ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Asymptotics.IsBigO.integrableAtFilter
theorem _root_.Asymptotics.IsBigO.integrableAtFilter [IsMeasurablyGenerated l]
    (hf : f =O[l] g) (hfm : StronglyMeasurableAtFilter f l μ) (hg : IntegrableAtFilter g l μ) :
    IntegrableAtFilter f l μ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : α → F} {μ : MeasureTheory.Measure α} {l : Filter α} [l.IsMeasurablyGenerated] (hf : f =O[l] g) (hfm : StronglyMeasurableAtFilter f l μ) (hg : MeasureTheory.IntegrableAtFilter g l μ)
Goal: MeasureTheory.IntegrableAtFilter f l μ
-- Asymptotics.IsBigO.integrable
theorem _root_.Asymptotics.IsBigO.integrable (hfm : AEStronglyMeasurable f μ)
    (hf : f =O[⊤] g) (hg : Integrable g μ) : Integrable f μ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : α → F} {μ : MeasureTheory.Measure α} (hfm : MeasureTheory.AEStronglyMeasurable f μ) (hf : f =O[⊤] g) (hg : MeasureTheory.Integrable g μ)
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.LocallyIntegrable.integrable_of_isBigO_cocompact
theorem LocallyIntegrable.integrable_of_isBigO_cocompact [IsMeasurablyGenerated (cocompact α)]
    (hf : LocallyIntegrable f μ) (ho : f =O[cocompact α] g)
    (hg : IntegrableAtFilter g (cocompact α) μ) : Integrable f μ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : α → F} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [SecondCountableTopology α] [(Filter.cocompact α).IsMeasurablyGenerated] (hf : MeasureTheory.LocallyIntegrable f μ) (ho : f =O[Filter.cocompact α] g) (hg : MeasureTheory.IntegrableAtFilter g (Filter.cocompact α) μ)
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.LocallyIntegrable.integrable_of_isBigO_atBot_atTop
theorem LocallyIntegrable.integrable_of_isBigO_atBot_atTop
    [IsMeasurablyGenerated (atBot (α := α))] [IsMeasurablyGenerated (atTop (α := α))]
    (hf : LocallyIntegrable f μ)
    (ho : f =O[atBot] g) (hg : IntegrableAtFilter g atBot μ)
    (ho' : f =O[atTop] g') (hg' : IntegrableAtFilter g' atTop μ) : Integrable f μ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : α → F} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [SecondCountableTopology α] [LinearOrder α] [CompactIccSpace α] {g' : α → F} [Filter.atBot.IsMeasurablyGenerated] [Filter.atTop.IsMeasurablyGenerated] (hf : MeasureTheory.LocallyIntegrable f μ) (ho : f =O[Filter.atBot] g) (hg : MeasureTheory.IntegrableAtFilter g Filter.atBot μ) (ho' : f =O[Filter.atTop] g') (hg' : MeasureTheory.IntegrableAtFilter g' Filter.atTop μ)
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.LocallyIntegrableOn.integrableOn_of_isBigO_atBot
theorem LocallyIntegrableOn.integrableOn_of_isBigO_atBot [IsMeasurablyGenerated (atBot (α := α))]
    (hf : LocallyIntegrableOn f (Iic a) μ) (ho : f =O[atBot] g)
