ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Ideal.FinrankQuotientMap.linearIndependent_of_nontrivial
theorem FinrankQuotientMap.linearIndependent_of_nontrivial [IsDedekindDomain R]
    (hRS : RingHom.ker (algebraMap R S) ≠ ⊤) (f : V'' →ₗ[R] V) (hf : Function.Injective f)
    (f' : V'' →ₗ[R] V') {ι : Type*} {b : ι → V''} (hb' : LinearIndependent S (f' ∘ b)) :
    LinearIndependent K (f ∘ b) 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (K : Type u_1) [Field K] [Algebra R K] [hRK : IsFractionRing R K] {V : Type u_3} {V' : Type u_4} {V'' : Type u_5} [AddCommGroup V] [Module R V] [Module K V] [IsScalarTower R K V] [AddCommGroup V'] [Module R V'] [Module S V'] [IsScalarTower R S V'] [AddCommGroup V''] [Module R V''] [IsDedekindDomain R] (hRS : RingHom.ker (algebraMap R S) ≠ ⊤) (f : V'' →ₗ[R] V) (hf : Function.Injective ⇑f) (f' : V'' →ₗ[R] V') {ι : Type u_6} {b : ι → V''} (hb' : LinearIndependent S (⇑f' ∘ b))
Goal: LinearIndependent K (⇑f ∘ b)
-- Ideal.FinrankQuotientMap.span_eq_top
theorem FinrankQuotientMap.span_eq_top [IsDomain R] [IsDomain S] [Algebra K L] [IsNoetherian R S]
    [Algebra R L] [IsScalarTower R S L] [IsScalarTower R K L] [IsIntegralClosure S R L]
    [NoZeroSMulDivisors R K] (hp : p ≠ ⊤) (b : Set S)
    (hb' : Submodule.span R b ⊔ (p.map (algebraMap R S)).restrictScalars R = ⊤) :
    Submodule.span K (algebraMap S L '' b) = ⊤ 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] (p : Ideal R) [Algebra R S] {K : Type u_1} [Field K] [Algebra R K] {L : Type u_2} [Field L] [Algebra S L] [IsFractionRing S L] [IsDomain R] [IsDomain S] [Algebra K L] [IsNoetherian R S] [Algebra R L] [IsScalarTower R S L] [IsScalarTower R K L] [IsIntegralClosure S R L] [NoZeroSMulDivisors R K] (hp : p ≠ ⊤) (b : Set S) (hb' : Submodule.span R b ⊔ Submodule.restrictScalars R (Ideal.map (algebraMap R S) p) = ⊤)
Goal: Submodule.span K (⇑(algebraMap S L) '' b) = ⊤
-- Ideal.finrank_quotient_map
theorem finrank_quotient_map [IsDomain S] [IsDedekindDomain R] [Algebra K L]
    [Algebra R L] [IsScalarTower R K L] [IsScalarTower R S L] [IsIntegralClosure S R L]
    [hp : p.IsMaximal] [IsNoetherian R S] :
    finrank (R ⧸ p) (S ⧸ map (algebraMap R S) p) = finrank K L 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] (p : Ideal R) [Algebra R S] (K : Type u_1) [Field K] [Algebra R K] [hRK : IsFractionRing R K] (L : Type u_2) [Field L] [Algebra S L] [IsFractionRing S L] [IsDomain S] [IsDedekindDomain R] [Algebra K L] [Algebra R L] [IsScalarTower R K L] [IsScalarTower R S L] [IsIntegralClosure S R L] [hp : p.IsMaximal] [IsNoetherian R S]
Goal: FiniteDimensional.finrank (R ⧸ p) (S ⧸ Ideal.map (algebraMap R S) p) = FiniteDimensional.finrank K L
-- Ideal.Quotient.algebraMap_quotient_pow_ramificationIdx
@[simp]
theorem Quotient.algebraMap_quotient_pow_ramificationIdx (x : R) :
    algebraMap (R ⧸ p) (S ⧸ P ^ e) (Ideal.Quotient.mk p x) = Ideal.Quotient.mk (P ^ e) (f x) 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] (f : R →+* S) (p : Ideal R) (P : Ideal S) (x : R)
