ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Ideal.mul_add_mem_pow_succ_inj
theorem Ideal.mul_add_mem_pow_succ_inj (P : Ideal S) {i : ℕ} (a d d' e e' : S) (a_mem : a ∈ P ^ i)
    (e_mem : e ∈ P ^ (i + 1)) (e'_mem : e' ∈ P ^ (i + 1)) (h : d - d' ∈ P) :
    a * d + e - (a * d' + e') ∈ P ^ (i + 1) 
Arguments: {S : Type u_1} [CommRing S] (P : Ideal S) {i : ℕ} (a d d' e e' : S) (a_mem : a ∈ P ^ i) (e_mem : e ∈ P ^ (i + 1)) (e'_mem : e' ∈ P ^ (i + 1)) (h : d - d' ∈ P)
Goal: a * d + e - (a * d' + e') ∈ P ^ (i + 1)
-- Ideal.exists_mul_add_mem_pow_succ
theorem Ideal.exists_mul_add_mem_pow_succ [IsDedekindDomain S] {i : ℕ} (a c : S) (a_mem : a ∈ P ^ i)
    (a_not_mem : a ∉ P ^ (i + 1)) (c_mem : c ∈ P ^ i) :
    ∃ d : S, ∃ e ∈ P ^ (i + 1), a * d + e = c 
Arguments: {S : Type u_1} [CommRing S] {P : Ideal S} [P_prime : P.IsPrime] (hP : P ≠ ⊥) [IsDedekindDomain S] {i : ℕ} (a c : S) (a_mem : a ∈ P ^ i) (a_not_mem : a ∉ P ^ (i + 1)) (c_mem : c ∈ P ^ i)
Goal: ∃ d, ∃ e ∈ P ^ (i + 1), a * d + e = c
-- Ideal.mem_prime_of_mul_mem_pow
theorem Ideal.mem_prime_of_mul_mem_pow [IsDedekindDomain S] {P : Ideal S} [P_prime : P.IsPrime]
    (hP : P ≠ ⊥) {i : ℕ} {a b : S} (a_not_mem : a ∉ P ^ (i + 1)) (ab_mem : a * b ∈ P ^ (i + 1)) :
    b ∈ P 
Arguments: {S : Type u_1} [CommRing S] [IsDedekindDomain S] {P : Ideal S} [P_prime : P.IsPrime] (hP : P ≠ ⊥) {i : ℕ} {a b : S} (a_not_mem : a ∉ P ^ (i + 1)) (ab_mem : a * b ∈ P ^ (i + 1))
Goal: b ∈ P
-- Ideal.mul_add_mem_pow_succ_unique
theorem Ideal.mul_add_mem_pow_succ_unique [IsDedekindDomain S] {i : ℕ} (a d d' e e' : S)
    (a_not_mem : a ∉ P ^ (i + 1)) (e_mem : e ∈ P ^ (i + 1)) (e'_mem : e' ∈ P ^ (i + 1))
    (h : a * d + e - (a * d' + e') ∈ P ^ (i + 1)) : d - d' ∈ P 
Arguments: {S : Type u_1} [CommRing S] {P : Ideal S} [P_prime : P.IsPrime] (hP : P ≠ ⊥) [IsDedekindDomain S] {i : ℕ} (a d d' e e' : S) (a_not_mem : a ∉ P ^ (i + 1)) (e_mem : e ∈ P ^ (i + 1)) (e'_mem : e' ∈ P ^ (i + 1)) (h : a * d + e - (a * d' + e') ∈ P ^ (i + 1))
Goal: d - d' ∈ P
-- cardQuot_pow_of_prime
theorem cardQuot_pow_of_prime [IsDedekindDomain S] {i : ℕ} :
    cardQuot (P ^ i) = cardQuot P ^ i 
Arguments: {S : Type u_1} [CommRing S] {P : Ideal S} [P_prime : P.IsPrime] (hP : P ≠ ⊥) [IsDedekindDomain S] {i : ℕ}
Goal: Submodule.cardQuot (P ^ i) = Submodule.cardQuot P ^ i
-- cardQuot_mul
theorem cardQuot_mul [IsDedekindDomain S] [Module.Free ℤ S] (I J : Ideal S) :
    cardQuot (I * J) = cardQuot I * cardQuot J 
Arguments: {S : Type u_1} [CommRing S] [IsDomain S] [IsDedekindDomain S] [Module.Free ℤ S] (I J : Ideal S)
Goal: Submodule.cardQuot (I * J) = Submodule.cardQuot I * Submodule.cardQuot J
-- Ideal.absNorm_apply
theorem absNorm_apply (I : Ideal S) : absNorm I = cardQuot I 
Arguments: {S : Type u_1} [CommRing S] [IsDomain S] [Nontrivial S] [IsDedekindDomain S] [Module.Free ℤ S] (I : Ideal S)
Goal: Ideal.absNorm I = Submodule.cardQuot I
-- Ideal.absNorm_bot
@[simp]
theorem absNorm_bot : absNorm (⊥ : Ideal S) = 0 
Arguments: {S : Type u_1} [CommRing S] [IsDomain S] [Nontrivial S] [IsDedekindDomain S] [Module.Free ℤ S]
