ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- _private.Mathlib.RingTheory.WittVector.Basic.0.WittVector.ghostFun_zsmul
private lemma ghostFun_zsmul (m : ℤ) (x : WittVector p R) : ghostFun (m • x) = m • ghostFun x 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (m : ℤ) (x : 𝕎 R)
Goal: WittVector.ghostFun (m • x) = m • WittVector.ghostFun x
-- _private.Mathlib.RingTheory.WittVector.Basic.0.WittVector.ghostFun_pow
private theorem ghostFun_pow (m : ℕ) : ghostFun (x ^ m) = ghostFun x ^ m 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x : 𝕎 R) (m : ℕ)
Goal: WittVector.ghostFun (x ^ m) = WittVector.ghostFun x ^ m
-- WittVector.map_injective
theorem map_injective (f : R →+* S) (hf : Injective f) : Injective (map f : 𝕎 R → 𝕎 S) 
Arguments: {p : ℕ} {R : Type u_1} {S : Type u_2} [hp : Fact (Nat.Prime p)] [CommRing R] [CommRing S] (f : R →+* S) (hf : Function.Injective ⇑f)
Goal: Function.Injective ⇑(WittVector.map f)
-- WittVector.map_surjective
theorem map_surjective (f : R →+* S) (hf : Surjective f) : Surjective (map f : 𝕎 R → 𝕎 S) 
Arguments: {p : ℕ} {R : Type u_1} {S : Type u_2} [hp : Fact (Nat.Prime p)] [CommRing R] [CommRing S] (f : R →+* S) (hf : Function.Surjective ⇑f)
Goal: Function.Surjective ⇑(WittVector.map f)
-- WittVector.map_coeff
@[simp]
theorem map_coeff (f : R →+* S) (x : 𝕎 R) (n : ℕ) : (map f x).coeff n = f (x.coeff n) 
Arguments: {p : ℕ} {R : Type u_1} {S : Type u_2} [hp : Fact (Nat.Prime p)] [CommRing R] [CommRing S] (f : R →+* S) (x : 𝕎 R) (n : ℕ)
Goal: ((WittVector.map f) x).coeff n = f (x.coeff n)
-- WittVector.ghostComponent_apply
theorem ghostComponent_apply (n : ℕ) (x : 𝕎 R) : ghostComponent n x = aeval x.coeff (W_ ℤ n) 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (n : ℕ) (x : 𝕎 R)
Goal: (WittVector.ghostComponent n) x = (MvPolynomial.aeval x.coeff) (W_ ℤ n)
-- WittVector.ghostMap_apply
@[simp]
theorem ghostMap_apply (x : 𝕎 R) (n : ℕ) : ghostMap x n = ghostComponent n x 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x : 𝕎 R) (n : ℕ)
Goal: WittVector.ghostMap x n = (WittVector.ghostComponent n) x
-- WittVector.ghostEquiv_coe
@[simp]
theorem ghostEquiv_coe : (ghostEquiv p R : 𝕎 R →+* ℕ → R) = ghostMap 
Arguments: (p : ℕ) (R : Type u_1) [hp : Fact (Nat.Prime p)] [CommRing R] [Invertible ↑p]
Goal: ↑(WittVector.ghostEquiv p R) = WittVector.ghostMap
-- WittVector.ghostMap.bijective_of_invertible
theorem ghostMap.bijective_of_invertible : Function.Bijective (ghostMap : 𝕎 R → ℕ → R) 
Arguments: (p : ℕ) (R : Type u_1) [hp : Fact (Nat.Prime p)] [CommRing R] [Invertible ↑p]
Goal: Function.Bijective ⇑WittVector.ghostMap
-- WittVector.constantCoeff_apply
@[simps]
noncomputable def constantCoeff : 𝕎 R →+* R 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x : 𝕎 R)
Goal: WittVector.constantCoeff x = x.coeff 0
-- ValuationSubring.mem_or_inv_mem'
structure ValuationSubring extends Subring K where
  mem_or_inv_mem' : ∀ x : K, x ∈ carrier ∨ x⁻¹ ∈ carrier
Arguments: {K : Type u} [Field K] (self : ValuationSubring K) (x : K)