Goal: (algebraMap (R ⧸ p) (S ⧸ P ^ Ideal.ramificationIdx f p P)) ((Ideal.Quotient.mk p) x) =
  (Ideal.Quotient.mk (P ^ Ideal.ramificationIdx f p P)) (f x)
-- Ideal.Quotient.algebraMap_quotient_of_ramificationIdx_neZero
@[simp]
theorem Quotient.algebraMap_quotient_of_ramificationIdx_neZero (x : R) :
    algebraMap (R ⧸ p) (S ⧸ P) (Ideal.Quotient.mk p x) = Ideal.Quotient.mk P (f x) 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] (f : R →+* S) (p : Ideal R) (P : Ideal S) [hfp : NeZero (Ideal.ramificationIdx f p P)] (x : R)
Goal: (algebraMap (R ⧸ p) (S ⧸ P)) ((Ideal.Quotient.mk p) x) = (Ideal.Quotient.mk P) (f x)
-- Ideal.powQuotSuccInclusion_apply_coe
@[simps]
def powQuotSuccInclusion (i : ℕ) :
    Ideal.map (Ideal.Quotient.mk (P ^ e)) (P ^ (i + 1)) →ₗ[R ⧸ p]
    Ideal.map (Ideal.Quotient.mk (P ^ e)) (P ^ i) 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] (f : R →+* S) (p : Ideal R) (P : Ideal S) (i : ℕ) (x : ↥(Ideal.map (Ideal.Quotient.mk (P ^ Ideal.ramificationIdx f p P)) (P ^ (i + 1))))
Goal: ↑((Ideal.powQuotSuccInclusion f p P i) x) = ↑x
-- Ideal.powQuotSuccInclusion_injective
theorem powQuotSuccInclusion_injective (i : ℕ) :
    Function.Injective (powQuotSuccInclusion f p P i) 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] (f : R →+* S) (p : Ideal R) (P : Ideal S) (i : ℕ)
Goal: Function.Injective ⇑(Ideal.powQuotSuccInclusion f p P i)
-- Ideal.quotientToQuotientRangePowQuotSuccAux_mk
theorem quotientToQuotientRangePowQuotSuccAux_mk {i : ℕ} {a : S} (a_mem : a ∈ P ^ i) (x : S) :
    quotientToQuotientRangePowQuotSuccAux f p P a_mem (Submodule.Quotient.mk x) =
      Submodule.Quotient.mk ⟨_, Ideal.mem_map_of_mem _ (Ideal.mul_mem_right x _ a_mem)⟩ 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] (f : R →+* S) (p : Ideal R) (P : Ideal S) {i : ℕ} {a : S} (a_mem : a ∈ P ^ i) (x : S)
Goal: Ideal.quotientToQuotientRangePowQuotSuccAux f p P a_mem (Submodule.Quotient.mk x) =
  Submodule.Quotient.mk ⟨(Ideal.Quotient.mk (P ^ Ideal.ramificationIdx f p P)) (a * x), ⋯⟩
-- Ideal.quotientToQuotientRangePowQuotSucc_mk
theorem quotientToQuotientRangePowQuotSucc_mk {i : ℕ} {a : S} (a_mem : a ∈ P ^ i) (x : S) :
    quotientToQuotientRangePowQuotSucc f p P a_mem (Submodule.Quotient.mk x) =
      Submodule.Quotient.mk ⟨_, Ideal.mem_map_of_mem _ (Ideal.mul_mem_right x _ a_mem)⟩ 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] (f : R →+* S) (p : Ideal R) (P : Ideal S) [hfp : NeZero (Ideal.ramificationIdx f p P)] {i : ℕ} {a : S} (a_mem : a ∈ P ^ i) (x : S)
Goal: (Ideal.quotientToQuotientRangePowQuotSucc f p P a_mem) (Submodule.Quotient.mk x) =
  Submodule.Quotient.mk ⟨(Ideal.Quotient.mk (P ^ Ideal.ramificationIdx f p P)) (a * x), ⋯⟩
-- Ideal.quotientToQuotientRangePowQuotSucc_injective
theorem quotientToQuotientRangePowQuotSucc_injective [IsDedekindDomain S] [P.IsPrime]
    {i : ℕ} (hi : i < e) {a : S} (a_mem : a ∈ P ^ i) (a_not_mem : a ∉ P ^ (i + 1)) :