Goal: Ideal.absNorm ⊥ = 0
-- Ideal.absNorm_top
@[simp]
theorem absNorm_top : absNorm (⊤ : Ideal S) = 1 
Arguments: {S : Type u_1} [CommRing S] [IsDomain S] [Nontrivial S] [IsDedekindDomain S] [Module.Free ℤ S]
Goal: Ideal.absNorm ⊤ = 1
-- Ideal.absNorm_eq_one_iff
@[simp]
theorem absNorm_eq_one_iff {I : Ideal S} : absNorm I = 1 ↔ I = ⊤ 
Arguments: {S : Type u_1} [CommRing S] [IsDomain S] [Nontrivial S] [IsDedekindDomain S] [Module.Free ℤ S] {I : Ideal S}
Goal: Ideal.absNorm I = 1 ↔ I = ⊤
-- Ideal.absNorm_ne_zero_iff
theorem absNorm_ne_zero_iff (I : Ideal S) : Ideal.absNorm I ≠ 0 ↔ Finite (S ⧸ I) 
Arguments: {S : Type u_1} [CommRing S] [IsDomain S] [Nontrivial S] [IsDedekindDomain S] [Module.Free ℤ S] (I : Ideal S)
Goal: Ideal.absNorm I ≠ 0 ↔ Finite (S ⧸ I)
-- Ideal.natAbs_det_equiv
theorem natAbs_det_equiv (I : Ideal S) {E : Type*} [EquivLike E S I] [AddEquivClass E S I] (e : E) :
    Int.natAbs
        (LinearMap.det
          ((Submodule.subtype I).restrictScalars ℤ ∘ₗ AddMonoidHom.toIntLinearMap (e : S →+ I))) =
      Ideal.absNorm I 
Arguments: {S : Type u_1} [CommRing S] [IsDomain S] [Nontrivial S] [IsDedekindDomain S] [Module.Free ℤ S] [Module.Finite ℤ S] (I : Ideal S) {E : Type u_2} [EquivLike E S ↥I] [AddEquivClass E S ↥I] (e : E)
Goal: (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = Ideal.absNorm I
-- Ideal.natAbs_det_basis_change
theorem natAbs_det_basis_change {ι : Type*} [Fintype ι] [DecidableEq ι] (b : Basis ι ℤ S)
    (I : Ideal S) (bI : Basis ι ℤ I) : (b.det ((↑) ∘ bI)).natAbs = Ideal.absNorm I 
Arguments: {S : Type u_1} [CommRing S] [IsDomain S] [Nontrivial S] [IsDedekindDomain S] [Module.Free ℤ S] [Module.Finite ℤ S] {ι : Type u_2} [Fintype ι] [DecidableEq ι] (b : Basis ι ℤ S) (I : Ideal S) (bI : Basis ι ℤ ↥I)
Goal: (b.det (Subtype.val ∘ ⇑bI)).natAbs = Ideal.absNorm I
-- Ideal.absNorm_span_singleton
@[simp]
theorem absNorm_span_singleton (r : S) :
    absNorm (span ({r} : Set S)) = (Algebra.norm ℤ r).natAbs 
Arguments: {S : Type u_1} [CommRing S] [IsDomain S] [Nontrivial S] [IsDedekindDomain S] [Module.Free ℤ S] [Module.Finite ℤ S] (r : S)
Goal: Ideal.absNorm (Ideal.span {r}) = ((Algebra.norm ℤ) r).natAbs
-- Ideal.absNorm_dvd_absNorm_of_le
theorem absNorm_dvd_absNorm_of_le {I J : Ideal S} (h : J ≤ I) : Ideal.absNorm I ∣ Ideal.absNorm J 
Arguments: {S : Type u_1} [CommRing S] [IsDomain S] [Nontrivial S] [IsDedekindDomain S] [Module.Free ℤ S] {I J : Ideal S} (h : J ≤ I)
Goal: Ideal.absNorm I ∣ Ideal.absNorm J
-- Ideal.absNorm_dvd_norm_of_mem
theorem absNorm_dvd_norm_of_mem {I : Ideal S} {x : S} (h : x ∈ I) :
    ↑(Ideal.absNorm I) ∣ Algebra.norm ℤ x 
Arguments: {S : Type u_1} [CommRing S] [IsDomain S] [Nontrivial S] [IsDedekindDomain S] [Module.Free ℤ S] [Module.Finite ℤ S] {I : Ideal S} {x : S} (h : x ∈ I)
Goal: ↑(Ideal.absNorm I) ∣ (Algebra.norm ℤ) x
-- Ideal.absNorm_span_insert
@[simp]
theorem absNorm_span_insert (r : S) (s : Set S) :
    absNorm (span (insert r s)) ∣ gcd (absNorm (span s)) (Algebra.norm ℤ r).natAbs 
