ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- FirstOrder.Language.ElementarySubstructure.coe_top
@[simp]
theorem coe_top : ((⊤ : L.ElementarySubstructure M) : Set M) = Set.univ 
Arguments: {L : FirstOrder.Language} {M : Type u_1} [L.Structure M]
Goal: ↑⊤ = Set.univ
-- FirstOrder.Language.ElementarySubstructure.realize_sentence
@[simp]
theorem realize_sentence (S : L.ElementarySubstructure M) (φ : L.Sentence) : S ⊨ φ ↔ M ⊨ φ 
Arguments: {L : FirstOrder.Language} {M : Type u_1} [L.Structure M] (S : L.ElementarySubstructure M) (φ : L.Sentence)
Goal: ↥S ⊨ φ ↔ M ⊨ φ
-- FirstOrder.Language.ElementarySubstructure.theory_model_iff
@[simp]
theorem theory_model_iff (S : L.ElementarySubstructure M) (T : L.Theory) : S ⊨ T ↔ M ⊨ T 
Arguments: {L : FirstOrder.Language} {M : Type u_1} [L.Structure M] (S : L.ElementarySubstructure M) (T : L.Theory)
Goal: ↥S ⊨ T ↔ M ⊨ T
-- FirstOrder.Language.ElementarySubstructure.elementarilyEquivalent
theorem elementarilyEquivalent (S : L.ElementarySubstructure M) : S ≅[L] M 
Arguments: {L : FirstOrder.Language} {M : Type u_1} [L.Structure M] (S : L.ElementarySubstructure M)
Goal: ↥S ≅[L] M
-- FirstOrder.Language.Substructure.isElementary_of_exists
theorem isElementary_of_exists (S : L.Substructure M)
    (htv :
      ∀ (n : ℕ) (φ : L.BoundedFormula Empty (n + 1)) (x : Fin n → S) (a : M),
        φ.Realize default (Fin.snoc ((↑) ∘ x) a : _ → M) →
          ∃ b : S, φ.Realize default (Fin.snoc ((↑) ∘ x) b : _ → M)) :
    S.IsElementary 
Arguments: {L : FirstOrder.Language} {M : Type u_1} [L.Structure M] (S : L.Substructure M) (htv :
  ∀ (n : ℕ) (φ : L.BoundedFormula Empty (n + 1)) (x : Fin n → ↥S) (a : M),
    φ.Realize default (Fin.snoc (Subtype.val ∘ x) a) → ∃ b, φ.Realize default (Fin.snoc (Subtype.val ∘ x) ↑b))
Goal: S.IsElementary
-- FirstOrder.Language.Substructure.toElementarySubstructure_toSubstructure
@[simps]
def toElementarySubstructure (S : L.Substructure M)
    (htv :
      ∀ (n : ℕ) (φ : L.BoundedFormula Empty (n + 1)) (x : Fin n → S) (a : M),
        φ.Realize default (Fin.snoc ((↑) ∘ x) a : _ → M) →
          ∃ b : S, φ.Realize default (Fin.snoc ((↑) ∘ x) b : _ → M)) :
    L.ElementarySubstructure M 
Arguments: {L : FirstOrder.Language} {M : Type u_1} [L.Structure M] (S : L.Substructure M) (htv :
  ∀ (n : ℕ) (φ : L.BoundedFormula Empty (n + 1)) (x : Fin n → ↥S) (a : M),
    φ.Realize default (Fin.snoc (Subtype.val ∘ x) a) → ∃ b, φ.Realize default (Fin.snoc (Subtype.val ∘ x) ↑b))
Goal: ↑(S.toElementarySubstructure htv) = S
-- NumberField.coe_discr
theorem coe_discr : (discr K : ℚ) = Algebra.discr ℚ (integralBasis K) 
Arguments: (K : Type u_1) [Field K] [NumberField K]
Goal: ↑(NumberField.discr K) = Algebra.discr ℚ ⇑(NumberField.integralBasis K)
-- NumberField.discr_ne_zero
theorem discr_ne_zero : discr K ≠ 0 
Arguments: (K : Type u_1) [Field K] [NumberField K]
Goal: NumberField.discr K ≠ 0