Goal: x ∈ self.carrier ∨ x⁻¹ ∈ self.carrier
-- ValuationSubring.mem_carrier
@[simp, nolint simpNF] -- Porting note (#10959): simp cannot prove that
theorem mem_carrier (x : K) : x ∈ A.carrier ↔ x ∈ A 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (x : K)
Goal: x ∈ A.carrier ↔ x ∈ A
-- ValuationSubring.mem_toSubring
@[simp]
theorem mem_toSubring (x : K) : x ∈ A.toSubring ↔ x ∈ A 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (x : K)
Goal: x ∈ A.toSubring ↔ x ∈ A
-- ValuationSubring.ext
@[ext]
theorem ext (A B : ValuationSubring K) (h : ∀ x, x ∈ A ↔ x ∈ B) : A = B 
Arguments: {K : Type u} [Field K] (A B : ValuationSubring K) (h : ∀ (x : K), x ∈ A ↔ x ∈ B)
Goal: A = B
-- ValuationSubring.zero_mem
theorem zero_mem : (0 : K) ∈ A 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K)
Goal: 0 ∈ A
-- ValuationSubring.one_mem
theorem one_mem : (1 : K) ∈ A 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K)
Goal: 1 ∈ A
-- ValuationSubring.add_mem
theorem add_mem (x y : K) : x ∈ A → y ∈ A → x + y ∈ A 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (x y : K)
Goal: x ∈ A → y ∈ A → x + y ∈ A
-- ValuationSubring.mul_mem
theorem mul_mem (x y : K) : x ∈ A → y ∈ A → x * y ∈ A 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (x y : K)
Goal: x ∈ A → y ∈ A → x * y ∈ A
-- ValuationSubring.neg_mem
theorem neg_mem (x : K) : x ∈ A → -x ∈ A 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (x : K)
Goal: x ∈ A → -x ∈ A
-- ValuationSubring.mem_or_inv_mem
theorem mem_or_inv_mem (x : K) : x ∈ A ∨ x⁻¹ ∈ A 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (x : K)
Goal: x ∈ A ∨ x⁻¹ ∈ A
-- ValuationSubring.toSubring_injective
theorem toSubring_injective : Function.Injective (toSubring : ValuationSubring K → Subring K) 
Arguments: {K : Type u} [Field K]
Goal: Function.Injective ValuationSubring.toSubring
-- ValuationSubring.mem_top
theorem mem_top (x : K) : x ∈ (⊤ : ValuationSubring K) 
Arguments: {K : Type u} [Field K] (x : K)
Goal: x ∈ ⊤
-- ValuationSubring.le_top
theorem le_top : A ≤ ⊤ 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K)
Goal: A ≤ ⊤
-- ValuationSubring.algebraMap_apply
@[simp]
theorem algebraMap_apply (a : A) : algebraMap A K a = a 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (a : ↥A)
Goal: (algebraMap (↥A) K) a = ↑a
-- ValuationSubring.valuation_le_one
theorem valuation_le_one (a : A) : A.valuation a ≤ 1 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (a : ↥A)
Goal: A.valuation ↑a ≤ 1
-- ValuationSubring.mem_of_valuation_le_one
theorem mem_of_valuation_le_one (x : K) (h : A.valuation x ≤ 1) : x ∈ A 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (x : K) (h : A.valuation x ≤ 1)
Goal: x ∈ A
-- ValuationSubring.valuation_le_one_iff
theorem valuation_le_one_iff (x : K) : A.valuation x ≤ 1 ↔ x ∈ A 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (x : K)
Goal: A.valuation x ≤ 1 ↔ x ∈ A
-- ValuationSubring.valuation_eq_iff
theorem valuation_eq_iff (x y : K) : A.valuation x = A.valuation y ↔ ∃ a : Aˣ, (a : K) * y = x 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (x y : K)