    Function.Injective (quotientToQuotientRangePowQuotSucc f p P a_mem) 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] (f : R →+* S) (p : Ideal R) (P : Ideal S) [hfp : NeZero (Ideal.ramificationIdx f p P)] [IsDedekindDomain S] [P.IsPrime] {i : ℕ} (hi : i < Ideal.ramificationIdx f p P) {a : S} (a_mem : a ∈ P ^ i) (a_not_mem : a ∉ P ^ (i + 1))
Goal: Function.Injective ⇑(Ideal.quotientToQuotientRangePowQuotSucc f p P a_mem)
-- Ideal.quotientToQuotientRangePowQuotSucc_surjective
theorem quotientToQuotientRangePowQuotSucc_surjective [IsDedekindDomain S]
    (hP0 : P ≠ ⊥) [hP : P.IsPrime] {i : ℕ} (hi : i < e) {a : S} (a_mem : a ∈ P ^ i)
    (a_not_mem : a ∉ P ^ (i + 1)) :
    Function.Surjective (quotientToQuotientRangePowQuotSucc f p P a_mem) 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] (f : R →+* S) (p : Ideal R) (P : Ideal S) [hfp : NeZero (Ideal.ramificationIdx f p P)] [IsDedekindDomain S] (hP0 : P ≠ ⊥) [hP : P.IsPrime] {i : ℕ} (hi : i < Ideal.ramificationIdx f p P) {a : S} (a_mem : a ∈ P ^ i) (a_not_mem : a ∉ P ^ (i + 1))
Goal: Function.Surjective ⇑(Ideal.quotientToQuotientRangePowQuotSucc f p P a_mem)
-- Ideal.rank_pow_quot_aux
theorem rank_pow_quot_aux [IsDedekindDomain S] [p.IsMaximal] [P.IsPrime] (hP0 : P ≠ ⊥)
    {i : ℕ} (hi : i < e) :
    Module.rank (R ⧸ p) (Ideal.map (Ideal.Quotient.mk (P ^ e)) (P ^ i)) =
      Module.rank (R ⧸ p) (S ⧸ P) +
        Module.rank (R ⧸ p) (Ideal.map (Ideal.Quotient.mk (P ^ e)) (P ^ (i + 1))) 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] (f : R →+* S) (p : Ideal R) (P : Ideal S) [hfp : NeZero (Ideal.ramificationIdx f p P)] [IsDedekindDomain S] [p.IsMaximal] [P.IsPrime] (hP0 : P ≠ ⊥) {i : ℕ} (hi : i < Ideal.ramificationIdx f p P)
Goal: Module.rank (R ⧸ p) ↥(Ideal.map (Ideal.Quotient.mk (P ^ Ideal.ramificationIdx f p P)) (P ^ i)) =
  Module.rank (R ⧸ p) (S ⧸ P) +
    Module.rank (R ⧸ p) ↥(Ideal.map (Ideal.Quotient.mk (P ^ Ideal.ramificationIdx f p P)) (P ^ (i + 1)))
-- Ideal.rank_pow_quot
theorem rank_pow_quot [IsDedekindDomain S] [p.IsMaximal] [P.IsPrime] (hP0 : P ≠ ⊥)
    (i : ℕ) (hi : i ≤ e) :
    Module.rank (R ⧸ p) (Ideal.map (Ideal.Quotient.mk (P ^ e)) (P ^ i)) =
      (e - i) • Module.rank (R ⧸ p) (S ⧸ P) 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] (f : R →+* S) (p : Ideal R) (P : Ideal S) [hfp : NeZero (Ideal.ramificationIdx f p P)] [IsDedekindDomain S] [p.IsMaximal] [P.IsPrime] (hP0 : P ≠ ⊥) (i : ℕ) (hi : i ≤ Ideal.ramificationIdx f p P)
Goal: Module.rank (R ⧸ p) ↥(Ideal.map (Ideal.Quotient.mk (P ^ Ideal.ramificationIdx f p P)) (P ^ i)) =
  (Ideal.ramificationIdx f p P - i) • Module.rank (R ⧸ p) (S ⧸ P)
-- Ideal.rank_prime_pow_ramificationIdx
theorem rank_prime_pow_ramificationIdx [IsDedekindDomain S] [p.IsMaximal] [P.IsPrime]
    (hP0 : P ≠ ⊥) (he : e ≠ 0) :
    Module.rank (R ⧸ p) (S ⧸ P ^ e) =
      e •
        @Module.rank (R ⧸ p) (S ⧸ P) _ _
          (@Algebra.toModule _ _ _ _ <|