    (hg : IntegrableAtFilter g atBot μ) : IntegrableOn f (Iic a) μ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : α → F} {a : α} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [SecondCountableTopology α] [LinearOrder α] [CompactIccSpace α] [Filter.atBot.IsMeasurablyGenerated] (hf : MeasureTheory.LocallyIntegrableOn f (Set.Iic a) μ) (ho : f =O[Filter.atBot] g) (hg : MeasureTheory.IntegrableAtFilter g Filter.atBot μ)
Goal: MeasureTheory.IntegrableOn f (Set.Iic a) μ
-- MeasureTheory.LocallyIntegrableOn.integrableOn_of_isBigO_atTop
theorem LocallyIntegrableOn.integrableOn_of_isBigO_atTop [IsMeasurablyGenerated (atTop (α := α))]
    (hf : LocallyIntegrableOn f (Ici a) μ) (ho : f =O[atTop] g)
    (hg : IntegrableAtFilter g atTop μ) : IntegrableOn f (Ici a) μ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : α → F} {a : α} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [SecondCountableTopology α] [LinearOrder α] [CompactIccSpace α] [Filter.atTop.IsMeasurablyGenerated] (hf : MeasureTheory.LocallyIntegrableOn f (Set.Ici a) μ) (ho : f =O[Filter.atTop] g) (hg : MeasureTheory.IntegrableAtFilter g Filter.atTop μ)
Goal: MeasureTheory.IntegrableOn f (Set.Ici a) μ
-- MeasureTheory.LocallyIntegrable.integrable_of_isBigO_atBot
theorem LocallyIntegrable.integrable_of_isBigO_atBot [IsMeasurablyGenerated (atBot (α := α))]
    [OrderTop α] (hf : LocallyIntegrable f μ) (ho : f =O[atBot] g)
    (hg : IntegrableAtFilter g atBot μ) : Integrable f μ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : α → F} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [SecondCountableTopology α] [LinearOrder α] [CompactIccSpace α] [Filter.atBot.IsMeasurablyGenerated] [OrderTop α] (hf : MeasureTheory.LocallyIntegrable f μ) (ho : f =O[Filter.atBot] g) (hg : MeasureTheory.IntegrableAtFilter g Filter.atBot μ)
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.LocallyIntegrable.integrable_of_isBigO_atTop
theorem LocallyIntegrable.integrable_of_isBigO_atTop [IsMeasurablyGenerated (atTop (α := α))]
    [OrderBot α] (hf : LocallyIntegrable f μ) (ho : f =O[atTop] g)
    (hg : IntegrableAtFilter g atTop μ) : Integrable f μ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : α → F} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [SecondCountableTopology α] [LinearOrder α] [CompactIccSpace α] [Filter.atTop.IsMeasurablyGenerated] [OrderBot α] (hf : MeasureTheory.LocallyIntegrable f μ) (ho : f =O[Filter.atTop] g) (hg : MeasureTheory.IntegrableAtFilter g Filter.atTop μ)
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.LocallyIntegrable.integrable_of_isBigO_atTop_of_norm_isNegInvariant
theorem LocallyIntegrable.integrable_of_isBigO_atTop_of_norm_isNegInvariant
    [IsMeasurablyGenerated (atTop (α := α))] [MeasurableNeg α] [μ.IsNegInvariant]
    (hf : LocallyIntegrable f μ) (hsymm : norm ∘ f =ᵐ[μ] norm ∘ f ∘ Neg.neg) (ho : f =O[atTop] g)
