne_zero
theorem mul_ne_zero_of_pow_p_ne_zero {x y : ModP K v O hv p} (hx : x ^ p ≠ 0) (hy : y ^ p ≠ 0) :
    x * y ≠ 0 
Arguments: {K : Type u₁} [Field K] {v : Valuation K ℝ≥0} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} [hp : Fact (Nat.Prime p)] {x y : ModP K v O hv p} (hx : x ^ p ≠ 0) (hy : y ^ p ≠ 0)
Goal: x * y ≠ 0
-- PreTilt.coeff_nat_find_add_ne_zero
theorem coeff_nat_find_add_ne_zero {f : PreTilt K v O hv p} {h : ∃ n, coeff _ _ n f ≠ 0} (k : ℕ) :
    coeff _ _ (Nat.find h + k) f ≠ 0 
Arguments: {K : Type u₁} [Field K] {v : Valuation K ℝ≥0} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} [hp : Fact (Nat.Prime p)] [hvp : Fact (v ↑p ≠ 1)] {f : PreTilt K v O hv p} {h : ∃ n, (Perfection.coeff (ModP K v O hv p) p n) f ≠ 0} (k : ℕ)
Goal: (Perfection.coeff (ModP K v O hv p) p (Nat.find h + k)) f ≠ 0
-- PreTilt.valAux_eq
theorem valAux_eq {f : PreTilt K v O hv p} {n : ℕ} (hfn : coeff _ _ n f ≠ 0) :
    valAux K v O hv p f = ModP.preVal K v O hv p (coeff _ _ n f) ^ p ^ n 
Arguments: {K : Type u₁} [Field K] {v : Valuation K ℝ≥0} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} [hp : Fact (Nat.Prime p)] [hvp : Fact (v ↑p ≠ 1)] {f : PreTilt K v O hv p} {n : ℕ} (hfn : (Perfection.coeff (ModP K v O hv p) p n) f ≠ 0)
Goal: PreTilt.valAux K v O hv p f = ModP.preVal K v O hv p ((Perfection.coeff (ModP K v O hv p) p n) f) ^ p ^ n
-- PreTilt.valAux_zero
theorem valAux_zero : valAux K v O hv p 0 = 0 
Arguments: {K : Type u₁} [Field K] {v : Valuation K ℝ≥0} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} [hp : Fact (Nat.Prime p)] [hvp : Fact (v ↑p ≠ 1)]
Goal: PreTilt.valAux K v O hv p 0 = 0
-- PreTilt.valAux_one
theorem valAux_one : valAux K v O hv p 1 = 1 
Arguments: {K : Type u₁} [Field K] {v : Valuation K ℝ≥0} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} [hp : Fact (Nat.Prime p)] [hvp : Fact (v ↑p ≠ 1)]
Goal: PreTilt.valAux K v O hv p 1 = 1
-- PreTilt.valAux_mul
theorem valAux_mul (f g : PreTilt K v O hv p) :
    valAux K v O hv p (f * g) = valAux K v O hv p f * valAux K v O hv p g 
Arguments: {K : Type u₁} [Field K] {v : Valuation K ℝ≥0} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} [hp : Fact (Nat.Prime p)] [hvp : Fact (v ↑p ≠ 1)] (f g : PreTilt K v O hv p)
Goal: PreTilt.valAux K v O hv p (f * g) = PreTilt.valAux K v O hv p f * PreTilt.valAux K v O hv p g
-- PreTilt.valAux_add
theorem valAux_add (f g : PreTilt K v O hv p) :
    valAux K v O hv p (f + g) ≤ max (valAux K v O hv p f) (valAux K v O hv p g) 
Arguments: {K : Type u₁} [Field K] {v : Valuation K ℝ≥0} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} [hp : Fact (Nat.Prime p)] [hvp : Fact (v ↑p ≠ 1)] (f g : PreTilt K v O hv p)
Goal: PreTilt.valAux K v O hv p (f + g) ≤ max (PreTilt.valAux K v O hv p f) (PreTilt.valAux K v O hv p g)
-- PreTilt.map_eq_zero
theorem map_eq_zero {f : PreTilt K v O hv p} : val K v O hv p f = 0 ↔ f = 0 
Arguments: {K : Type u₁} [Field K] {v : Valuation K ℝ≥0} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} [hp : Fact (Nat.Prime p)] [hvp : Fact (v ↑p ≠ 1)] {f : PreTilt K v O hv p}