            @Quotient.algebraQuotientOfRamificationIdxNeZero _ _ _ _ _ _ _ ⟨he⟩) 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] (f : R →+* S) (p : Ideal R) (P : Ideal S) [IsDedekindDomain S] [p.IsMaximal] [P.IsPrime] (hP0 : P ≠ ⊥) (he : Ideal.ramificationIdx f p P ≠ 0)
Goal: Module.rank (R ⧸ p) (S ⧸ P ^ Ideal.ramificationIdx f p P) = Ideal.ramificationIdx f p P • Module.rank (R ⧸ p) (S ⧸ P)
-- Ideal.finrank_prime_pow_ramificationIdx
theorem finrank_prime_pow_ramificationIdx [IsDedekindDomain S] (hP0 : P ≠ ⊥)
    [p.IsMaximal] [P.IsPrime] (he : e ≠ 0) :
    finrank (R ⧸ p) (S ⧸ P ^ e) =
      e *
        @finrank (R ⧸ p) (S ⧸ P) _ _
          (@Algebra.toModule _ _ _ _ <|
            @Quotient.algebraQuotientOfRamificationIdxNeZero _ _ _ _ _ _ _ ⟨he⟩) 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] (f : R →+* S) (p : Ideal R) (P : Ideal S) [IsDedekindDomain S] (hP0 : P ≠ ⊥) [p.IsMaximal] [P.IsPrime] (he : Ideal.ramificationIdx f p P ≠ 0)
Goal: FiniteDimensional.finrank (R ⧸ p) (S ⧸ P ^ Ideal.ramificationIdx f p P) =
  Ideal.ramificationIdx f p P * FiniteDimensional.finrank (R ⧸ p) (S ⧸ P)
-- Ideal.Factors.ne_bot
theorem Factors.ne_bot (P : (factors (map (algebraMap R S) p)).toFinset) : (P : Ideal S) ≠ ⊥ 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] (p : Ideal R) [IsDedekindDomain S] [Algebra R S] (P : { x // x ∈ (UniqueFactorizationMonoid.factors (Ideal.map (algebraMap R S) p)).toFinset })
Goal: ↑P ≠ ⊥
-- Ideal.Factors.ramificationIdx_ne_zero
theorem Factors.ramificationIdx_ne_zero (P : (factors (map (algebraMap R S) p)).toFinset) :
    ramificationIdx (algebraMap R S) p P ≠ 0 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] (p : Ideal R) [IsDedekindDomain S] [Algebra R S] (P : { x // x ∈ (UniqueFactorizationMonoid.factors (Ideal.map (algebraMap R S) p)).toFinset })
Goal: Ideal.ramificationIdx (algebraMap R S) p ↑P ≠ 0
-- Ideal.Factors.finrank_pow_ramificationIdx
theorem Factors.finrank_pow_ramificationIdx [p.IsMaximal]
    (P : (factors (map (algebraMap R S) p)).toFinset) :
    finrank (R ⧸ p) (S ⧸ (P : Ideal S) ^ ramificationIdx (algebraMap R S) p P) =
      ramificationIdx (algebraMap R S) p P * inertiaDeg (algebraMap R S) p P 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] (p : Ideal R) [IsDedekindDomain S] [Algebra R S] [p.IsMaximal] (P : { x // x ∈ (UniqueFactorizationMonoid.factors (Ideal.map (algebraMap R S) p)).toFinset })
Goal: FiniteDimensional.finrank (R ⧸ p) (S ⧸ ↑P ^ Ideal.ramificationIdx (algebraMap R S) p ↑P) =
  Ideal.ramificationIdx (algebraMap R S) p ↑P * Ideal.inertiaDeg (algebraMap R S) p ↑P
-- Ideal.Factors.inertiaDeg_ne_zero
theorem Factors.inertiaDeg_ne_zero [IsNoetherian R S] [p.IsMaximal]
    (P : (factors (map (algebraMap R S) p)).toFinset) : inertiaDeg (algebraMap R S) p P ≠ 0 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] (p : Ideal R) [IsDedekindDomain S] [Algebra R S] [IsNoetherian R S] [p.IsMaximal] (P : { x // x ∈ (UniqueFactorizationMonoid.factors (Ideal.map (algebraMap R S) p)).toFinset })
Goal: Ideal.inertiaDeg (algebraMap R S) p ↑P ≠ 0