    (hg : IntegrableAtFilter g atTop μ) : Integrable f μ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : α → F} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [SecondCountableTopology α] [LinearOrderedAddCommGroup α] [CompactIccSpace α] [Filter.atTop.IsMeasurablyGenerated] [MeasurableNeg α] [μ.IsNegInvariant] (hf : MeasureTheory.LocallyIntegrable f μ) (hsymm : norm ∘ f =ᵐ[μ] norm ∘ f ∘ Neg.neg) (ho : f =O[Filter.atTop] g) (hg : MeasureTheory.IntegrableAtFilter g Filter.atTop μ)
Goal: MeasureTheory.Integrable f μ
-- FirstOrder.Language.Theory.Model.isSatisfiable
theorem Model.isSatisfiable (M : Type w) [Nonempty M] [L.Structure M] [M ⊨ T] :
    T.IsSatisfiable 
Arguments: {L : FirstOrder.Language} {T : L.Theory} (M : Type w) [Nonempty M] [L.Structure M] [M ⊨ T]
Goal: T.IsSatisfiable
-- FirstOrder.Language.Theory.IsSatisfiable.mono
theorem IsSatisfiable.mono (h : T'.IsSatisfiable) (hs : T ⊆ T') : T.IsSatisfiable 
Arguments: {L : FirstOrder.Language} {T T' : L.Theory} (h : T'.IsSatisfiable) (hs : T ⊆ T')
Goal: T.IsSatisfiable
-- FirstOrder.Language.Theory.isSatisfiable_empty
theorem isSatisfiable_empty (L : Language.{u, v}) : IsSatisfiable (∅ : L.Theory) 
Arguments: (L : FirstOrder.Language)
Goal: ∅.IsSatisfiable
-- FirstOrder.Language.Theory.isSatisfiable_of_isSatisfiable_onTheory
theorem isSatisfiable_of_isSatisfiable_onTheory {L' : Language.{w, w'}} (φ : L →ᴸ L')
    (h : (φ.onTheory T).IsSatisfiable) : T.IsSatisfiable 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {L' : FirstOrder.Language} (φ : L →ᴸ L') (h : (φ.onTheory T).IsSatisfiable)
Goal: T.IsSatisfiable
-- FirstOrder.Language.Theory.isSatisfiable_onTheory_iff
theorem isSatisfiable_onTheory_iff {L' : Language.{w, w'}} {φ : L →ᴸ L'} (h : φ.Injective) :
    (φ.onTheory T).IsSatisfiable ↔ T.IsSatisfiable 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {L' : FirstOrder.Language} {φ : L →ᴸ L'} (h : φ.Injective)
Goal: (φ.onTheory T).IsSatisfiable ↔ T.IsSatisfiable
-- FirstOrder.Language.Theory.IsSatisfiable.isFinitelySatisfiable
theorem IsSatisfiable.isFinitelySatisfiable (h : T.IsSatisfiable) : T.IsFinitelySatisfiable 
Arguments: {L : FirstOrder.Language} {T : L.Theory} (h : T.IsSatisfiable)
Goal: T.IsFinitelySatisfiable
-- FirstOrder.Language.Theory.isSatisfiable_iff_isFinitelySatisfiable
theorem isSatisfiable_iff_isFinitelySatisfiable {T : L.Theory} :
    T.IsSatisfiable ↔ T.IsFinitelySatisfiable 
Arguments: {L : FirstOrder.Language} {T : L.Theory}
Goal: T.IsSatisfiable ↔ T.IsFinitelySatisfiable
-- FirstOrder.Language.Theory.isSatisfiable_directed_union_iff
theorem isSatisfiable_directed_union_iff {ι : Type*} [Nonempty ι] {T : ι → L.Theory}
    (h : Directed (· ⊆ ·) T) : Theory.IsSatisfiable (⋃ i, T i) ↔ ∀ i, (T i).IsSatisfiable 
Arguments: {L : FirstOrder.Language} {ι : Type u_1} [Nonempty ι] {T : ι → L.Theory} (h : Directed (fun x x_1 => x ⊆ x_1) T)
Goal: FirstOrder.Language.Theory.IsSatisfiable (⋃ i, T i) ↔ ∀ (i : ι), (T i).IsSatisfiable
-- FirstOrder.Language.Theory.isSatisfiable_union_distinctConstantsTheory_of_card_le
theorem isSatisfiable_union_distinctConstantsTheory_of_card_le (T : L.Theory) (s : Set α)