Goal: (PreTilt.val K v O hv p) f = 0 ↔ f = 0
-- Submodule.mem_colon
theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N 
Arguments: {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {N P : Submodule R M} {r : R}
Goal: r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N
-- Submodule.mem_colon'
theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N 
Arguments: {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {N P : Submodule R M} {r : R}
Goal: r ∈ N.colon P ↔ P ≤ Submodule.comap (r • LinearMap.id) N
-- Submodule.colon_top
@[simp]
theorem colon_top {I : Ideal R} : I.colon ⊤ = I 
Arguments: {R : Type u_1} [CommRing R] {I : Ideal R}
Goal: Submodule.colon I ⊤ = I
-- Submodule.colon_bot
@[simp]
theorem colon_bot : colon ⊥ N = N.annihilator 
Arguments: {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {N : Submodule R M}
Goal: ⊥.colon N = N.annihilator
-- Submodule.colon_mono
theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ 
Arguments: {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {N₁ N₂ P₁ P₂ : Submodule R M} (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂)
Goal: N₁.colon P₂ ≤ N₂.colon P₁
-- Submodule.iInf_colon_iSup
theorem iInf_colon_iSup (ι₁ : Sort*) (f : ι₁ → Submodule R M) (ι₂ : Sort*)
    (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) 
Arguments: {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (ι₁ : Sort u_6) (f : ι₁ → Submodule R M) (ι₂ : Sort u_7) (g : ι₂ → Submodule R M)
Goal: (⨅ i, f i).colon (⨆ j, g j) = ⨅ i, ⨅ j, (f i).colon (g j)
-- Submodule.mem_colon_singleton
@[simp]
theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} :
    r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N 
Arguments: {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {N : Submodule R M} {x : M} {r : R}
Goal: r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N
-- Ideal.mem_colon_singleton
@[simp]
theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} :
    r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I 
Arguments: {R : Type u_1} [CommRing R] {I : Ideal R} {x r : R}
Goal: r ∈ Submodule.colon I (Ideal.span {x}) ↔ r * x ∈ I
-- Submodule.annihilator_quotient
theorem annihilator_quotient {N : Submodule R M} :
    Module.annihilator R (M ⧸ N) = N.colon ⊤ 
Arguments: {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {N : Submodule R M}
Goal: Module.annihilator R (M ⧸ N) = N.colon ⊤
-- Ideal.annihilator_quotient
theorem _root_.Ideal.annihilator_quotient {I : Ideal R} : Module.annihilator R (R ⧸ I) = I 
Arguments: {R : Type u_1} [CommRing R] {I : Ideal R}
Goal: Module.annihilator R (R ⧸ I) = I
-- AssociatePrimes.mem_iff
theorem AssociatePrimes.mem_iff : I ∈ associatedPrimes R M ↔ IsAssociatedPrime I M 
Arguments: {R : Type u_1} [CommRing R] {I : Ideal R} {M : Type u_2} [AddCommGroup M] [Module R M]
Goal: I ∈ associatedPrimes R M ↔ IsAssociatedPrime I M
-- IsAssociatedPrime.isPrime
theorem IsAssociatedPrime.isPrime (h : IsAssociatedPrime I M) : I.IsPrime 
Arguments: {R : Type u_1} [CommRing R] {I : Ideal R} {M : Type u_2} [AddCommGroup M] [Module R M] (h : IsAssociatedPrime I M)