-- NumberField.discr_eq_discr
theorem discr_eq_discr {ι : Type*} [Fintype ι] [DecidableEq ι] (b : Basis ι ℤ (𝓞 K)) :
    Algebra.discr ℤ b = discr K 
Arguments: (K : Type u_1) [Field K] [NumberField K] {ι : Type u_2} [Fintype ι] [DecidableEq ι] (b : Basis ι ℤ (NumberField.RingOfIntegers K))
Goal: Algebra.discr ℤ ⇑b = NumberField.discr K
-- NumberField.discr_eq_discr_of_algEquiv
theorem discr_eq_discr_of_algEquiv {L : Type*} [Field L] [NumberField L] (f : K ≃ₐ[ℚ] L) :
    discr K = discr L 
Arguments: (K : Type u_1) [Field K] [NumberField K] {L : Type u_2} [Field L] [NumberField L] (f : K ≃ₐ[ℚ] L)
Goal: NumberField.discr K = NumberField.discr L
-- NumberField.mixedEmbedding.volume_fundamentalDomain_latticeBasis
theorem _root_.NumberField.mixedEmbedding.volume_fundamentalDomain_latticeBasis :
    volume (fundamentalDomain (latticeBasis K)) =
      (2 : ℝ≥0∞)⁻¹ ^ NrComplexPlaces K * sqrt ‖discr K‖₊ 
Arguments: (K : Type u_1) [Field K] [NumberField K]
Goal: MeasureTheory.volume (Zspan.fundamentalDomain (NumberField.mixedEmbedding.latticeBasis K)) =
  2⁻¹ ^ NumberField.InfinitePlace.NrComplexPlaces K * ↑(NNReal.sqrt ‖NumberField.discr K‖₊)
-- NumberField.exists_ne_zero_mem_ideal_of_norm_le_mul_sqrt_discr
theorem exists_ne_zero_mem_ideal_of_norm_le_mul_sqrt_discr (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ) :
    ∃ a ∈ (I : FractionalIdeal (𝓞 K)⁰ K), a ≠ 0 ∧
      |Algebra.norm ℚ (a : K)| ≤ FractionalIdeal.absNorm I.1 * (4 / π) ^ NrComplexPlaces K *
        (finrank ℚ K).factorial / (finrank ℚ K) ^ (finrank ℚ K) * Real.sqrt |discr K| 
Arguments: (K : Type u_1) [Field K] [NumberField K] (I : (FractionalIdeal (nonZeroDivisors (NumberField.RingOfIntegers K)) K)ˣ)
Goal: ∃ a ∈ ↑I,
  a ≠ 0 ∧
    ↑|(Algebra.norm ℚ) a| ≤
      ↑(FractionalIdeal.absNorm ↑I) * (4 / Real.pi) ^ NumberField.InfinitePlace.NrComplexPlaces K *
            ↑(FiniteDimensional.finrank ℚ K).factorial /
          ↑(FiniteDimensional.finrank ℚ K) ^ FiniteDimensional.finrank ℚ K *
        √|↑(NumberField.discr K)|
-- NumberField.exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr
theorem exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr :
    ∃ (a : 𝓞 K), a ≠ 0 ∧
      |Algebra.norm ℚ (a : K)| ≤ (4 / π) ^ NrComplexPlaces K *
        (finrank ℚ K).factorial / (finrank ℚ K) ^ (finrank ℚ K) * Real.sqrt |discr K| 
Arguments: (K : Type u_1) [Field K] [NumberField K]
Goal: ∃ a,
  a ≠ 0 ∧
    ↑|(Algebra.norm ℚ) ↑a| ≤
      (4 / Real.pi) ^ NumberField.InfinitePlace.NrComplexPlaces K * ↑(FiniteDimensional.finrank ℚ K).factorial /
          ↑(FiniteDimensional.finrank ℚ K) ^ FiniteDimensional.finrank ℚ K *
        √|↑(NumberField.discr K)|
-- NumberField.abs_discr_ge
theorem abs_discr_ge (h : 1 < finrank ℚ K) :
    (4 / 9 : ℝ) * (3 * π / 4) ^ finrank ℚ K ≤ |discr K| 
Arguments: {K : Type u_1} [Field K] [NumberField K] (h : 1 < FiniteDimensional.finrank ℚ K)