-- Ideal.Factors.piQuotientEquiv_mk
@[simp]
theorem Factors.piQuotientEquiv_mk (p : Ideal R) (hp : map (algebraMap R S) p ≠ ⊥) (x : S) :
    Factors.piQuotientEquiv p hp (Ideal.Quotient.mk _ x) = fun _ => Ideal.Quotient.mk _ x 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] [IsDedekindDomain S] [Algebra R S] (p : Ideal R) (hp : Ideal.map (algebraMap R S) p ≠ ⊥) (x : S)
Goal: (Ideal.Factors.piQuotientEquiv p hp) ((Ideal.Quotient.mk (Ideal.map (algebraMap R S) p)) x) = fun x_1 =>
  (Ideal.Quotient.mk (↑x_1 ^ Ideal.ramificationIdx (algebraMap R S) p ↑x_1)) x
-- Ideal.Factors.piQuotientEquiv_map
@[simp]
theorem Factors.piQuotientEquiv_map (p : Ideal R) (hp : map (algebraMap R S) p ≠ ⊥) (x : R) :
    Factors.piQuotientEquiv p hp (algebraMap _ _ x) = fun _ =>
      Ideal.Quotient.mk _ (algebraMap _ _ x) 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] [IsDedekindDomain S] [Algebra R S] (p : Ideal R) (hp : Ideal.map (algebraMap R S) p ≠ ⊥) (x : R)
Goal: (Ideal.Factors.piQuotientEquiv p hp) ((algebraMap R (S ⧸ Ideal.map (algebraMap R S) p)) x) = fun x_1 =>
  (Ideal.Quotient.mk (↑x_1 ^ Ideal.ramificationIdx (algebraMap R S) p ↑x_1)) ((algebraMap R S) x)
-- Ideal.sum_ramification_inertia
theorem sum_ramification_inertia (K L : Type*) [Field K] [Field L] [IsDedekindDomain R]
    [Algebra R K] [IsFractionRing R K] [Algebra S L] [IsFractionRing S L] [Algebra K L]
    [Algebra R L] [IsScalarTower R S L] [IsScalarTower R K L] [IsNoetherian R S]
    [IsIntegralClosure S R L] [p.IsMaximal] (hp0 : p ≠ ⊥) :
    (∑ P ∈ (factors (map (algebraMap R S) p)).toFinset,
        ramificationIdx (algebraMap R S) p P * inertiaDeg (algebraMap R S) p P) =
      finrank K L 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] (p : Ideal R) [IsDedekindDomain S] [Algebra R S] (K : Type u_1) (L : Type u_2) [Field K] [Field L] [IsDedekindDomain R] [Algebra R K] [IsFractionRing R K] [Algebra S L] [IsFractionRing S L] [Algebra K L] [Algebra R L] [IsScalarTower R S L] [IsScalarTower R K L] [IsNoetherian R S] [IsIntegralClosure S R L] [p.IsMaximal] (hp0 : p ≠ ⊥)
Goal: ∑ P ∈ (UniqueFactorizationMonoid.factors (Ideal.map (algebraMap R S) p)).toFinset,
    Ideal.ramificationIdx (algebraMap R S) p P * Ideal.inertiaDeg (algebraMap R S) p P =
  FiniteDimensional.finrank K L
-- FirstOrder.Language.skolem₁_Functions
@[simps]
def skolem₁ : Language 
Arguments: (L : FirstOrder.Language) (n : ℕ)
Goal: L.skolem₁.Functions n = L.BoundedFormula Empty (n + 1)
-- FirstOrder.Language.skolem₁_Relations
@[simps]
def skolem₁ : Language 
Arguments: (L : FirstOrder.Language)
Goal: ∀ (x : ℕ), L.skolem₁.Relations x = Empty
-- FirstOrder.Language.card_functions_sum_skolem₁
theorem card_functions_sum_skolem₁ :
    #(Σ n, (L.sum L.skolem₁).Functions n) = #(Σ n, L.BoundedFormula Empty (n + 1)) 
Arguments: {L : FirstOrder.Language}
Goal: Cardinal.mk ((n : ℕ) × (L.sum L.skolem₁).Functions n) = Cardinal.mk ((n : ℕ) × L.BoundedFormula Empty (n + 1))
-- FirstOrder.Language.card_functions_sum_skolem₁_le