Goal: I.IsPrime
-- IsAssociatedPrime.map_of_injective
theorem IsAssociatedPrime.map_of_injective (h : IsAssociatedPrime I M) (hf : Function.Injective f) :
    IsAssociatedPrime I M' 
Arguments: {R : Type u_1} [CommRing R] {I : Ideal R} {M : Type u_2} [AddCommGroup M] [Module R M] {M' : Type u_3} [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M') (h : IsAssociatedPrime I M) (hf : Function.Injective ⇑f)
Goal: IsAssociatedPrime I M'
-- LinearEquiv.isAssociatedPrime_iff
theorem LinearEquiv.isAssociatedPrime_iff (l : M ≃ₗ[R] M') :
    IsAssociatedPrime I M ↔ IsAssociatedPrime I M' 
Arguments: {R : Type u_1} [CommRing R] {I : Ideal R} {M : Type u_2} [AddCommGroup M] [Module R M] {M' : Type u_3} [AddCommGroup M'] [Module R M'] (l : M ≃ₗ[R] M')
Goal: IsAssociatedPrime I M ↔ IsAssociatedPrime I M'
-- not_isAssociatedPrime_of_subsingleton
theorem not_isAssociatedPrime_of_subsingleton [Subsingleton M] : ¬IsAssociatedPrime I M 
Arguments: {R : Type u_1} [CommRing R] {I : Ideal R} {M : Type u_2} [AddCommGroup M] [Module R M] [Subsingleton M]
Goal: ¬IsAssociatedPrime I M
-- exists_le_isAssociatedPrime_of_isNoetherianRing
theorem exists_le_isAssociatedPrime_of_isNoetherianRing [H : IsNoetherianRing R] (x : M)
    (hx : x ≠ 0) : ∃ P : Ideal R, IsAssociatedPrime P M ∧ (R ∙ x).annihilator ≤ P 
Arguments: (R : Type u_1) [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [H : IsNoetherianRing R] (x : M) (hx : x ≠ 0)
Goal: ∃ P, IsAssociatedPrime P M ∧ (Submodule.span R {x}).annihilator ≤ P
-- associatedPrimes.subset_of_injective
theorem associatedPrimes.subset_of_injective (hf : Function.Injective f) :
    associatedPrimes R M ⊆ associatedPrimes R M' 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {M' : Type u_3} [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M') (hf : Function.Injective ⇑f)
Goal: associatedPrimes R M ⊆ associatedPrimes R M'
-- LinearEquiv.AssociatedPrimes.eq
theorem LinearEquiv.AssociatedPrimes.eq (l : M ≃ₗ[R] M') :
    associatedPrimes R M = associatedPrimes R M' 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {M' : Type u_3} [AddCommGroup M'] [Module R M'] (l : M ≃ₗ[R] M')
Goal: associatedPrimes R M = associatedPrimes R M'
-- associatedPrimes.eq_empty_of_subsingleton
theorem associatedPrimes.eq_empty_of_subsingleton [Subsingleton M] : associatedPrimes R M = ∅ 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [Subsingleton M]
Goal: associatedPrimes R M = ∅
-- associatedPrimes.nonempty
theorem associatedPrimes.nonempty [IsNoetherianRing R] [Nontrivial M] :
    (associatedPrimes R M).Nonempty 
Arguments: (R : Type u_1) [CommRing R] (M : Type u_2) [AddCommGroup M] [Module R M] [IsNoetherianRing R] [Nontrivial M]
Goal: (associatedPrimes R M).Nonempty
-- biUnion_associatedPrimes_eq_zero_divisors
theorem biUnion_associatedPrimes_eq_zero_divisors [IsNoetherianRing R] :
    ⋃ p ∈ associatedPrimes R M, p = { r : R | ∃ x : M, x ≠ 0 ∧ r • x = 0 } 
Arguments: (R : Type u_1) [CommRing R] (M : Type u_2) [AddCommGroup M] [Module R M] [IsNoetherianRing R]
Goal: ⋃ p ∈ associatedPrimes R M, ↑p = {r | ∃ x, x ≠ 0 ∧ r • x = 0}