Goal: A.valuation x = A.valuation y ↔ ∃ a, ↑↑a * y = x
-- ValuationSubring.valuation_le_iff
theorem valuation_le_iff (x y : K) : A.valuation x ≤ A.valuation y ↔ ∃ a : A, (a : K) * y = x 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (x y : K)
Goal: A.valuation x ≤ A.valuation y ↔ ∃ a, ↑a * y = x
-- ValuationSubring.valuation_surjective
theorem valuation_surjective : Function.Surjective A.valuation 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K)
Goal: Function.Surjective ⇑A.valuation
-- ValuationSubring.valuation_unit
theorem valuation_unit (a : Aˣ) : A.valuation a = 1 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (a : (↥A)ˣ)
Goal: A.valuation ↑↑a = 1
-- ValuationSubring.valuation_eq_one_iff
theorem valuation_eq_one_iff (a : A) : IsUnit a ↔ A.valuation a = 1 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (a : ↥A)
Goal: IsUnit a ↔ A.valuation ↑a = 1
-- ValuationSubring.valuation_lt_one_or_eq_one
theorem valuation_lt_one_or_eq_one (a : A) : A.valuation a < 1 ∨ A.valuation a = 1 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (a : ↥A)
Goal: A.valuation ↑a < 1 ∨ A.valuation ↑a = 1
-- ValuationSubring.valuation_lt_one_iff
theorem valuation_lt_one_iff (a : A) : a ∈ LocalRing.maximalIdeal A ↔ A.valuation a < 1 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (a : ↥A)
Goal: a ∈ LocalRing.maximalIdeal ↥A ↔ A.valuation ↑a < 1
-- ValuationSubring.mem_ofSubring
@[simp]
theorem mem_ofSubring (R : Subring K) (hR : ∀ x : K, x ∈ R ∨ x⁻¹ ∈ R) (x : K) :
    x ∈ ofSubring R hR ↔ x ∈ R 
Arguments: {K : Type u} [Field K] (R : Subring K) (hR : ∀ (x : K), x ∈ R ∨ x⁻¹ ∈ R) (x : K)
Goal: x ∈ ValuationSubring.ofSubring R hR ↔ x ∈ R
-- ValuationSubring.monotone_mapOfLE
@[mono]
theorem monotone_mapOfLE (R S : ValuationSubring K) (h : R ≤ S) : Monotone (R.mapOfLE S h) 
Arguments: {K : Type u} [Field K] (R S : ValuationSubring K) (h : R ≤ S)
Goal: Monotone ⇑(R.mapOfLE S h)
-- ValuationSubring.mapOfLE_comp_valuation
@[simp]
theorem mapOfLE_comp_valuation (R S : ValuationSubring K) (h : R ≤ S) :
    R.mapOfLE S h ∘ R.valuation = S.valuation 
Arguments: {K : Type u} [Field K] (R S : ValuationSubring K) (h : R ≤ S)
Goal: ⇑(R.mapOfLE S h) ∘ ⇑R.valuation = ⇑S.valuation
-- ValuationSubring.mapOfLE_valuation_apply
@[simp]
theorem mapOfLE_valuation_apply (R S : ValuationSubring K) (h : R ≤ S) (x : K) :
    R.mapOfLE S h (R.valuation x) = S.valuation x 
Arguments: {K : Type u} [Field K] (R S : ValuationSubring K) (h : R ≤ S) (x : K)
Goal: (R.mapOfLE S h) (R.valuation x) = S.valuation x
-- ValuationSubring.le_ofPrime
theorem le_ofPrime (A : ValuationSubring K) (P : Ideal A) [P.IsPrime] : A ≤ ofPrime A P 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (P : Ideal ↥A) [P.IsPrime]
Goal: A ≤ A.ofPrime P
-- ValuationSubring.ofPrime_valuation_eq_one_iff_mem_primeCompl
theorem ofPrime_valuation_eq_one_iff_mem_primeCompl (A : ValuationSubring K) (P : Ideal A)