Arguments: {S : Type u_1} [CommRing S] [IsDomain S] [Nontrivial S] [IsDedekindDomain S] [Module.Free ℤ S] [Module.Finite ℤ S] (r : S) (s : Set S)
Goal: Ideal.absNorm (Ideal.span (insert r s)) ∣ gcd (Ideal.absNorm (Ideal.span s)) ((Algebra.norm ℤ) r).natAbs
-- Ideal.absNorm_eq_zero_iff
theorem absNorm_eq_zero_iff {I : Ideal S} : Ideal.absNorm I = 0 ↔ I = ⊥ 
Arguments: {S : Type u_1} [CommRing S] [IsDomain S] [Nontrivial S] [IsDedekindDomain S] [Module.Free ℤ S] [Module.Finite ℤ S] {I : Ideal S}
Goal: Ideal.absNorm I = 0 ↔ I = ⊥
-- Ideal.absNorm_ne_zero_of_nonZeroDivisors
theorem absNorm_ne_zero_of_nonZeroDivisors (I : (Ideal S)⁰) : Ideal.absNorm (I : Ideal S) ≠ 0 
Arguments: {S : Type u_1} [CommRing S] [IsDomain S] [Nontrivial S] [IsDedekindDomain S] [Module.Free ℤ S] [Module.Finite ℤ S] (I : ↥(Ideal S)⁰)
Goal: Ideal.absNorm ↑I ≠ 0
-- Ideal.irreducible_of_irreducible_absNorm
theorem irreducible_of_irreducible_absNorm {I : Ideal S} (hI : Irreducible (Ideal.absNorm I)) :
    Irreducible I 
Arguments: {S : Type u_1} [CommRing S] [IsDomain S] [Nontrivial S] [IsDedekindDomain S] [Module.Free ℤ S] {I : Ideal S} (hI : Irreducible (Ideal.absNorm I))
Goal: Irreducible I
-- Ideal.isPrime_of_irreducible_absNorm
theorem isPrime_of_irreducible_absNorm {I : Ideal S} (hI : Irreducible (Ideal.absNorm I)) :
    I.IsPrime 
Arguments: {S : Type u_1} [CommRing S] [IsDomain S] [Nontrivial S] [IsDedekindDomain S] [Module.Free ℤ S] {I : Ideal S} (hI : Irreducible (Ideal.absNorm I))
Goal: I.IsPrime
-- Ideal.prime_of_irreducible_absNorm_span
theorem prime_of_irreducible_absNorm_span {a : S} (ha : a ≠ 0)
    (hI : Irreducible (Ideal.absNorm (Ideal.span ({a} : Set S)))) : Prime a 
Arguments: {S : Type u_1} [CommRing S] [IsDomain S] [Nontrivial S] [IsDedekindDomain S] [Module.Free ℤ S] {a : S} (ha : a ≠ 0) (hI : Irreducible (Ideal.absNorm (Ideal.span {a})))
Goal: Prime a
-- Ideal.absNorm_mem
theorem absNorm_mem (I : Ideal S) : ↑(Ideal.absNorm I) ∈ I 
Arguments: {S : Type u_1} [CommRing S] [IsDomain S] [Nontrivial S] [IsDedekindDomain S] [Module.Free ℤ S] (I : Ideal S)
Goal: ↑(Ideal.absNorm I) ∈ I
-- Ideal.span_singleton_absNorm_le
theorem span_singleton_absNorm_le (I : Ideal S) : Ideal.span {(Ideal.absNorm I : S)} ≤ I 
Arguments: {S : Type u_1} [CommRing S] [IsDomain S] [Nontrivial S] [IsDedekindDomain S] [Module.Free ℤ S] (I : Ideal S)
Goal: Ideal.span {↑(Ideal.absNorm I)} ≤ I
-- Ideal.span_singleton_absNorm
theorem span_singleton_absNorm {I : Ideal S} (hI : (Ideal.absNorm I).Prime) :
    Ideal.span (singleton (Ideal.absNorm I : ℤ)) = I.comap (algebraMap ℤ S) 
Arguments: {S : Type u_1} [CommRing S] [IsDomain S] [Nontrivial S] [IsDedekindDomain S] [Module.Free ℤ S] {I : Ideal S} (hI : Nat.Prime (Ideal.absNorm I))
Goal: Ideal.span {↑(Ideal.absNorm I)} = Ideal.comap (algebraMap ℤ S) I
-- Ideal.finite_setOf_absNorm_eq
theorem finite_setOf_absNorm_eq [CharZero S] {n : ℕ} (hn : 0 < n) :
    {I : Ideal S | Ideal.absNorm I = n}.Finite 