-- IsAssociatedPrime.annihilator_le
theorem IsAssociatedPrime.annihilator_le (h : IsAssociatedPrime I M) :
    (⊤ : Submodule R M).annihilator ≤ I 
Arguments: {R : Type u_1} [CommRing R] {I : Ideal R} {M : Type u_2} [AddCommGroup M] [Module R M] (h : IsAssociatedPrime I M)
Goal: ⊤.annihilator ≤ I
-- IsAssociatedPrime.eq_radical
theorem IsAssociatedPrime.eq_radical (hI : I.IsPrimary) (h : IsAssociatedPrime J (R ⧸ I)) :
    J = I.radical 
Arguments: {R : Type u_1} [CommRing R] {I J : Ideal R} (hI : I.IsPrimary) (h : IsAssociatedPrime J (R ⧸ I))
Goal: J = I.radical
-- associatedPrimes.eq_singleton_of_isPrimary
theorem associatedPrimes.eq_singleton_of_isPrimary [IsNoetherianRing R] (hI : I.IsPrimary) :
    associatedPrimes R (R ⧸ I) = {I.radical} 
Arguments: {R : Type u_1} [CommRing R] {I : Ideal R} [IsNoetherianRing R] (hI : I.IsPrimary)
Goal: associatedPrimes R (R ⧸ I) = {I.radical}
-- Ideal.pointwise_smul_def
theorem pointwise_smul_def {a : M} (S : Ideal R) :
    a • S = S.map (MulSemiringAction.toRingHom _ _ a) 
Arguments: {M : Type u_1} {R : Type u_2} [Monoid M] [CommRing R] [MulSemiringAction M R] {a : M} (S : Ideal R)
Goal: a • S = Ideal.map (MulSemiringAction.toRingHom M R a) S
-- Ideal.smul_mem_pointwise_smul
theorem smul_mem_pointwise_smul (m : M) (r : R) (S : Ideal R) : r ∈ S → m • r ∈ m • S 
Arguments: {M : Type u_1} {R : Type u_2} [Monoid M] [CommRing R] [MulSemiringAction M R] (m : M) (r : R) (S : Ideal R)
Goal: r ∈ S → m • r ∈ m • S
-- Ideal.smul_bot
@[simp]
theorem smul_bot (a : M) : a • (⊥ : Ideal R) = ⊥ 
Arguments: {M : Type u_1} {R : Type u_2} [Monoid M] [CommRing R] [MulSemiringAction M R] (a : M)
Goal: a • ⊥ = ⊥
-- Ideal.smul_sup
theorem smul_sup (a : M) (S T : Ideal R) : a • (S ⊔ T) = a • S ⊔ a • T 
Arguments: {M : Type u_1} {R : Type u_2} [Monoid M] [CommRing R] [MulSemiringAction M R] (a : M) (S T : Ideal R)
Goal: a • (S ⊔ T) = a • S ⊔ a • T
-- Ideal.smul_closure
theorem smul_closure (a : M) (s : Set R) : a • span s = span (a • s) 
Arguments: {M : Type u_1} {R : Type u_2} [Monoid M] [CommRing R] [MulSemiringAction M R] (a : M) (s : Set R)
Goal: a • Ideal.span s = Ideal.span (a • s)
-- Ideal.pointwise_smul_eq_comap
theorem pointwise_smul_eq_comap {a : M} (S : Ideal R) :
    a • S = S.comap (MulSemiringAction.toRingAut _ _ a).symm 
Arguments: {M : Type u_1} {R : Type u_2} [Group M] [CommRing R] [MulSemiringAction M R] {a : M} (S : Ideal R)
Goal: a • S = Ideal.comap (RingEquiv.symm ((MulSemiringAction.toRingAut M R) a)) S
-- Ideal.smul_mem_pointwise_smul_iff
@[simp]
theorem smul_mem_pointwise_smul_iff {a : M} {S : Ideal R} {x : R} : a • x ∈ a • S ↔ x ∈ S 
Arguments: {M : Type u_1} {R : Type u_2} [Group M] [CommRing R] [MulSemiringAction M R] {a : M} {S : Ideal R} {x : R}
Goal: a • x ∈ a • S ↔ x ∈ S
-- Ideal.mem_pointwise_smul_iff_inv_smul_mem
theorem mem_pointwise_smul_iff_inv_smul_mem {a : M} {S : Ideal R} {x : R} :
    x ∈ a • S ↔ a⁻¹ • x ∈ S 
Arguments: {M : Type u_1} {R : Type u_2} [Group M] [CommRing R] [MulSemiringAction M R] {a : M} {S : Ideal R} {x : R}
Goal: x ∈ a • S ↔ a⁻¹ • x ∈ S