theorem card_functions_sum_skolem₁_le : #(Σ n, (L.sum L.skolem₁).Functions n) ≤ max ℵ₀ L.card 
Arguments: {L : FirstOrder.Language}
Goal: Cardinal.mk ((n : ℕ) × (L.sum L.skolem₁).Functions n) ≤ max Cardinal.aleph0 L.card
-- FirstOrder.Language.Substructure.skolem₁_reduct_isElementary
theorem skolem₁_reduct_isElementary (S : (L.sum L.skolem₁).Substructure M) :
    (LHom.sumInl.substructureReduct S).IsElementary 
Arguments: {L : FirstOrder.Language} {M : Type w} [Nonempty M] [L.Structure M] (S : (L.sum L.skolem₁).Substructure M)
Goal: (FirstOrder.Language.LHom.sumInl.substructureReduct S).IsElementary
-- FirstOrder.Language.Substructure.coeSort_elementarySkolem₁Reduct
theorem coeSort_elementarySkolem₁Reduct (S : (L.sum L.skolem₁).Substructure M) :
    (S.elementarySkolem₁Reduct : Type w) = S 
Arguments: {L : FirstOrder.Language} {M : Type w} [Nonempty M] [L.Structure M] (S : (L.sum L.skolem₁).Substructure M)
Goal: ↥S.elementarySkolem₁Reduct = ↥S
-- FirstOrder.Language.exists_small_elementarySubstructure
theorem exists_small_elementarySubstructure : ∃ S : L.ElementarySubstructure M, Small.{max u v} S 
Arguments: (L : FirstOrder.Language) (M : Type w) [Nonempty M] [L.Structure M]
Goal: ∃ S, Small.{max u v, w} ↥S
-- FirstOrder.Language.exists_elementarySubstructure_card_eq
theorem exists_elementarySubstructure_card_eq (s : Set M) (κ : Cardinal.{w'}) (h1 : ℵ₀ ≤ κ)
    (h2 : Cardinal.lift.{w'} #s ≤ Cardinal.lift.{w} κ)
    (h3 : Cardinal.lift.{w'} L.card ≤ Cardinal.lift.{max u v} κ)
    (h4 : Cardinal.lift.{w} κ ≤ Cardinal.lift.{w'} #M) :
    ∃ S : L.ElementarySubstructure M, s ⊆ S ∧ Cardinal.lift.{w'} #S = Cardinal.lift.{w} κ 
Arguments: (L : FirstOrder.Language) {M : Type w} [Nonempty M] [L.Structure M] (s : Set M) (κ : Cardinal.{w'}) (h1 : Cardinal.aleph0 ≤ κ) (h2 : Cardinal.lift.{w', w} (Cardinal.mk ↑s) ≤ Cardinal.lift.{w, w'} κ) (h3 : Cardinal.lift.{w', max u v} L.card ≤ Cardinal.lift.{max u v, w'} κ) (h4 : Cardinal.lift.{w, w'} κ ≤ Cardinal.lift.{w', w} (Cardinal.mk M))
Goal: ∃ S, s ⊆ ↑S ∧ Cardinal.lift.{w', w} (Cardinal.mk ↥S) = Cardinal.lift.{w, w'} κ
-- FirstOrder.Language.Theory.CompleteType.subset'
structure CompleteType where
  toTheory : L[[α]].Theory
  subset' : (L.lhomWithConstants α).onTheory T ⊆ toTheory
  isMaximal' : toTheory.IsMaximal
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} (self : T.CompleteType α)
Goal: (L.lhomWithConstants α).onTheory T ⊆ ↑self
-- FirstOrder.Language.Theory.CompleteType.isMaximal'
structure CompleteType where
  toTheory : L[[α]].Theory
  subset' : (L.lhomWithConstants α).onTheory T ⊆ toTheory
  isMaximal' : toTheory.IsMaximal
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} (self : T.CompleteType α)
Goal: (↑self).IsMaximal
-- FirstOrder.Language.Theory.CompleteType.isMaximal
theorem isMaximal (p : T.CompleteType α) : IsMaximal (p : L[[α]].Theory) 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} (p : T.CompleteType α)
Goal: FirstOrder.Language.Theory.IsMaximal ↑p