    (M : Type w') [Nonempty M] [L.Structure M] [M ⊨ T]
    (h : Cardinal.lift.{w'} #s ≤ Cardinal.lift.{w} #M) :
    ((L.lhomWithConstants α).onTheory T ∪ L.distinctConstantsTheory s).IsSatisfiable 
Arguments: {L : FirstOrder.Language} {α : Type w} (T : L.Theory) (s : Set α) (M : Type w') [Nonempty M] [L.Structure M] [M ⊨ T] (h : Cardinal.lift.{w', w} #↑s ≤ Cardinal.lift.{w, w'} #M)
Goal: ((L.lhomWithConstants α).onTheory T ∪ L.distinctConstantsTheory s).IsSatisfiable
-- FirstOrder.Language.Theory.isSatisfiable_union_distinctConstantsTheory_of_infinite
theorem isSatisfiable_union_distinctConstantsTheory_of_infinite (T : L.Theory) (s : Set α)
    (M : Type w') [L.Structure M] [M ⊨ T] [Infinite M] :
    ((L.lhomWithConstants α).onTheory T ∪ L.distinctConstantsTheory s).IsSatisfiable 
Arguments: {L : FirstOrder.Language} {α : Type w} (T : L.Theory) (s : Set α) (M : Type w') [L.Structure M] [M ⊨ T] [Infinite M]
Goal: ((L.lhomWithConstants α).onTheory T ∪ L.distinctConstantsTheory s).IsSatisfiable
-- FirstOrder.Language.Theory.exists_large_model_of_infinite_model
theorem exists_large_model_of_infinite_model (T : L.Theory) (κ : Cardinal.{w}) (M : Type w')
    [L.Structure M] [M ⊨ T] [Infinite M] :
    ∃ N : ModelType.{_, _, max u v w} T, Cardinal.lift.{max u v w} κ ≤ #N 
Arguments: {L : FirstOrder.Language} (T : L.Theory) (κ : Cardinal.{w}) (M : Type w') [L.Structure M] [M ⊨ T] [Infinite M]
Goal: ∃ N, Cardinal.lift.{max u v w, w} κ ≤ #↑N
-- FirstOrder.Language.Theory.isSatisfiable_iUnion_iff_isSatisfiable_iUnion_finset
theorem isSatisfiable_iUnion_iff_isSatisfiable_iUnion_finset {ι : Type*} (T : ι → L.Theory) :
    IsSatisfiable (⋃ i, T i) ↔ ∀ s : Finset ι, IsSatisfiable (⋃ i ∈ s, T i) 
Arguments: {L : FirstOrder.Language} {ι : Type u_1} (T : ι → L.Theory)
Goal: FirstOrder.Language.Theory.IsSatisfiable (⋃ i, T i) ↔
  ∀ (s : Finset ι), FirstOrder.Language.Theory.IsSatisfiable (⋃ i ∈ s, T i)
-- FirstOrder.Language.exists_elementaryEmbedding_card_eq_of_le
theorem exists_elementaryEmbedding_card_eq_of_le (M : Type w') [L.Structure M] [Nonempty M]
    (κ : Cardinal.{w}) (h1 : ℵ₀ ≤ κ) (h2 : lift.{w} L.card ≤ Cardinal.lift.{max u v} κ)
    (h3 : lift.{w'} κ ≤ Cardinal.lift.{w} #M) :
    ∃ N : Bundled L.Structure, Nonempty (N ↪ₑ[L] M) ∧ #N = κ 
Arguments: (L : FirstOrder.Language) (M : Type w') [L.Structure M] [Nonempty M] (κ : Cardinal.{w}) (h1 : ℵ₀ ≤ κ) (h2 : Cardinal.lift.{w, max u v} L.card ≤ Cardinal.lift.{max u v, w} κ) (h3 : Cardinal.lift.{w', w} κ ≤ Cardinal.lift.{w, w'} #M)
Goal: ∃ N, Nonempty (↑N ↪ₑ[L] M) ∧ #↑N = κ
-- FirstOrder.Language.exists_elementaryEmbedding_card_eq_of_ge
theorem exists_elementaryEmbedding_card_eq_of_ge (M : Type w') [L.Structure M] [iM : Infinite M]
    (κ : Cardinal.{w}) (h1 : Cardinal.lift.{w} L.card ≤ Cardinal.lift.{max u v} κ)
    (h2 : Cardinal.lift.{w} #M ≤ Cardinal.lift.{w'} κ) :