Goal: 4 / 9 * (3 * Real.pi / 4) ^ FiniteDimensional.finrank ℚ K ≤ ↑|NumberField.discr K|
-- NumberField.abs_discr_gt_two
theorem abs_discr_gt_two (h : 1 < finrank ℚ K) : 2 < |discr K| 
Arguments: {K : Type u_1} [Field K] [NumberField K] (h : 1 < FiniteDimensional.finrank ℚ K)
Goal: 2 < |NumberField.discr K|
-- NumberField.hermiteTheorem.finite_of_finite_generating_set
theorem finite_of_finite_generating_set {p : IntermediateField ℚ A → Prop}
    (S : Set {F : IntermediateField ℚ A // p F}) {T : Set A}
    (hT : T.Finite) (h : ∀ F ∈ S, ∃ x ∈ T, F = ℚ⟮x⟯) :
    S.Finite 
Arguments: (A : Type u_2) [Field A] [CharZero A] {p : IntermediateField ℚ A → Prop} (S : Set { F // p F }) {T : Set A} (hT : T.Finite) (h : ∀ F ∈ S, ∃ x ∈ T, ↑F = ℚ⟮x⟯)
Goal: S.Finite
-- NumberField.hermiteTheorem.rank_le_rankOfDiscrBdd
theorem rank_le_rankOfDiscrBdd :
    finrank ℚ K ≤ rankOfDiscrBdd N 
Arguments: {K : Type u_1} [Field K] [NumberField K] {N : ℕ} (hK : |NumberField.discr K| ≤ ↑N)
Goal: FiniteDimensional.finrank ℚ K ≤ NumberField.hermiteTheorem.rankOfDiscrBdd N
-- NumberField.hermiteTheorem.minkowskiBound_lt_boundOfDiscBdd
theorem minkowskiBound_lt_boundOfDiscBdd : minkowskiBound K ↑1 < boundOfDiscBdd N 
Arguments: {K : Type u_1} [Field K] [NumberField K] {N : ℕ} (hK : |NumberField.discr K| ≤ ↑N)
Goal: NumberField.mixedEmbedding.minkowskiBound K 1 < ↑(NumberField.hermiteTheorem.boundOfDiscBdd N)
-- NumberField.hermiteTheorem.natDegree_le_rankOfDiscrBdd
theorem natDegree_le_rankOfDiscrBdd (a : 𝓞 K) (h : ℚ⟮(a : K)⟯ = ⊤) :
    natDegree (minpoly ℤ (a : K)) ≤ rankOfDiscrBdd N 
Arguments: {K : Type u_1} [Field K] [NumberField K] {N : ℕ} (hK : |NumberField.discr K| ≤ ↑N) (a : NumberField.RingOfIntegers K) (h : ℚ⟮↑a⟯ = ⊤)
Goal: (minpoly ℤ ↑a).natDegree ≤ NumberField.hermiteTheorem.rankOfDiscrBdd N
-- NumberField.hermiteTheorem.finite_of_discr_bdd_of_isReal
theorem finite_of_discr_bdd_of_isReal :
    {K : { F : IntermediateField ℚ A // FiniteDimensional ℚ F} |
      haveI :  NumberField K := @NumberField.mk _ _ inferInstance K.prop
      {w : InfinitePlace K | IsReal w}.Nonempty ∧ |discr K| ≤ N }.Finite 
Arguments: (A : Type u_2) [Field A] [CharZero A] (N : ℕ)
Goal: {K | {w | w.IsReal}.Nonempty ∧ |NumberField.discr ↥↑K| ≤ ↑N}.Finite
-- NumberField.hermiteTheorem.finite_of_discr_bdd_of_isComplex
theorem finite_of_discr_bdd_of_isComplex :
    {K : { F : IntermediateField ℚ A // FiniteDimensional ℚ F} |
      haveI :  NumberField K := @NumberField.mk _ _ inferInstance K.prop
      {w : InfinitePlace K | IsComplex w}.Nonempty ∧ |discr K| ≤ N }.Finite 
Arguments: (A : Type u_2) [Field A] [CharZero A] (N : ℕ)
Goal: {K | {w | w.IsComplex}.Nonempty ∧ |NumberField.discr ↥↑K| ≤ ↑N}.Finite
-- NumberField.finite_of_discr_bdd
theorem _root_.NumberField.finite_of_discr_bdd :
    {K : { F : IntermediateField ℚ A // FiniteDimensional ℚ F} |