Arguments: {S : Type u_1} [CommRing S] [IsDomain S] [Nontrivial S] [IsDedekindDomain S] [Module.Free ℤ S] [Module.Finite ℤ S] [CharZero S] {n : ℕ} (hn : 0 < n)
Goal: {I | Ideal.absNorm I = n}.Finite
-- Ideal.norm_dvd_iff
theorem norm_dvd_iff {x : S} (hx : Prime (Algebra.norm ℤ x)) {y : ℤ} :
    Algebra.norm ℤ x ∣ y ↔ x ∣ y 
Arguments: {S : Type u_1} [CommRing S] [IsDomain S] [Nontrivial S] [IsDedekindDomain S] [Module.Free ℤ S] [Module.Finite ℤ S] {x : S} (hx : Prime ((Algebra.norm ℤ) x)) {y : ℤ}
Goal: (Algebra.norm ℤ) x ∣ y ↔ x ∣ ↑y
-- Ideal.spanNorm_bot
@[simp]
theorem spanNorm_bot [Nontrivial S] [Module.Free R S] [Module.Finite R S] :
    spanNorm R (⊥ : Ideal S) = ⊥ 
Arguments: (R : Type u_1) [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [Nontrivial S] [Module.Free R S] [Module.Finite R S]
Goal: Ideal.spanNorm R ⊥ = ⊥
-- Ideal.spanNorm_eq_bot_iff
@[simp]
theorem spanNorm_eq_bot_iff [IsDomain R] [IsDomain S] [Module.Free R S] [Module.Finite R S]
    {I : Ideal S} : spanNorm R I = ⊥ ↔ I = ⊥ 
Arguments: {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [IsDomain R] [IsDomain S] [Module.Free R S] [Module.Finite R S] {I : Ideal S}
Goal: Ideal.spanNorm R I = ⊥ ↔ I = ⊥
-- Ideal.norm_mem_spanNorm
theorem norm_mem_spanNorm {I : Ideal S} (x : S) (hx : x ∈ I) : Algebra.norm R x ∈ I.spanNorm R 
Arguments: (R : Type u_1) [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] {I : Ideal S} (x : S) (hx : x ∈ I)
Goal: (Algebra.norm R) x ∈ Ideal.spanNorm R I
-- Ideal.spanNorm_singleton
@[simp]
theorem spanNorm_singleton {r : S} : spanNorm R (span ({r} : Set S)) = span {Algebra.norm R r} 
Arguments: (R : Type u_1) [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] {r : S}
Goal: Ideal.spanNorm R (Ideal.span {r}) = Ideal.span {(Algebra.norm R) r}
-- Ideal.spanNorm_top
@[simp]
theorem spanNorm_top : spanNorm R (⊤ : Ideal S) = ⊤ 
Arguments: (R : Type u_1) [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S]
Goal: Ideal.spanNorm R ⊤ = ⊤
-- Ideal.map_spanNorm
theorem map_spanNorm (I : Ideal S) {T : Type*} [CommRing T] (f : R →+* T) :
    map f (spanNorm R I) = span (f ∘ Algebra.norm R '' (I : Set S)) 
Arguments: (R : Type u_1) [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] (I : Ideal S) {T : Type u_3} [CommRing T] (f : R →+* T)
Goal: Ideal.map f (Ideal.spanNorm R I) = Ideal.span (⇑f ∘ ⇑(Algebra.norm R) '' ↑I)
-- Ideal.spanNorm_mono
@[mono]
theorem spanNorm_mono {I J : Ideal S} (h : I ≤ J) : spanNorm R I ≤ spanNorm R J 
Arguments: (R : Type u_1) [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] {I J : Ideal S} (h : I ≤ J)
Goal: Ideal.spanNorm R I ≤ Ideal.spanNorm R J
-- Ideal.spanNorm_localization
theorem spanNorm_localization (I : Ideal S) [Module.Finite R S] [Module.Free R S] (M : Submonoid R)
    {Rₘ : Type*} (Sₘ : Type*) [CommRing Rₘ] [Algebra R Rₘ] [CommRing Sₘ] [Algebra S Sₘ]
    [Algebra Rₘ Sₘ] [Algebra R Sₘ] [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ]
    [IsLocalization M Rₘ] [IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ] :
    spanNorm Rₘ (I.map (algebraMap S Sₘ)) = (spanNorm R I).map (algebraMap R Rₘ) 