    ∃ N : Bundled L.Structure, Nonempty (M ↪ₑ[L] N) ∧ #N = κ 
Arguments: (L : FirstOrder.Language) (M : Type w') [L.Structure M] [iM : Infinite M] (κ : Cardinal.{w}) (h1 : Cardinal.lift.{w, max u v} L.card ≤ Cardinal.lift.{max u v, w} κ) (h2 : Cardinal.lift.{w, w'} #M ≤ Cardinal.lift.{w', w} κ)
Goal: ∃ N, Nonempty (M ↪ₑ[L] ↑N) ∧ #↑N = κ
-- FirstOrder.Language.exists_elementaryEmbedding_card_eq
theorem exists_elementaryEmbedding_card_eq (M : Type w') [L.Structure M] [iM : Infinite M]
    (κ : Cardinal.{w}) (h1 : ℵ₀ ≤ κ) (h2 : lift.{w} L.card ≤ Cardinal.lift.{max u v} κ) :
    ∃ N : Bundled L.Structure, (Nonempty (N ↪ₑ[L] M) ∨ Nonempty (M ↪ₑ[L] N)) ∧ #N = κ 
Arguments: (L : FirstOrder.Language) (M : Type w') [L.Structure M] [iM : Infinite M] (κ : Cardinal.{w}) (h1 : ℵ₀ ≤ κ) (h2 : Cardinal.lift.{w, max u v} L.card ≤ Cardinal.lift.{max u v, w} κ)
Goal: ∃ N, (Nonempty (↑N ↪ₑ[L] M) ∨ Nonempty (M ↪ₑ[L] ↑N)) ∧ #↑N = κ
-- FirstOrder.Language.exists_elementarilyEquivalent_card_eq
theorem exists_elementarilyEquivalent_card_eq (M : Type w') [L.Structure M] [Infinite M]
    (κ : Cardinal.{w}) (h1 : ℵ₀ ≤ κ) (h2 : lift.{w} L.card ≤ Cardinal.lift.{max u v} κ) :
    ∃ N : CategoryTheory.Bundled L.Structure, (M ≅[L] N) ∧ #N = κ 
Arguments: (L : FirstOrder.Language) (M : Type w') [L.Structure M] [Infinite M] (κ : Cardinal.{w}) (h1 : ℵ₀ ≤ κ) (h2 : Cardinal.lift.{w, max u v} L.card ≤ Cardinal.lift.{max u v, w} κ)
Goal: ∃ N, (M ≅[L] ↑N) ∧ #↑N = κ
-- FirstOrder.Language.Theory.exists_model_card_eq
theorem exists_model_card_eq (h : ∃ M : ModelType.{u, v, max u v} T, Infinite M) (κ : Cardinal.{w})
    (h1 : ℵ₀ ≤ κ) (h2 : Cardinal.lift.{w} L.card ≤ Cardinal.lift.{max u v} κ) :
    ∃ N : ModelType.{u, v, w} T, #N = κ 
Arguments: {L : FirstOrder.Language} {T : L.Theory} (h : ∃ M, Infinite ↑M) (κ : Cardinal.{w}) (h1 : ℵ₀ ≤ κ) (h2 : Cardinal.lift.{w, max u v} L.card ≤ Cardinal.lift.{max u v, w} κ)
Goal: ∃ N, #↑N = κ
-- FirstOrder.Language.Theory.models_formula_iff
theorem models_formula_iff {φ : L.Formula α} :
    T ⊨ᵇ φ ↔ ∀ (M : ModelType.{u, v, max u v} T) (v : α → M), φ.Realize v 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} {φ : L.Formula α}
Goal: T ⊨ᵇ φ ↔ ∀ (M : T.ModelType) (v : α → ↑M), φ.Realize v
-- FirstOrder.Language.Theory.models_sentence_iff
theorem models_sentence_iff {φ : L.Sentence} : T ⊨ᵇ φ ↔ ∀ M : ModelType.{u, v, max u v} T, M ⊨ φ 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {φ : L.Sentence}
Goal: T ⊨ᵇ φ ↔ ∀ (M : T.ModelType), ↑M ⊨ φ
-- FirstOrder.Language.Theory.models_sentence_of_mem
theorem models_sentence_of_mem {φ : L.Sentence} (h : φ ∈ T) : T ⊨ᵇ φ 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {φ : L.Sentence} (h : φ ∈ T)
Goal: T ⊨ᵇ φ
-- FirstOrder.Language.Theory.models_iff_not_satisfiable
theorem models_iff_not_satisfiable (φ : L.Sentence) : T ⊨ᵇ φ ↔ ¬IsSatisfiable (T ∪ {φ.not}) 
Arguments: {L : FirstOrder.Language} {T : L.Theory} (φ : L.Sentence)