    [P.IsPrime] (x : A) : (ofPrime A P).valuation x = 1 ↔ x ∈ P.primeCompl 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (P : Ideal ↥A) [P.IsPrime] (x : ↥A)
Goal: (A.ofPrime P).valuation ↑x = 1 ↔ x ∈ P.primeCompl
-- ValuationSubring.idealOfLE_ofPrime
@[simp]
theorem idealOfLE_ofPrime (A : ValuationSubring K) (P : Ideal A) [P.IsPrime] :
    idealOfLE A (ofPrime A P) (le_ofPrime A P) = P 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (P : Ideal ↥A) [P.IsPrime]
Goal: A.idealOfLE (A.ofPrime P) ⋯ = P
-- ValuationSubring.ofPrime_idealOfLE
@[simp]
theorem ofPrime_idealOfLE (R S : ValuationSubring K) (h : R ≤ S) :
    ofPrime R (idealOfLE R S h) = S 
Arguments: {K : Type u} [Field K] (R S : ValuationSubring K) (h : R ≤ S)
Goal: R.ofPrime (R.idealOfLE S h) = S
-- ValuationSubring.ofPrime_le_of_le
theorem ofPrime_le_of_le (P Q : Ideal A) [P.IsPrime] [Q.IsPrime] (h : P ≤ Q) :
    ofPrime A Q ≤ ofPrime A P 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (P Q : Ideal ↥A) [P.IsPrime] [Q.IsPrime] (h : P ≤ Q)
Goal: A.ofPrime Q ≤ A.ofPrime P
-- ValuationSubring.idealOfLE_le_of_le
theorem idealOfLE_le_of_le (R S : ValuationSubring K) (hR : A ≤ R) (hS : A ≤ S) (h : R ≤ S) :
    idealOfLE A S hS ≤ idealOfLE A R hR 
Arguments: {K : Type u} [Field K] (A R S : ValuationSubring K) (hR : A ≤ R) (hS : A ≤ S) (h : R ≤ S)
Goal: A.idealOfLE S hS ≤ A.idealOfLE R hR
-- ValuationSubring.primeSpectrumEquiv_apply_coe
@[simps]
def primeSpectrumEquiv : PrimeSpectrum A ≃ {S // A ≤ S} 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (P : PrimeSpectrum ↥A)
Goal: ↑(A.primeSpectrumEquiv P) = A.ofPrime P.asIdeal
-- ValuationSubring.primeSpectrumEquiv_symm_apply_asIdeal
@[simps]
def primeSpectrumEquiv : PrimeSpectrum A ≃ {S // A ≤ S} 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (S : { S // A ≤ S })
Goal: (A.primeSpectrumEquiv.symm S).asIdeal = A.idealOfLE ↑S ⋯
-- ValuationSubring.primeSpectrumOrderEquiv_apply_coe_carrier
@[simps!]
def primeSpectrumOrderEquiv : (PrimeSpectrum A)ᵒᵈ ≃o {S // A ≤ S} 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (P : PrimeSpectrum ↥A)
Goal: ↑↑(A.primeSpectrumOrderEquiv P) =
  {x | ∃ a ∈ A, ∃ a_1, (∃ (x : a_1 ∈ A), ⟨a_1, ⋯⟩ ∈ P.asIdeal.primeCompl) ∧ x = a * a_1⁻¹}
-- ValuationSubring.primeSpectrumOrderEquiv_symm_apply_asIdeal_carrier
@[simps!]
def primeSpectrumOrderEquiv : (PrimeSpectrum A)ᵒᵈ ≃o {S // A ≤ S} 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (S : { S // A ≤ S })
Goal: ↑((RelIso.symm A.primeSpectrumOrderEquiv) S).asIdeal = ⇑(A.inclusion ↑S ⋯) ⁻¹' ↑(LocalRing.maximalIdeal ↥↑S)
-- Valuation.mem_valuationSubring_iff
@[simp]
theorem mem_valuationSubring_iff (x : K) : x ∈ v.valuationSubring ↔ v x ≤ 1 
Arguments: {K : Type u} [Field K] {Γ : Type u_1} [LinearOrderedCommGroupWithZero Γ] (v : Valuation K Γ) (x : K)
Goal: x ∈ v.valuationSubring ↔ v x ≤ 1
-- Valuation.isEquiv_iff_valuationSubring
theorem isEquiv_iff_valuationSubring :
    v₁.IsEquiv v₂ ↔ v₁.valuationSubring = v₂.valuationSubring 