      haveI :  NumberField K := @NumberField.mk _ _ inferInstance K.prop
      |discr K| ≤ N }.Finite 
Arguments: (A : Type u_2) [Field A] [CharZero A] (N : ℕ)
Goal: {K | |NumberField.discr ↥↑K| ≤ ↑N}.Finite
-- Rat.numberField_discr
@[simp]
theorem numberField_discr : discr ℚ = 1 
Goal: NumberField.discr ℚ = 1
-- Algebra.discr_eq_discr_of_toMatrix_coeff_isIntegral
theorem Algebra.discr_eq_discr_of_toMatrix_coeff_isIntegral [NumberField K]
    {b : Basis ι ℚ K} {b' : Basis ι' ℚ K} (h : ∀ i j, IsIntegral ℤ (b.toMatrix b' i j))
    (h' : ∀ i j, IsIntegral ℤ (b'.toMatrix b i j)) : discr ℚ b = discr ℚ b' 
Arguments: {ι : Type u_2} {ι' : Type u_3} (K : Type u_1) [Field K] [DecidableEq ι] [DecidableEq ι'] [Fintype ι] [Fintype ι'] [NumberField K] {b : Basis ι ℚ K} {b' : Basis ι' ℚ K} (h : ∀ (i : ι) (j : ι'), IsIntegral ℤ (b.toMatrix (⇑b') i j)) (h' : ∀ (i : ι') (j : ι), IsIntegral ℤ (b'.toMatrix (⇑b) i j))
Goal: Algebra.discr ℚ ⇑b = Algebra.discr ℚ ⇑b'
-- FirstOrder.Language.Theory.simpleGraph_model_iff
@[simp]
theorem Theory.simpleGraph_model_iff [Language.graph.Structure V] :
    V ⊨ Theory.simpleGraph ↔
      (Irreflexive fun x y : V => RelMap adj ![x, y]) ∧
        Symmetric fun x y : V => RelMap adj ![x, y] 
Arguments: {V : Type w'} [FirstOrder.Language.graph.Structure V]
Goal: V ⊨ FirstOrder.Language.Theory.simpleGraph ↔
  (Irreflexive fun x y => FirstOrder.Language.Structure.RelMap FirstOrder.Language.adj ![x, y]) ∧
    Symmetric fun x y => FirstOrder.Language.Structure.RelMap FirstOrder.Language.adj ![x, y]
-- FirstOrder.Language.simpleGraphOfStructure_adj
@[simps]
def simpleGraphOfStructure [Language.graph.Structure V] [V ⊨ Theory.simpleGraph] :
    SimpleGraph V 
Arguments: (V : Type w') [FirstOrder.Language.graph.Structure V] [V ⊨ FirstOrder.Language.Theory.simpleGraph] (x y : V)
Goal: (FirstOrder.Language.simpleGraphOfStructure V).Adj x y =
  FirstOrder.Language.Structure.RelMap FirstOrder.Language.adj ![x, y]
-- SimpleGraph.simpleGraphOfStructure
@[simp]
theorem _root_.SimpleGraph.simpleGraphOfStructure (G : SimpleGraph V) :
    @simpleGraphOfStructure V G.structure _ = G 
Arguments: {V : Type w'} (G : SimpleGraph V)
Goal: FirstOrder.Language.simpleGraphOfStructure V = G
-- FirstOrder.Language.structure_simpleGraphOfStructure
@[simp]
theorem structure_simpleGraphOfStructure [S : Language.graph.Structure V] [V ⊨ Theory.simpleGraph] :
    (simpleGraphOfStructure V).structure = S 
Arguments: {V : Type w'} [S : FirstOrder.Language.graph.Structure V] [V ⊨ FirstOrder.Language.Theory.simpleGraph]
Goal: (FirstOrder.Language.simpleGraphOfStructure V).structure = S
-- FirstOrder.Language.Theory.simpleGraph_isSatisfiable
theorem Theory.simpleGraph_isSatisfiable : Theory.IsSatisfiable Theory.simpleGraph 
Goal: FirstOrder.Language.Theory.simpleGraph.IsSatisfiable
-- groupCohomology.linearYonedaObjResolution_d_apply
theorem linearYonedaObjResolution_d_apply {A : Rep k G} (i j : ℕ) (x : (resolution k G).X i ⟶ A) :