-- FirstOrder.Language.Theory.CompleteType.subset
theorem subset (p : T.CompleteType α) : (L.lhomWithConstants α).onTheory T ⊆ (p : L[[α]].Theory) 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} (p : T.CompleteType α)
Goal: (L.lhomWithConstants α).onTheory T ⊆ ↑p
-- FirstOrder.Language.Theory.CompleteType.mem_or_not_mem
theorem mem_or_not_mem (p : T.CompleteType α) (φ : L[[α]].Sentence) : φ ∈ p ∨ φ.not ∈ p 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} (p : T.CompleteType α) (φ : L[[α]].Sentence)
Goal: φ ∈ p ∨ FirstOrder.Language.Formula.not φ ∈ p
-- FirstOrder.Language.Theory.CompleteType.mem_of_models
theorem mem_of_models (p : T.CompleteType α) {φ : L[[α]].Sentence}
    (h : (L.lhomWithConstants α).onTheory T ⊨ᵇ φ) : φ ∈ p 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} (p : T.CompleteType α) {φ : L[[α]].Sentence} (h : (L.lhomWithConstants α).onTheory T ⊨ᵇ φ)
Goal: φ ∈ p
-- FirstOrder.Language.Theory.CompleteType.not_mem_iff
theorem not_mem_iff (p : T.CompleteType α) (φ : L[[α]].Sentence) : φ.not ∈ p ↔ ¬φ ∈ p 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} (p : T.CompleteType α) (φ : L[[α]].Sentence)
Goal: FirstOrder.Language.Formula.not φ ∈ p ↔ φ ∉ p
-- FirstOrder.Language.Theory.CompleteType.compl_setOf_mem
@[simp]
theorem compl_setOf_mem {φ : L[[α]].Sentence} :
    { p : T.CompleteType α | φ ∈ p }ᶜ = { p : T.CompleteType α | φ.not ∈ p } 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} {φ : L[[α]].Sentence}
Goal: {p | φ ∈ p}ᶜ = {p | FirstOrder.Language.Formula.not φ ∈ p}
-- FirstOrder.Language.Theory.CompleteType.setOf_subset_eq_empty_iff
theorem setOf_subset_eq_empty_iff (S : L[[α]].Theory) :
    { p : T.CompleteType α | S ⊆ ↑p } = ∅ ↔
      ¬((L.lhomWithConstants α).onTheory T ∪ S).IsSatisfiable 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} (S : L[[α]].Theory)
Goal: {p | S ⊆ ↑p} = ∅ ↔ ¬((L.lhomWithConstants α).onTheory T ∪ S).IsSatisfiable
-- FirstOrder.Language.Theory.CompleteType.setOf_mem_eq_univ_iff
theorem setOf_mem_eq_univ_iff (φ : L[[α]].Sentence) :
    { p : T.CompleteType α | φ ∈ p } = Set.univ ↔ (L.lhomWithConstants α).onTheory T ⊨ᵇ φ 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} (φ : L[[α]].Sentence)
Goal: {p | φ ∈ p} = Set.univ ↔ (L.lhomWithConstants α).onTheory T ⊨ᵇ φ
-- FirstOrder.Language.Theory.CompleteType.setOf_subset_eq_univ_iff
theorem setOf_subset_eq_univ_iff (S : L[[α]].Theory) :
    { p : T.CompleteType α | S ⊆ ↑p } = Set.univ ↔
      ∀ φ, φ ∈ S → (L.lhomWithConstants α).onTheory T ⊨ᵇ φ 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} (S : L[[α]].Theory)
Goal: {p | S ⊆ ↑p} = Set.univ ↔ ∀ φ ∈ S, (L.lhomWithConstants α).onTheory T ⊨ᵇ φ
-- FirstOrder.Language.Theory.CompleteType.nonempty_iff
theorem nonempty_iff : Nonempty (T.CompleteType α) ↔ T.IsSatisfiable 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w}
Goal: Nonempty (T.CompleteType α) ↔ T.IsSatisfiable
-- FirstOrder.Language.Theory.CompleteType.iInter_setOf_subset