Arguments: {K : Type u} [Field K] {Γ₁ : Type u_2} {Γ₂ : Type u_3} [LinearOrderedCommGroupWithZero Γ₁] [LinearOrderedCommGroupWithZero Γ₂] (v₁ : Valuation K Γ₁) (v₂ : Valuation K Γ₂)
Goal: v₁.IsEquiv v₂ ↔ v₁.valuationSubring = v₂.valuationSubring
-- Valuation.isEquiv_valuation_valuationSubring
theorem isEquiv_valuation_valuationSubring : v.IsEquiv v.valuationSubring.valuation 
Arguments: {K : Type u} [Field K] {Γ : Type u_1} [LinearOrderedCommGroupWithZero Γ] (v : Valuation K Γ)
Goal: v.IsEquiv v.valuationSubring.valuation
-- ValuationSubring.valuationSubring_valuation
@[simp]
theorem valuationSubring_valuation : A.valuation.valuationSubring = A 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K)
Goal: A.valuation.valuationSubring = A
-- ValuationSubring.mem_unitGroup_iff
@[simp]
theorem mem_unitGroup_iff (x : Kˣ) : x ∈ A.unitGroup ↔ A.valuation x = 1 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (x : Kˣ)
Goal: x ∈ A.unitGroup ↔ A.valuation ↑x = 1
-- ValuationSubring.coe_unitGroupMulEquiv_apply
@[simp]
theorem coe_unitGroupMulEquiv_apply (a : A.unitGroup) :
    ((A.unitGroupMulEquiv a : A) : K) = ((a : Kˣ) : K) 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (a : ↥A.unitGroup)
Goal: ↑↑(A.unitGroupMulEquiv a) = ↑↑a
-- ValuationSubring.coe_unitGroupMulEquiv_symm_apply
@[simp]
theorem coe_unitGroupMulEquiv_symm_apply (a : Aˣ) : ((A.unitGroupMulEquiv.symm a : Kˣ) : K) = a 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (a : (↥A)ˣ)
Goal: ↑↑(A.unitGroupMulEquiv.symm a) = ↑↑a
-- ValuationSubring.unitGroup_le_unitGroup
theorem unitGroup_le_unitGroup {A B : ValuationSubring K} : A.unitGroup ≤ B.unitGroup ↔ A ≤ B 
Arguments: {K : Type u} [Field K] {A B : ValuationSubring K}
Goal: A.unitGroup ≤ B.unitGroup ↔ A ≤ B
-- ValuationSubring.unitGroup_injective
theorem unitGroup_injective : Function.Injective (unitGroup : ValuationSubring K → Subgroup _) 
Arguments: {K : Type u} [Field K]
Goal: Function.Injective ValuationSubring.unitGroup
-- ValuationSubring.eq_iff_unitGroup
theorem eq_iff_unitGroup {A B : ValuationSubring K} : A = B ↔ A.unitGroup = B.unitGroup 
Arguments: {K : Type u} [Field K] {A B : ValuationSubring K}
Goal: A = B ↔ A.unitGroup = B.unitGroup
-- ValuationSubring.unitGroup_strictMono
theorem unitGroup_strictMono : StrictMono (unitGroup : ValuationSubring K → Subgroup _) 
Arguments: {K : Type u} [Field K]
Goal: StrictMono ValuationSubring.unitGroup
-- ValuationSubring.mem_nonunits_iff
theorem mem_nonunits_iff {x : K} : x ∈ A.nonunits ↔ A.valuation x < 1 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) {x : K}
Goal: x ∈ A.nonunits ↔ A.valuation x < 1
-- ValuationSubring.nonunits_le_nonunits
theorem nonunits_le_nonunits {A B : ValuationSubring K} : B.nonunits ≤ A.nonunits ↔ A ≤ B 
Arguments: {K : Type u} [Field K] {A B : ValuationSubring K}
Goal: B.nonunits ≤ A.nonunits ↔ A ≤ B
-- ValuationSubring.nonunits_injective
theorem nonunits_injective : Function.Injective (nonunits : ValuationSubring K → Subsemigroup _) 
Arguments: {K : Type u} [Field K]
Goal: Function.Injective ValuationSubring.nonunits