    (linearYonedaObjResolution A).d i j x = (resolution k G).d j i ≫ x 
Arguments: {k G : Type u} [CommRing k] [Monoid G] {A : Rep k G} (i j : ℕ) (x : (groupCohomology.resolution k G).X i ⟶ A)
Goal: ((groupCohomology.linearYonedaObjResolution A).d i j) x = (groupCohomology.resolution k G).d j i ≫ x
-- inhomogeneousCochains.d_apply
@[simps]
def d [Monoid G] (n : ℕ) (A : Rep k G) : ((Fin n → G) → A) →ₗ[k] (Fin (n + 1) → G) → A 
Arguments: {k G : Type u} [CommRing k] [Monoid G] (n : ℕ) (A : Rep k G) (f : (Fin n → G) → CoeSort.coe A) (g : Fin (n + 1) → G)
Goal: (inhomogeneousCochains.d n A) f g =
  (A.ρ (g 0)) (f fun i => g i.succ) + ∑ j : Fin (n + 1), (-1) ^ (↑j + 1) • f (j.contractNth (fun x x_1 => x * x_1) g)
-- inhomogeneousCochains.d_eq
@[nolint checkType] theorem d_eq :
    d n A =
      (diagonalHomEquiv n A).toModuleIso.inv ≫
        (linearYonedaObjResolution A).d n (n + 1) ≫
          (diagonalHomEquiv (n + 1) A).toModuleIso.hom 
Arguments: {k G : Type u} [CommRing k] (n : ℕ) [Group G] (A : Rep k G)
Goal: inhomogeneousCochains.d n A =
  (Rep.diagonalHomEquiv n A).toModuleIso.inv ≫
    (groupCohomology.linearYonedaObjResolution A).d n (n + 1) ≫ (Rep.diagonalHomEquiv (n + 1) A).toModuleIso.hom
-- groupCohomology.inhomogeneousCochains.d_def
@[simp]
theorem inhomogeneousCochains.d_def (n : ℕ) :
    (inhomogeneousCochains A).d n (n + 1) = inhomogeneousCochains.d n A 
Arguments: {k G : Type u} [CommRing k] [Group G] (A : Rep k G) (n : ℕ)
Goal: (groupCohomology.inhomogeneousCochains A).d n (n + 1) = inhomogeneousCochains.d n A
-- FirstOrder.Language.orderLHom_leSymb
@[simp]
theorem orderLHom_leSymb [L.IsOrdered] :
    (orderLHom L).onRelation leSymb = (leSymb : L.Relations 2) 
Arguments: {L : FirstOrder.Language} [L.IsOrdered]
Goal: L.orderLHom.onRelation FirstOrder.Language.leSymb = FirstOrder.Language.leSymb
-- FirstOrder.Language.orderLHom_order
@[simp]
theorem orderLHom_order : orderLHom Language.order = LHom.id Language.order 
Goal: FirstOrder.Language.order.orderLHom = FirstOrder.Language.LHom.id FirstOrder.Language.order
-- FirstOrder.Language.orderedStructure_iff
@[simp]
theorem orderedStructure_iff [IsOrdered L] [LE M] [L.Structure M] :
    L.OrderedStructure M ↔ LHom.IsExpansionOn (orderLHom L) M 
Arguments: {L : FirstOrder.Language} {M : Type w'} [L.IsOrdered] [LE M] [L.Structure M]
Goal: L.OrderedStructure M ↔ L.orderLHom.IsExpansionOn M
-- FirstOrder.Language.relMap_leSymb
@[simp]
theorem relMap_leSymb [LE M] [L.OrderedStructure M] {a b : M} :
    RelMap (leSymb : L.Relations 2) ![a, b] ↔ a ≤ b 
Arguments: {L : FirstOrder.Language} {M : Type w'} [L.IsOrdered] [L.Structure M] [LE M] [L.OrderedStructure M] {a b : M}
Goal: FirstOrder.Language.Structure.RelMap FirstOrder.Language.leSymb ![a, b] ↔ a ≤ b
-- FirstOrder.Language.Term.realize_le
@[simp]