Goal: T ⊨ᵇ φ ↔ ¬(T ∪ {FirstOrder.Language.Formula.not φ}).IsSatisfiable
-- FirstOrder.Language.Theory.ModelsBoundedFormula.realize_sentence
theorem ModelsBoundedFormula.realize_sentence {φ : L.Sentence} (h : T ⊨ᵇ φ) (M : Type*)
    [L.Structure M] [M ⊨ T] [Nonempty M] : M ⊨ φ 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {φ : L.Sentence} (h : T ⊨ᵇ φ) (M : Type u_1) [L.Structure M] [M ⊨ T] [Nonempty M]
Goal: M ⊨ φ
-- FirstOrder.Language.Theory.models_of_models_theory
theorem models_of_models_theory {T' : L.Theory}
    (h : ∀ φ : L.Sentence, φ ∈ T' → T ⊨ᵇ φ)
    {φ : L.Formula α} (hφ : T' ⊨ᵇ φ) : T ⊨ᵇ φ 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} {T' : L.Theory} (h : ∀ φ ∈ T', T ⊨ᵇ φ) {φ : L.Formula α} (hφ : T' ⊨ᵇ φ)
Goal: T ⊨ᵇ φ
-- FirstOrder.Language.Theory.models_iff_finset_models
theorem models_iff_finset_models {φ : L.Sentence} :
    T ⊨ᵇ φ ↔ ∃ T0 : Finset L.Sentence, (T0 : L.Theory) ⊆ T ∧ (T0 : L.Theory) ⊨ᵇ φ 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {φ : L.Sentence}
Goal: T ⊨ᵇ φ ↔ ∃ T0, ↑T0 ⊆ T ∧ ↑T0 ⊨ᵇ φ
-- FirstOrder.Language.Theory.IsComplete.models_not_iff
theorem models_not_iff (h : T.IsComplete) (φ : L.Sentence) : T ⊨ᵇ φ.not ↔ ¬T ⊨ᵇ φ 
Arguments: {L : FirstOrder.Language} {T : L.Theory} (h : T.IsComplete) (φ : L.Sentence)
Goal: T ⊨ᵇ FirstOrder.Language.Formula.not φ ↔ ¬T ⊨ᵇ φ
-- FirstOrder.Language.Theory.IsComplete.realize_sentence_iff
theorem realize_sentence_iff (h : T.IsComplete) (φ : L.Sentence) (M : Type*) [L.Structure M]
    [M ⊨ T] [Nonempty M] : M ⊨ φ ↔ T ⊨ᵇ φ 
Arguments: {L : FirstOrder.Language} {T : L.Theory} (h : T.IsComplete) (φ : L.Sentence) (M : Type u_1) [L.Structure M] [M ⊨ T] [Nonempty M]
Goal: M ⊨ φ ↔ T ⊨ᵇ φ
-- FirstOrder.Language.Theory.IsMaximal.isComplete
theorem IsMaximal.isComplete (h : T.IsMaximal) : T.IsComplete 
Arguments: {L : FirstOrder.Language} {T : L.Theory} (h : T.IsMaximal)
Goal: T.IsComplete
-- FirstOrder.Language.Theory.IsMaximal.mem_or_not_mem
theorem IsMaximal.mem_or_not_mem (h : T.IsMaximal) (φ : L.Sentence) : φ ∈ T ∨ φ.not ∈ T 
Arguments: {L : FirstOrder.Language} {T : L.Theory} (h : T.IsMaximal) (φ : L.Sentence)
Goal: φ ∈ T ∨ FirstOrder.Language.Formula.not φ ∈ T
-- FirstOrder.Language.Theory.IsMaximal.mem_of_models
theorem IsMaximal.mem_of_models (h : T.IsMaximal) {φ : L.Sentence} (hφ : T ⊨ᵇ φ) : φ ∈ T 
Arguments: {L : FirstOrder.Language} {T : L.Theory} (h : T.IsMaximal) {φ : L.Sentence} (hφ : T ⊨ᵇ φ)
Goal: φ ∈ T
-- FirstOrder.Language.Theory.IsMaximal.mem_iff_models
theorem IsMaximal.mem_iff_models (h : T.IsMaximal) (φ : L.Sentence) : φ ∈ T ↔ T ⊨ᵇ φ 
Arguments: {L : FirstOrder.Language} {T : L.Theory} (h : T.IsMaximal) (φ : L.Sentence)
Goal: φ ∈ T ↔ T ⊨ᵇ φ
-- FirstOrder.Language.Theory.SemanticallyEquivalent.refl
@[refl]
theorem SemanticallyEquivalent.refl (φ : L.BoundedFormula α n) : T.SemanticallyEquivalent φ φ 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} {n : ℕ} (φ : L.BoundedFormula α n)