-- ValuationSubring.nonunits_inj
theorem nonunits_inj {A B : ValuationSubring K} : A.nonunits = B.nonunits ↔ A = B 
Arguments: {K : Type u} [Field K] {A B : ValuationSubring K}
Goal: A.nonunits = B.nonunits ↔ A = B
-- ValuationSubring.coe_mem_nonunits_iff
theorem coe_mem_nonunits_iff {a : A} : (a : K) ∈ A.nonunits ↔ a ∈ LocalRing.maximalIdeal A 
Arguments: {K : Type u} [Field K] {A : ValuationSubring K} {a : ↥A}
Goal: ↑a ∈ A.nonunits ↔ a ∈ LocalRing.maximalIdeal ↥A
-- ValuationSubring.nonunits_le
theorem nonunits_le : A.nonunits ≤ A.toSubring.toSubmonoid.toSubsemigroup 
Arguments: {K : Type u} [Field K] {A : ValuationSubring K}
Goal: A.nonunits ≤ A.toSubsemigroup
-- ValuationSubring.nonunits_subset
theorem nonunits_subset : (A.nonunits : Set K) ⊆ A 
Arguments: {K : Type u} [Field K] {A : ValuationSubring K}
Goal: ↑A.nonunits ⊆ ↑A
-- ValuationSubring.mem_nonunits_iff_exists_mem_maximalIdeal
theorem mem_nonunits_iff_exists_mem_maximalIdeal {a : K} :
    a ∈ A.nonunits ↔ ∃ ha, (⟨a, ha⟩ : A) ∈ LocalRing.maximalIdeal A 
Arguments: {K : Type u} [Field K] {A : ValuationSubring K} {a : K}
Goal: a ∈ A.nonunits ↔ ∃ (ha : a ∈ A), ⟨a, ha⟩ ∈ LocalRing.maximalIdeal ↥A
-- ValuationSubring.image_maximalIdeal
theorem image_maximalIdeal : ((↑) : A → K) '' LocalRing.maximalIdeal A = A.nonunits 
Arguments: {K : Type u} [Field K] {A : ValuationSubring K}
Goal: Subtype.val '' ↑(LocalRing.maximalIdeal ↥A) = ↑A.nonunits
-- ValuationSubring.principal_units_le_units
theorem principal_units_le_units : A.principalUnitGroup ≤ A.unitGroup 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K)
Goal: A.principalUnitGroup ≤ A.unitGroup
-- ValuationSubring.mem_principalUnitGroup_iff
theorem mem_principalUnitGroup_iff (x : Kˣ) :
    x ∈ A.principalUnitGroup ↔ A.valuation ((x : K) - 1) < 1 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (x : Kˣ)
Goal: x ∈ A.principalUnitGroup ↔ A.valuation (↑x - 1) < 1
-- ValuationSubring.principalUnitGroup_le_principalUnitGroup
theorem principalUnitGroup_le_principalUnitGroup {A B : ValuationSubring K} :
    B.principalUnitGroup ≤ A.principalUnitGroup ↔ A ≤ B 
Arguments: {K : Type u} [Field K] {A B : ValuationSubring K}
Goal: B.principalUnitGroup ≤ A.principalUnitGroup ↔ A ≤ B
-- ValuationSubring.principalUnitGroup_injective
theorem principalUnitGroup_injective :
    Function.Injective (principalUnitGroup : ValuationSubring K → Subgroup _) 
Arguments: {K : Type u} [Field K]
Goal: Function.Injective ValuationSubring.principalUnitGroup
-- ValuationSubring.eq_iff_principalUnitGroup
theorem eq_iff_principalUnitGroup {A B : ValuationSubring K} :
    A = B ↔ A.principalUnitGroup = B.principalUnitGroup 
Arguments: {K : Type u} [Field K] {A B : ValuationSubring K}
Goal: A = B ↔ A.principalUnitGroup = B.principalUnitGroup
-- ValuationSubring.coe_mem_principalUnitGroup_iff
theorem coe_mem_principalUnitGroup_iff {x : A.unitGroup} :
    (x : Kˣ) ∈ A.principalUnitGroup ↔
      A.unitGroupMulEquiv x ∈ (Units.map (LocalRing.residue A).toMonoidHom).ker 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) {x : ↥A.unitGroup}