theorem iInter_setOf_subset {ι : Type*} (S : ι → L[[α]].Theory) :
    ⋂ i : ι, { p : T.CompleteType α | S i ⊆ p } =
      { p : T.CompleteType α | ⋃ i : ι, S i ⊆ p } 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} {ι : Type u_1} (S : ι → L[[α]].Theory)
Goal: ⋂ i, {p | S i ⊆ ↑p} = {p | ⋃ i, S i ⊆ ↑p}
-- FirstOrder.Language.Theory.CompleteType.toList_foldr_inf_mem
theorem toList_foldr_inf_mem {p : T.CompleteType α} {t : Finset (L[[α]]).Sentence} :
    t.toList.foldr (· ⊓ ·) ⊤ ∈ p ↔ (t : L[[α]].Theory) ⊆ ↑p 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} {p : T.CompleteType α} {t : Finset L[[α]].Sentence}
Goal: List.foldr (fun x x_1 => x ⊓ x_1) ⊤ t.toList ∈ p ↔ ↑t ⊆ ↑p
-- FirstOrder.Language.Theory.CompleteType.mem_typeOf
@[simp]
theorem mem_typeOf {φ : L[[α]].Sentence} :
    φ ∈ T.typeOf v ↔ (Formula.equivSentence.symm φ).Realize v 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} {M : Type w'} [L.Structure M] [Nonempty M] [M ⊨ T] {v : α → M} {φ : L[[α]].Sentence}
Goal: φ ∈ T.typeOf v ↔ (FirstOrder.Language.Formula.equivSentence.symm φ).Realize v
-- FirstOrder.Language.Theory.CompleteType.formula_mem_typeOf
theorem formula_mem_typeOf {φ : L.Formula α} :
    Formula.equivSentence φ ∈ T.typeOf v ↔ φ.Realize v 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} {M : Type w'} [L.Structure M] [Nonempty M] [M ⊨ T] {v : α → M} {φ : L.Formula α}
Goal: FirstOrder.Language.Formula.equivSentence φ ∈ T.typeOf v ↔ φ.Realize v
-- FirstOrder.Language.Theory.exists_modelType_is_realized_in
theorem exists_modelType_is_realized_in (p : T.CompleteType α) :
    ∃ M : Theory.ModelType.{u, v, max u v w} T, p ∈ T.realizedTypes M α 
Arguments: {L : FirstOrder.Language} (T : L.Theory) {α : Type w} (p : T.CompleteType α)
Goal: ∃ M, p ∈ T.realizedTypes (↑M) α
-- mem_approxOrderOf_iff
@[to_additive mem_approx_add_orderOf_iff]
theorem mem_approxOrderOf_iff {A : Type*} [SeminormedGroup A] {n : ℕ} {δ : ℝ} {a : A} :
    a ∈ approxOrderOf A n δ ↔ ∃ b : A, orderOf b = n ∧ a ∈ ball b δ 
Arguments: {A : Type u_1} [SeminormedGroup A] {n : ℕ} {δ : ℝ} {a : A}
Goal: a ∈ approxOrderOf A n δ ↔ ∃ b, orderOf b = n ∧ a ∈ Metric.ball b δ
-- mem_approx_add_orderOf_iff
@[to_additive mem_approx_add_orderOf_iff]
theorem mem_approxOrderOf_iff {A : Type*} [SeminormedGroup A] {n : ℕ} {δ : ℝ} {a : A} :
    a ∈ approxOrderOf A n δ ↔ ∃ b : A, orderOf b = n ∧ a ∈ ball b δ 
Arguments: {A : Type u_1} [SeminormedAddGroup A] {n : ℕ} {δ : ℝ} {a : A}
Goal: a ∈ approxAddOrderOf A n δ ↔ ∃ b, addOrderOf b = n ∧ a ∈ Metric.ball b δ
-- mem_wellApproximable_iff
@[to_additive mem_add_wellApproximable_iff]
theorem mem_wellApproximable_iff {A : Type*} [SeminormedGroup A] {δ : ℕ → ℝ} {a : A} :
    a ∈ wellApproximable A δ ↔
      a ∈ blimsup (fun n => approxOrderOf A n (δ n)) atTop fun n => 0 < n 
Arguments: {A : Type u_1} [SeminormedGroup A] {δ : ℕ → ℝ} {a : A}
Goal: a ∈ wellApproximable A δ ↔ a ∈ Filter.blimsup (fun n => approxOrderOf A n (δ n)) Filter.atTop fun n => 0 < n