Goal: T.SemanticallyEquivalent φ φ
-- FirstOrder.Language.Theory.SemanticallyEquivalent.symm
@[symm]
theorem SemanticallyEquivalent.symm {φ ψ : L.BoundedFormula α n}
    (h : T.SemanticallyEquivalent φ ψ) : T.SemanticallyEquivalent ψ φ 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} {n : ℕ} {φ ψ : L.BoundedFormula α n} (h : T.SemanticallyEquivalent φ ψ)
Goal: T.SemanticallyEquivalent ψ φ
-- FirstOrder.Language.Theory.SemanticallyEquivalent.trans
@[trans]
theorem SemanticallyEquivalent.trans {φ ψ θ : L.BoundedFormula α n}
    (h1 : T.SemanticallyEquivalent φ ψ) (h2 : T.SemanticallyEquivalent ψ θ) :
    T.SemanticallyEquivalent φ θ 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} {n : ℕ} {φ ψ θ : L.BoundedFormula α n} (h1 : T.SemanticallyEquivalent φ ψ) (h2 : T.SemanticallyEquivalent ψ θ)
Goal: T.SemanticallyEquivalent φ θ
-- FirstOrder.Language.Theory.SemanticallyEquivalent.realize_bd_iff
theorem SemanticallyEquivalent.realize_bd_iff {φ ψ : L.BoundedFormula α n} {M : Type max u v}
    [Nonempty M] [L.Structure M] [T.Model M] (h : T.SemanticallyEquivalent φ ψ)
    {v : α → M} {xs : Fin n → M} : φ.Realize v xs ↔ ψ.Realize v xs 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} {n : ℕ} {φ ψ : L.BoundedFormula α n} {M : Type (max u v)} [Nonempty M] [L.Structure M] [M ⊨ T] (h : T.SemanticallyEquivalent φ ψ) {v : α → M} {xs : Fin n → M}
Goal: φ.Realize v xs ↔ ψ.Realize v xs
-- FirstOrder.Language.Theory.SemanticallyEquivalent.realize_iff
theorem SemanticallyEquivalent.realize_iff {φ ψ : L.Formula α} {M : Type max u v} [Nonempty M]
    [L.Structure M] (_hM : T.Model M) (h : T.SemanticallyEquivalent φ ψ) {v : α → M} :
    φ.Realize v ↔ ψ.Realize v 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} {φ ψ : L.Formula α} {M : Type (max u v)} [Nonempty M] [L.Structure M] (_hM : M ⊨ T) (h : T.SemanticallyEquivalent φ ψ) {v : α → M}
Goal: φ.Realize v ↔ ψ.Realize v
-- FirstOrder.Language.Theory.SemanticallyEquivalent.all
protected theorem SemanticallyEquivalent.all {φ ψ : L.BoundedFormula α (n + 1)}
    (h : T.SemanticallyEquivalent φ ψ) : T.SemanticallyEquivalent φ.all ψ.all 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} {n : ℕ} {φ ψ : L.BoundedFormula α (n + 1)} (h : T.SemanticallyEquivalent φ ψ)
Goal: T.SemanticallyEquivalent (∀'φ) (∀'ψ)
-- FirstOrder.Language.Theory.SemanticallyEquivalent.ex
protected theorem SemanticallyEquivalent.ex {φ ψ : L.BoundedFormula α (n + 1)}
    (h : T.SemanticallyEquivalent φ ψ) : T.SemanticallyEquivalent φ.ex ψ.ex 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} {n : ℕ} {φ ψ : L.BoundedFormula α (n + 1)} (h : T.SemanticallyEquivalent φ ψ)
Goal: T.SemanticallyEquivalent (∃'φ) (∃'ψ)
-- FirstOrder.Language.Theory.SemanticallyEquivalent.not
protected theorem SemanticallyEquivalent.not {φ ψ : L.BoundedFormula α n}