Goal: ↑x ∈ A.principalUnitGroup ↔ A.unitGroupMulEquiv x ∈ (Units.map ↑(LocalRing.residue ↥A)).ker
-- ValuationSubring.principalUnitGroupEquiv_apply
theorem principalUnitGroupEquiv_apply (a : A.principalUnitGroup) :
    (((principalUnitGroupEquiv A a : Aˣ) : A) : K) = (a : Kˣ) 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (a : ↥A.principalUnitGroup)
Goal: ↑↑↑(A.principalUnitGroupEquiv a) = ↑↑a
-- ValuationSubring.principalUnitGroup_symm_apply
theorem principalUnitGroup_symm_apply (a : (Units.map (LocalRing.residue A).toMonoidHom).ker) :
    ((A.principalUnitGroupEquiv.symm a : Kˣ) : K) = ((a : Aˣ) : A) 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (a : ↥(Units.map ↑(LocalRing.residue ↥A)).ker)
Goal: ↑↑(A.principalUnitGroupEquiv.symm a) = ↑↑↑a
-- ValuationSubring.coe_unitGroupToResidueFieldUnits_apply
@[simp]
theorem coe_unitGroupToResidueFieldUnits_apply (x : A.unitGroup) :
    (A.unitGroupToResidueFieldUnits x : LocalRing.ResidueField A) =
      Ideal.Quotient.mk _ (A.unitGroupMulEquiv x : A) 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (x : ↥A.unitGroup)
Goal: ↑(A.unitGroupToResidueFieldUnits x) = (Ideal.Quotient.mk (LocalRing.maximalIdeal ↥A)) ↑(A.unitGroupMulEquiv x)
-- ValuationSubring.ker_unitGroupToResidueFieldUnits
theorem ker_unitGroupToResidueFieldUnits :
    A.unitGroupToResidueFieldUnits.ker = A.principalUnitGroup.comap A.unitGroup.subtype 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K)
Goal: A.unitGroupToResidueFieldUnits.ker = Subgroup.comap A.unitGroup.subtype A.principalUnitGroup
-- ValuationSubring.surjective_unitGroupToResidueFieldUnits
theorem surjective_unitGroupToResidueFieldUnits :
    Function.Surjective A.unitGroupToResidueFieldUnits 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K)
Goal: Function.Surjective ⇑A.unitGroupToResidueFieldUnits
-- ValuationSubring.unitsModPrincipalUnitsEquivResidueFieldUnits_comp_quotientGroup_mk
theorem unitsModPrincipalUnitsEquivResidueFieldUnits_comp_quotientGroup_mk :
    A.unitsModPrincipalUnitsEquivResidueFieldUnits.toMonoidHom.comp (QuotientGroup.mk' _) =
      A.unitGroupToResidueFieldUnits 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K)
Goal: A.unitsModPrincipalUnitsEquivResidueFieldUnits.toMonoidHom.comp
    (QuotientGroup.mk' (Subgroup.comap A.unitGroup.subtype A.principalUnitGroup)) =
  A.unitGroupToResidueFieldUnits
-- ValuationSubring.unitsModPrincipalUnitsEquivResidueFieldUnits_comp_quotientGroup_mk_apply
theorem unitsModPrincipalUnitsEquivResidueFieldUnits_comp_quotientGroup_mk_apply
    (x : A.unitGroup) :
    A.unitsModPrincipalUnitsEquivResidueFieldUnits.toMonoidHom (QuotientGroup.mk x) =
      A.unitGroupToResidueFieldUnits x 
Arguments: {K : Type u} [Field K] (A : ValuationSubring K) (x : ↥A.unitGroup)
Goal: A.unitsModPrincipalUnitsEquivResidueFieldUnits.toMonoidHom ↑x = A.unitGroupToResidueFieldUnits x
-- ValuationSubring.coe_pointwise_smul
@[simp]