Arguments: (R : Type u_1) [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] (I : Ideal S) [Module.Finite R S] [Module.Free R S] (M : Submonoid R) {Rₘ : Type u_3} (Sₘ : Type u_4) [CommRing Rₘ] [Algebra R Rₘ] [CommRing Sₘ] [Algebra S Sₘ] [Algebra Rₘ Sₘ] [Algebra R Sₘ] [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ] [IsLocalization M Rₘ] [IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ]
Goal: Ideal.spanNorm Rₘ (Ideal.map (algebraMap S Sₘ) I) = Ideal.map (algebraMap R Rₘ) (Ideal.spanNorm R I)
-- Ideal.spanNorm_mul_spanNorm_le
theorem spanNorm_mul_spanNorm_le (I J : Ideal S) :
    spanNorm R I * spanNorm R J ≤ spanNorm R (I * J) 
Arguments: (R : Type u_1) [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] (I J : Ideal S)
Goal: Ideal.spanNorm R I * Ideal.spanNorm R J ≤ Ideal.spanNorm R (I * J)
-- Ideal.spanNorm_mul_of_bot_or_top
theorem spanNorm_mul_of_bot_or_top [IsDomain R] [IsDomain S] [Module.Free R S] [Module.Finite R S]
    (eq_bot_or_top : ∀ I : Ideal R, I = ⊥ ∨ I = ⊤) (I J : Ideal S) :
    spanNorm R (I * J) = spanNorm R I * spanNorm R J 
Arguments: (R : Type u_1) [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [IsDomain R] [IsDomain S] [Module.Free R S] [Module.Finite R S] (eq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤) (I J : Ideal S)
Goal: Ideal.spanNorm R (I * J) = Ideal.spanNorm R I * Ideal.spanNorm R J
-- Ideal.spanNorm_mul_of_field
@[simp]
theorem spanNorm_mul_of_field {K : Type*} [Field K] [Algebra K S] [IsDomain S] [Module.Finite K S]
    (I J : Ideal S) : spanNorm K (I * J) = spanNorm K I * spanNorm K J 
Arguments: {S : Type u_2} [CommRing S] {K : Type u_3} [Field K] [Algebra K S] [IsDomain S] [Module.Finite K S] (I J : Ideal S)
Goal: Ideal.spanNorm K (I * J) = Ideal.spanNorm K I * Ideal.spanNorm K J
-- Ideal.spanNorm_mul
theorem spanNorm_mul (I J : Ideal S) : spanNorm R (I * J) = spanNorm R I * spanNorm R J 
Arguments: (R : Type u_1) [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [IsDomain R] [IsDomain S] [IsDedekindDomain R] [IsDedekindDomain S] [Module.Finite R S] [Module.Free R S] (I J : Ideal S)
Goal: Ideal.spanNorm R (I * J) = Ideal.spanNorm R I * Ideal.spanNorm R J
-- Ideal.relNorm_apply
theorem relNorm_apply (I : Ideal S) : relNorm R I = span (Algebra.norm R '' (I : Set S) : Set R) 
Arguments: (R : Type u_1) [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [IsDomain R] [IsDomain S] [IsDedekindDomain R] [IsDedekindDomain S] [Module.Finite R S] [Module.Free R S] (I : Ideal S)
Goal: (Ideal.relNorm R) I = Ideal.span (⇑(Algebra.norm R) '' ↑I)
-- Ideal.spanNorm_eq
@[simp]
theorem spanNorm_eq (I : Ideal S) : spanNorm R I = relNorm R I 
Arguments: (R : Type u_1) [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [IsDomain R] [IsDomain S] [IsDedekindDomain R] [IsDedekindDomain S] [Module.Finite R S] [Module.Free R S] (I : Ideal S)
Goal: Ideal.spanNorm R I = (Ideal.relNorm R) I
-- Ideal.relNorm_bot
@[simp]
theorem relNorm_bot : relNorm R (⊥ : Ideal S) = ⊥ 