    (h : T.SemanticallyEquivalent φ ψ) : T.SemanticallyEquivalent φ.not ψ.not 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} {n : ℕ} {φ ψ : L.BoundedFormula α n} (h : T.SemanticallyEquivalent φ ψ)
Goal: T.SemanticallyEquivalent (∼φ) ∼ψ
-- FirstOrder.Language.Theory.SemanticallyEquivalent.imp
protected theorem SemanticallyEquivalent.imp {φ ψ φ' ψ' : L.BoundedFormula α n}
    (h : T.SemanticallyEquivalent φ ψ) (h' : T.SemanticallyEquivalent φ' ψ') :
    T.SemanticallyEquivalent (φ.imp φ') (ψ.imp ψ') 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} {n : ℕ} {φ ψ φ' ψ' : L.BoundedFormula α n} (h : T.SemanticallyEquivalent φ ψ) (h' : T.SemanticallyEquivalent φ' ψ')
Goal: T.SemanticallyEquivalent (φ ⟹ φ') (ψ ⟹ ψ')
-- FirstOrder.Language.completeTheory.isSatisfiable
theorem isSatisfiable [Nonempty M] : (L.completeTheory M).IsSatisfiable 
Arguments: (L : FirstOrder.Language) (M : Type w) [L.Structure M] [Nonempty M]
Goal: (L.completeTheory M).IsSatisfiable
-- FirstOrder.Language.completeTheory.mem_or_not_mem
theorem mem_or_not_mem (φ : L.Sentence) : φ ∈ L.completeTheory M ∨ φ.not ∈ L.completeTheory M 
Arguments: (L : FirstOrder.Language) (M : Type w) [L.Structure M] (φ : L.Sentence)
Goal: φ ∈ L.completeTheory M ∨ FirstOrder.Language.Formula.not φ ∈ L.completeTheory M
-- FirstOrder.Language.completeTheory.isMaximal
theorem isMaximal [Nonempty M] : (L.completeTheory M).IsMaximal 
Arguments: (L : FirstOrder.Language) (M : Type w) [L.Structure M] [Nonempty M]
Goal: (L.completeTheory M).IsMaximal
-- FirstOrder.Language.completeTheory.isComplete
theorem isComplete [Nonempty M] : (L.completeTheory M).IsComplete 
Arguments: (L : FirstOrder.Language) (M : Type w) [L.Structure M] [Nonempty M]
Goal: (L.completeTheory M).IsComplete
-- FirstOrder.Language.BoundedFormula.semanticallyEquivalent_not_not
theorem semanticallyEquivalent_not_not : T.SemanticallyEquivalent φ φ.not.not 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} {n : ℕ} (φ : L.BoundedFormula α n)
Goal: T.SemanticallyEquivalent φ ∼∼φ
-- FirstOrder.Language.BoundedFormula.imp_semanticallyEquivalent_not_sup
theorem imp_semanticallyEquivalent_not_sup : T.SemanticallyEquivalent (φ.imp ψ) (φ.not ⊔ ψ) 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} {n : ℕ} (φ ψ : L.BoundedFormula α n)
Goal: T.SemanticallyEquivalent (φ ⟹ ψ) (∼φ ⊔ ψ)
-- FirstOrder.Language.BoundedFormula.sup_semanticallyEquivalent_not_inf_not
theorem sup_semanticallyEquivalent_not_inf_not :
    T.SemanticallyEquivalent (φ ⊔ ψ) (φ.not ⊓ ψ.not).not 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} {n : ℕ} (φ ψ : L.BoundedFormula α n)
Goal: T.SemanticallyEquivalent (φ ⊔ ψ) ∼(∼φ ⊓ ∼ψ)
-- FirstOrder.Language.BoundedFormula.inf_semanticallyEquivalent_not_sup_not
theorem inf_semanticallyEquivalent_not_sup_not :
    T.SemanticallyEquivalent (φ ⊓ ψ) (φ.not ⊔ ψ.not).not 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {α : Type w} {n : ℕ} (φ ψ : L.BoundedFormula α n)
Goal: T.SemanticallyEquivalent (φ ⊓ ψ) ∼(∼φ ⊔ ∼ψ)