theorem coe_pointwise_smul (g : G) (S : ValuationSubring K) : ↑(g • S) = g • (S : Set K) 
Arguments: {K : Type u} [Field K] {G : Type u_1} [Group G] [MulSemiringAction G K] (g : G) (S : ValuationSubring K)
Goal: ↑(g • S) = g • ↑S
-- ValuationSubring.pointwise_smul_toSubring
@[simp]
theorem pointwise_smul_toSubring (g : G) (S : ValuationSubring K) :
    (g • S).toSubring = g • S.toSubring 
Arguments: {K : Type u} [Field K] {G : Type u_1} [Group G] [MulSemiringAction G K] (g : G) (S : ValuationSubring K)
Goal: (g • S).toSubring = g • S.toSubring
-- ValuationSubring.smul_mem_pointwise_smul
theorem smul_mem_pointwise_smul (g : G) (x : K) (S : ValuationSubring K) : x ∈ S → g • x ∈ g • S 
Arguments: {K : Type u} [Field K] {G : Type u_1} [Group G] [MulSemiringAction G K] (g : G) (x : K) (S : ValuationSubring K)
Goal: x ∈ S → g • x ∈ g • S
-- ValuationSubring.mem_smul_pointwise_iff_exists
theorem mem_smul_pointwise_iff_exists (g : G) (x : K) (S : ValuationSubring K) :
    x ∈ g • S ↔ ∃ s : K, s ∈ S ∧ g • s = x 
Arguments: {K : Type u} [Field K] {G : Type u_1} [Group G] [MulSemiringAction G K] (g : G) (x : K) (S : ValuationSubring K)
Goal: x ∈ g • S ↔ ∃ s ∈ S, g • s = x
-- ValuationSubring.smul_mem_pointwise_smul_iff
@[simp]
theorem smul_mem_pointwise_smul_iff {g : G} {S : ValuationSubring K} {x : K} :
    g • x ∈ g • S ↔ x ∈ S 
Arguments: {K : Type u} [Field K] {G : Type u_1} [Group G] [MulSemiringAction G K] {g : G} {S : ValuationSubring K} {x : K}
Goal: g • x ∈ g • S ↔ x ∈ S
-- ValuationSubring.mem_pointwise_smul_iff_inv_smul_mem
theorem mem_pointwise_smul_iff_inv_smul_mem {g : G} {S : ValuationSubring K} {x : K} :
    x ∈ g • S ↔ g⁻¹ • x ∈ S 
Arguments: {K : Type u} [Field K] {G : Type u_1} [Group G] [MulSemiringAction G K] {g : G} {S : ValuationSubring K} {x : K}
Goal: x ∈ g • S ↔ g⁻¹ • x ∈ S
-- ValuationSubring.mem_inv_pointwise_smul_iff
theorem mem_inv_pointwise_smul_iff {g : G} {S : ValuationSubring K} {x : K} :
    x ∈ g⁻¹ • S ↔ g • x ∈ S 
Arguments: {K : Type u} [Field K] {G : Type u_1} [Group G] [MulSemiringAction G K] {g : G} {S : ValuationSubring K} {x : K}
Goal: x ∈ g⁻¹ • S ↔ g • x ∈ S
-- ValuationSubring.pointwise_smul_le_pointwise_smul_iff
@[simp]
theorem pointwise_smul_le_pointwise_smul_iff {g : G} {S T : ValuationSubring K} :
    g • S ≤ g • T ↔ S ≤ T 
Arguments: {K : Type u} [Field K] {G : Type u_1} [Group G] [MulSemiringAction G K] {g : G} {S T : ValuationSubring K}
Goal: g • S ≤ g • T ↔ S ≤ T
-- ValuationSubring.pointwise_smul_subset_iff
theorem pointwise_smul_subset_iff {g : G} {S T : ValuationSubring K} : g • S ≤ T ↔ S ≤ g⁻¹ • T 
Arguments: {K : Type u} [Field K] {G : Type u_1} [Group G] [MulSemiringAction G K] {g : G} {S T : ValuationSubring K}
Goal: g • S ≤ T ↔ S ≤ g⁻¹ • T
-- ValuationSubring.subset_pointwise_smul_iff
theorem subset_pointwise_smul_iff {g : G} {S T : ValuationSubring K} : S ≤ g • T ↔ g⁻¹ • S ≤ T 
Arguments: {K : Type u} [Field K] {G : Type u_1} [Group G] [MulSemiringAction G K] {g : G} {S T : ValuationSubring K}
Goal: S ≤ g • T ↔ g⁻¹ • S ≤ T