Arguments: (R : Type u_1) [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [IsDomain R] [IsDomain S] [IsDedekindDomain R] [IsDedekindDomain S] [Module.Finite R S] [Module.Free R S]
Goal: (Ideal.relNorm R) ⊥ = ⊥
-- Ideal.relNorm_top
@[simp]
theorem relNorm_top : relNorm R (⊤ : Ideal S) = ⊤ 
Arguments: (R : Type u_1) [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [IsDomain R] [IsDomain S] [IsDedekindDomain R] [IsDedekindDomain S] [Module.Finite R S] [Module.Free R S]
Goal: (Ideal.relNorm R) ⊤ = ⊤
-- Ideal.relNorm_eq_bot_iff
@[simp]
theorem relNorm_eq_bot_iff {I : Ideal S} : relNorm R I = ⊥ ↔ I = ⊥ 
Arguments: {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [IsDomain R] [IsDomain S] [IsDedekindDomain R] [IsDedekindDomain S] [Module.Finite R S] [Module.Free R S] {I : Ideal S}
Goal: (Ideal.relNorm R) I = ⊥ ↔ I = ⊥
-- Ideal.norm_mem_relNorm
theorem norm_mem_relNorm (I : Ideal S) {x : S} (hx : x ∈ I) : Algebra.norm R x ∈ relNorm R I 
Arguments: (R : Type u_1) [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [IsDomain R] [IsDomain S] [IsDedekindDomain R] [IsDedekindDomain S] [Module.Finite R S] [Module.Free R S] (I : Ideal S) {x : S} (hx : x ∈ I)
Goal: (Algebra.norm R) x ∈ (Ideal.relNorm R) I
-- Ideal.relNorm_singleton
@[simp]
theorem relNorm_singleton (r : S) : relNorm R (span ({r} : Set S)) = span {Algebra.norm R r} 
Arguments: (R : Type u_1) [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [IsDomain R] [IsDomain S] [IsDedekindDomain R] [IsDedekindDomain S] [Module.Finite R S] [Module.Free R S] (r : S)
Goal: (Ideal.relNorm R) (Ideal.span {r}) = Ideal.span {(Algebra.norm R) r}
-- Ideal.map_relNorm
theorem map_relNorm (I : Ideal S) {T : Type*} [CommRing T] (f : R →+* T) :
    map f (relNorm R I) = span (f ∘ Algebra.norm R '' (I : Set S)) 
Arguments: (R : Type u_1) [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [IsDomain R] [IsDomain S] [IsDedekindDomain R] [IsDedekindDomain S] [Module.Finite R S] [Module.Free R S] (I : Ideal S) {T : Type u_3} [CommRing T] (f : R →+* T)
Goal: Ideal.map f ((Ideal.relNorm R) I) = Ideal.span (⇑f ∘ ⇑(Algebra.norm R) '' ↑I)
-- Ideal.relNorm_mono
@[mono]
theorem relNorm_mono {I J : Ideal S} (h : I ≤ J) : relNorm R I ≤ relNorm R J 
Arguments: (R : Type u_1) [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [IsDomain R] [IsDomain S] [IsDedekindDomain R] [IsDedekindDomain S] [Module.Finite R S] [Module.Free R S] {I J : Ideal S} (h : I ≤ J)
Goal: (Ideal.relNorm R) I ≤ (Ideal.relNorm R) J
-- FiniteDimensional.rank_linearMap
theorem FiniteDimensional.rank_linearMap :
    Module.rank S (M →ₗ[R] N) = lift.{w} (Module.rank R M) * lift.{v} (Module.rank S N) 
Arguments: (R : Type u) (S : Type u') (M : Type v) (N : Type w) [Ring R] [Ring S] [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] [AddCommGroup N] [Module R N] [Module S N] [SMulCommClass R S N] [StrongRankCondition R] [StrongRankCondition S] [Module.Free S N]
Goal: Module.rank S (M →ₗ[R] N) = Cardinal.lift.{w, v} (Module.rank R M) * Cardinal.lift.{v, w} (Module.rank S N)
-- FiniteDimensional.finrank_linearMap