-- Ideal.pointwise_smul_toAddSubmonoid
@[simp]
theorem pointwise_smul_toAddSubmonoid (a : M) (S : Ideal R)
    (ha : Function.Surjective fun r : R => a • r) :
    (a • S).toAddSubmonoid = a • S.toAddSubmonoid 
Arguments: {M : Type u_1} {R : Type u_2} [Group M] [CommRing R] [MulSemiringAction M R] (a : M) (S : Ideal R) (ha : Function.Surjective fun r => a • r)
Goal: (a • S).toAddSubmonoid = a • S.toAddSubmonoid
-- Ideal.pointwise_smul_toAddSubGroup
@[simp]
theorem pointwise_smul_toAddSubGroup (a : M) (S : Ideal R)
    (ha : Function.Surjective fun r : R => a • r)  :
    (a • S).toAddSubgroup = a • S.toAddSubgroup 
Arguments: {M : Type u_1} {R : Type u_2} [Group M] [CommRing R] [MulSemiringAction M R] (a : M) (S : Ideal R) (ha : Function.Surjective fun r => a • r)
Goal: Submodule.toAddSubgroup (a • S) = a • Submodule.toAddSubgroup S
-- Ideal.mem_inv_pointwise_smul_iff
theorem mem_inv_pointwise_smul_iff {a : M} {S : Ideal R} {x : R} : x ∈ a⁻¹ • S ↔ a • x ∈ S 
Arguments: {M : Type u_1} {R : Type u_2} [Group M] [CommRing R] [MulSemiringAction M R] {a : M} {S : Ideal R} {x : R}
Goal: x ∈ a⁻¹ • S ↔ a • x ∈ S
-- Ideal.pointwise_smul_le_pointwise_smul_iff
@[simp]
theorem pointwise_smul_le_pointwise_smul_iff {a : M} {S T : Ideal R} : a • S ≤ a • T ↔ S ≤ T 
Arguments: {M : Type u_1} {R : Type u_2} [Group M] [CommRing R] [MulSemiringAction M R] {a : M} {S T : Ideal R}
Goal: a • S ≤ a • T ↔ S ≤ T
-- Ideal.pointwise_smul_subset_iff
theorem pointwise_smul_subset_iff {a : M} {S T : Ideal R} : a • S ≤ T ↔ S ≤ a⁻¹ • T 
Arguments: {M : Type u_1} {R : Type u_2} [Group M] [CommRing R] [MulSemiringAction M R] {a : M} {S T : Ideal R}
Goal: a • S ≤ T ↔ S ≤ a⁻¹ • T
-- Ideal.subset_pointwise_smul_iff
theorem subset_pointwise_smul_iff {a : M} {S T : Ideal R} : S ≤ a • T ↔ a⁻¹ • S ≤ T 
Arguments: {M : Type u_1} {R : Type u_2} [Group M] [CommRing R] [MulSemiringAction M R] {a : M} {S T : Ideal R}
Goal: S ≤ a • T ↔ a⁻¹ • S ≤ T
-- algebraicIndependent_iff_ker_eq_bot
theorem algebraicIndependent_iff_ker_eq_bot :
    AlgebraicIndependent R x ↔
      RingHom.ker (MvPolynomial.aeval x : MvPolynomial ι R →ₐ[R] A).toRingHom = ⊥ 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A]
Goal: AlgebraicIndependent R x ↔ RingHom.ker (MvPolynomial.aeval x).toRingHom = ⊥
-- algebraicIndependent_iff
theorem algebraicIndependent_iff :
    AlgebraicIndependent R x ↔
      ∀ p : MvPolynomial ι R, MvPolynomial.aeval (x : ι → A) p = 0 → p = 0 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A]
Goal: AlgebraicIndependent R x ↔ ∀ (p : MvPolynomial ι R), (MvPolynomial.aeval x) p = 0 → p = 0
-- AlgebraicIndependent.eq_zero_of_aeval_eq_zero
theorem AlgebraicIndependent.eq_zero_of_aeval_eq_zero (h : AlgebraicIndependent R x) :
    ∀ p : MvPolynomial ι R, MvPolynomial.aeval (x : ι → A) p = 0 → p = 0 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] (h : AlgebraicIndependent R x) (p : MvPolynomial ι R)
Goal: (MvPolynomial.aeval x) p = 0 → p = 0
-- algebraicIndependent_iff_injective_aeval
theorem algebraicIndependent_iff_injective_aeval :