theorem FiniteDimensional.finrank_linearMap :
    finrank S (M →ₗ[R] N) = finrank R M * finrank S N 
Arguments: (R : Type u) (S : Type u') (M : Type v) (N : Type w) [Ring R] [Ring S] [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] [AddCommGroup N] [Module R N] [Module S N] [SMulCommClass R S N] [StrongRankCondition R] [StrongRankCondition S] [Module.Free S N]
Goal: FiniteDimensional.finrank S (M →ₗ[R] N) = FiniteDimensional.finrank R M * FiniteDimensional.finrank S N
-- FiniteDimensional.rank_linearMap_self
theorem FiniteDimensional.rank_linearMap_self :
    Module.rank S (M →ₗ[R] S) = lift.{u'} (Module.rank R M) 
Arguments: (R : Type u) (S : Type u') (M : Type v) [Ring R] [Ring S] [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] [StrongRankCondition R] [StrongRankCondition S] [Module R S] [SMulCommClass R S S]
Goal: Module.rank S (M →ₗ[R] S) = Cardinal.lift.{u', v} (Module.rank R M)
-- FiniteDimensional.finrank_linearMap_self
theorem FiniteDimensional.finrank_linearMap_self : finrank S (M →ₗ[R] S) = finrank R M 
Arguments: (R : Type u) (S : Type u') (M : Type v) [Ring R] [Ring S] [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] [StrongRankCondition R] [StrongRankCondition S] [Module R S] [SMulCommClass R S S]
Goal: FiniteDimensional.finrank S (M →ₗ[R] S) = FiniteDimensional.finrank R M
-- cardinal_mk_algHom_le_rank
theorem cardinal_mk_algHom_le_rank : #(M →ₐ[K] L) ≤ lift.{v} (Module.rank K M) 
Arguments: (K : Type u_1) (M : Type u_2) (L : Type v) [CommRing K] [Ring M] [Algebra K M] [Module.Free K M] [Module.Finite K M] [CommRing L] [IsDomain L] [Algebra K L]
Goal: Cardinal.mk (M →ₐ[K] L) ≤ Cardinal.lift.{v, u_2} (Module.rank K M)
-- card_algHom_le_finrank
theorem card_algHom_le_finrank : Nat.card (M →ₐ[K] L) ≤ finrank K M 
Arguments: (K : Type u_1) (M : Type u_2) (L : Type v) [CommRing K] [Ring M] [Algebra K M] [Module.Free K M] [Module.Finite K M] [CommRing L] [IsDomain L] [Algebra K L]
Goal: Nat.card (M →ₐ[K] L) ≤ FiniteDimensional.finrank K M
-- Matrix.rank_vecMulVec
theorem Matrix.rank_vecMulVec {K m n : Type u} [CommRing K] [Fintype n]
    [DecidableEq n] (w : m → K) (v : n → K) : (Matrix.vecMulVec w v).toLin'.rank ≤ 1 
Arguments: {K m n : Type u} [CommRing K] [Fintype n] [DecidableEq n] (w : m → K) (v : n → K)
Goal: (Matrix.toLin' (Matrix.vecMulVec w v)).rank ≤ 1
-- OreLocalization.ind
@[to_additive (attr := elab_as_elim, cases_eliminator, induction_eliminator)]
protected theorem ind {β : X[S⁻¹] → Prop}
    (c : ∀ (r : X) (s : S), β (r /ₒ s)) : ∀ q, β q 
Arguments: {R : Type u_1} [Monoid R] {S : Submonoid R} [OreLocalization.OreSet S] {X : Type u_2} [MulAction R X] {β : OreLocalization S X → Prop} (c : ∀ (r : X) (s : ↥S), β (r /ₒ s)) (q : OreLocalization S X)
Goal: β q
-- AddOreLocalization.ind
@[to_additive (attr := elab_as_elim, cases_eliminator, induction_eliminator)]
protected theorem ind {β : X[S⁻¹] → Prop}
    (c : ∀ (r : X) (s : S), β (r /ₒ s)) : ∀ q, β q 