    AlgebraicIndependent R x ↔ Injective (MvPolynomial.aeval x : MvPolynomial ι R →ₐ[R] A) 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A]
Goal: AlgebraicIndependent R x ↔ Function.Injective ⇑(MvPolynomial.aeval x)
-- algebraicIndependent_empty_type_iff
@[simp]
theorem algebraicIndependent_empty_type_iff [IsEmpty ι] :
    AlgebraicIndependent R x ↔ Injective (algebraMap R A) 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] [IsEmpty ι]
Goal: AlgebraicIndependent R x ↔ Function.Injective ⇑(algebraMap R A)
-- AlgebraicIndependent.algebraMap_injective
theorem algebraMap_injective : Injective (algebraMap R A) 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] (hx : AlgebraicIndependent R x)
Goal: Function.Injective ⇑(algebraMap R A)
-- AlgebraicIndependent.linearIndependent
theorem linearIndependent : LinearIndependent R x 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] (hx : AlgebraicIndependent R x)
Goal: LinearIndependent R x
-- AlgebraicIndependent.injective
protected theorem injective [Nontrivial R] : Injective x 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] (hx : AlgebraicIndependent R x) [Nontrivial R]
Goal: Function.Injective x
-- AlgebraicIndependent.ne_zero
theorem ne_zero [Nontrivial R] (i : ι) : x i ≠ 0 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] (hx : AlgebraicIndependent R x) [Nontrivial R] (i : ι)
Goal: x i ≠ 0
-- AlgebraicIndependent.comp
theorem comp (f : ι' → ι) (hf : Function.Injective f) : AlgebraicIndependent R (x ∘ f) 
Arguments: {ι : Type u_1} {ι' : Type u_2} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] (hx : AlgebraicIndependent R x) (f : ι' → ι) (hf : Function.Injective f)
Goal: AlgebraicIndependent R (x ∘ f)
-- AlgebraicIndependent.coe_range
theorem coe_range : AlgebraicIndependent R ((↑) : range x → A) 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] (hx : AlgebraicIndependent R x)
Goal: AlgebraicIndependent R Subtype.val
-- AlgebraicIndependent.map
theorem map {f : A →ₐ[R] A'} (hf_inj : Set.InjOn f (adjoin R (range x))) :
    AlgebraicIndependent R (f ∘ x) 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {A' : Type u_6} {x : ι → A} [CommRing R] [CommRing A] [CommRing A'] [Algebra R A] [Algebra R A'] (hx : AlgebraicIndependent R x) {f : A →ₐ[R] A'} (hf_inj : Set.InjOn ⇑f ↑(Algebra.adjoin R (Set.range x)))
Goal: AlgebraicIndependent R (⇑f ∘ x)
-- AlgebraicIndependent.map'
theorem map' {f : A →ₐ[R] A'} (hf_inj : Injective f) : AlgebraicIndependent R (f ∘ x) 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {A' : Type u_6} {x : ι → A} [CommRing R] [CommRing A] [CommRing A'] [Algebra R A] [Algebra R A'] (hx : AlgebraicIndependent R x) {f : A →ₐ[R] A'} (hf_inj : Function.Injective ⇑f)
Goal: AlgebraicIndependent R (⇑f ∘ x)
-- AlgebraicIndependent.of_comp
theorem of_comp (f : A →ₐ[R] A') (hfv : AlgebraicIndependent R (f ∘ x)) :
    AlgebraicIndependent R x 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_5} {A' : Type u_6} {x : ι → A} [CommRing R] [CommRing A] [CommRing A'] [Algebra R A] [Algebra R A'] (f : A →ₐ[R] A') (hfv : AlgebraicIndependent R (⇑f ∘ x))
Goal: AlgebraicIndependent R x