Arguments: {R : Type u_1} [AddMonoid R] {S : AddSubmonoid R} [AddOreLocalization.AddOreSet S] {X : Type u_2} [AddAction R X] {β : AddOreLocalization S X → Prop} (c : ∀ (r : X) (s : ↥S), β (r -ₒ s)) (q : AddOreLocalization S X)
Goal: β q
-- AddOreLocalization.oreSub_eq_iff
@[to_additive]
theorem oreDiv_eq_iff {r₁ r₂ : X} {s₁ s₂ : S} :
    r₁ /ₒ s₁ = r₂ /ₒ s₂ ↔ ∃ (u : S) (v : R), u • r₂ = v • r₁ ∧ u * s₂ = v * s₁ 
Arguments: {R : Type u_1} [AddMonoid R] {S : AddSubmonoid R} [AddOreLocalization.AddOreSet S] {X : Type u_2} [AddAction R X] {r₁ r₂ : X} {s₁ s₂ : ↥S}
Goal: r₁ -ₒ s₁ = r₂ -ₒ s₂ ↔ ∃ u v, u +ᵥ r₂ = v +ᵥ r₁ ∧ ↑u + ↑s₂ = v + ↑s₁
-- OreLocalization.oreDiv_eq_iff
@[to_additive]
theorem oreDiv_eq_iff {r₁ r₂ : X} {s₁ s₂ : S} :
    r₁ /ₒ s₁ = r₂ /ₒ s₂ ↔ ∃ (u : S) (v : R), u • r₂ = v • r₁ ∧ u * s₂ = v * s₁ 
Arguments: {R : Type u_1} [Monoid R] {S : Submonoid R} [OreLocalization.OreSet S] {X : Type u_2} [MulAction R X] {r₁ r₂ : X} {s₁ s₂ : ↥S}
Goal: r₁ /ₒ s₁ = r₂ /ₒ s₂ ↔ ∃ u v, u • r₂ = v • r₁ ∧ ↑u * ↑s₂ = v * ↑s₁
-- OreLocalization.expand
@[to_additive "A difference `r -ₒ s` is equal to its expansion by an
arbitrary translation `t` if `t + s ∈ S`."]
protected theorem expand (r : X) (s : S) (t : R) (hst : t * (s : R) ∈ S) :
    r /ₒ s = t • r /ₒ ⟨t * s, hst⟩ 
Arguments: {R : Type u_1} [Monoid R] {S : Submonoid R} [OreLocalization.OreSet S] {X : Type u_2} [MulAction R X] (r : X) (s : ↥S) (t : R) (hst : t * ↑s ∈ S)
Goal: r /ₒ s = t • r /ₒ ⟨t * ↑s, hst⟩
-- AddOreLocalization.expand
@[to_additive "A difference `r -ₒ s` is equal to its expansion by an
arbitrary translation `t` if `t + s ∈ S`."]
protected theorem expand (r : X) (s : S) (t : R) (hst : t * (s : R) ∈ S) :
    r /ₒ s = t • r /ₒ ⟨t * s, hst⟩ 
Arguments: {R : Type u_1} [AddMonoid R] {S : AddSubmonoid R} [AddOreLocalization.AddOreSet S] {X : Type u_2} [AddAction R X] (r : X) (s : ↥S) (t : R) (hst : t + ↑s ∈ S)
Goal: r -ₒ s = t +ᵥ r -ₒ ⟨t + ↑s, hst⟩
-- OreLocalization.expand'
@[to_additive "A difference is equal to its expansion by a summand from `S`."]
protected theorem expand' (r : X) (s s' : S) : r /ₒ s = s' • r /ₒ (s' * s) 
Arguments: {R : Type u_1} [Monoid R] {S : Submonoid R} [OreLocalization.OreSet S] {X : Type u_2} [MulAction R X] (r : X) (s s' : ↥S)
Goal: r /ₒ s = s' • r /ₒ (s' * s)
-- AddOreLocalization.expand'
@[to_additive "A difference is equal to its expansion by a summand from `S`."]
protected theorem expand' (r : X) (s s' : S) : r /ₒ s = s' • r /ₒ (s' * s) 
Arguments: {R : Type u_1} [AddMonoid R] {S : AddSubmonoid R} [AddOreLocalization.AddOreSet S] {X : Type u_2} [AddAction R X] (r : X) (s s' : ↥S)
Goal: r -ₒ s = s' +ᵥ r -ₒ (s' + s)
-- OreLocalization.eq_of_num_factor_eq
@[to_additive "Differences whose minuends differ by a common summand can be proven equal if
those summands expand to equal elements of `R`."]
protected theorem eq_of_num_factor_eq {r r' r₁ r₂ : R} {s t : S} (h : t * r = t * r') :
    r₁ * r * r₂ /ₒ s = r₁ * r' * r₂ /ₒ s 
Arguments: {R : Type u_1} [Monoid R] {S : Submonoid R} [OreLocalization.OreSet S] {r r' r₁ r₂ : R} {s t : ↥S} (h : ↑t * r = ↑t * r')
Goal: r₁ * r * r₂ /ₒ s = r₁ * r' * r₂ /ₒ s