theorem Term.realize_le [LE M] [L.OrderedStructure M] {t₁ t₂ : L.Term (α ⊕ (Fin n))} {v : α → M}
    {xs : Fin n → M} :
    (t₁.le t₂).Realize v xs ↔ t₁.realize (Sum.elim v xs) ≤ t₂.realize (Sum.elim v xs) 
Arguments: {L : FirstOrder.Language} {α : Type w} {M : Type w'} {n : ℕ} [L.IsOrdered] [L.Structure M] [LE M] [L.OrderedStructure M] {t₁ t₂ : L.Term (α ⊕ Fin n)} {v : α → M} {xs : Fin n → M}
Goal: (t₁.le t₂).Realize v xs ↔
  FirstOrder.Language.Term.realize (Sum.elim v xs) t₁ ≤ FirstOrder.Language.Term.realize (Sum.elim v xs) t₂
-- FirstOrder.Language.Term.realize_lt
@[simp]
theorem Term.realize_lt [Preorder M] [L.OrderedStructure M] {t₁ t₂ : L.Term (α ⊕ (Fin n))}
    {v : α → M} {xs : Fin n → M} :
    (t₁.lt t₂).Realize v xs ↔ t₁.realize (Sum.elim v xs) < t₂.realize (Sum.elim v xs) 
Arguments: {L : FirstOrder.Language} {α : Type w} {M : Type w'} {n : ℕ} [L.IsOrdered] [L.Structure M] [Preorder M] [L.OrderedStructure M] {t₁ t₂ : L.Term (α ⊕ Fin n)} {v : α → M} {xs : Fin n → M}
Goal: (t₁.lt t₂).Realize v xs ↔
  FirstOrder.Language.Term.realize (Sum.elim v xs) t₁ < FirstOrder.Language.Term.realize (Sum.elim v xs) t₂
-- FirstOrder.Language.realize_noTopOrder_iff
theorem realize_noTopOrder_iff : M ⊨ Language.order.noTopOrderSentence ↔ NoTopOrder M 
Arguments: {M : Type w'} [LE M]
Goal: M ⊨ FirstOrder.Language.order.noTopOrderSentence ↔ NoTopOrder M
-- FirstOrder.Language.realize_noTopOrder
@[simp]
theorem realize_noTopOrder [h : NoTopOrder M] : M ⊨ Language.order.noTopOrderSentence 
Arguments: {M : Type w'} [LE M] [h : NoTopOrder M]
Goal: M ⊨ FirstOrder.Language.order.noTopOrderSentence
-- FirstOrder.Language.realize_noBotOrder_iff
theorem realize_noBotOrder_iff : M ⊨ Language.order.noBotOrderSentence ↔ NoBotOrder M 
Arguments: {M : Type w'} [LE M]
Goal: M ⊨ FirstOrder.Language.order.noBotOrderSentence ↔ NoBotOrder M
-- FirstOrder.Language.realize_noBotOrder
@[simp]
theorem realize_noBotOrder [h : NoBotOrder M] : M ⊨ Language.order.noBotOrderSentence 
Arguments: {M : Type w'} [LE M] [h : NoBotOrder M]
Goal: M ⊨ FirstOrder.Language.order.noBotOrderSentence
-- FirstOrder.Language.realize_denselyOrdered_iff
theorem realize_denselyOrdered_iff [Preorder M] :
    M ⊨ Language.order.denselyOrderedSentence ↔ DenselyOrdered M 
Arguments: {M : Type w'} [Preorder M]
Goal: M ⊨ FirstOrder.Language.order.denselyOrderedSentence ↔ DenselyOrdered M
-- FirstOrder.Language.realize_denselyOrdered
@[simp]
theorem realize_denselyOrdered [Preorder M] [h : DenselyOrdered M] :
    M ⊨ Language.order.denselyOrderedSentence 
Arguments: {M : Type w'} [Preorder M] [h : DenselyOrdered M]
Goal: M ⊨ FirstOrder.Language.order.denselyOrderedSentence
-- Equiv.bundledInduced_str
@[simps]
def bundledInduced : CategoryTheory.Bundled.{w'} L.Structure 
Arguments: (L : FirstOrder.Language) {M : Type w} [L.Structure M] {N : Type w'} (g : M ≃ N)
Goal: (Equiv.bundledInduced L g).str = g.inducedStructure
-- Equiv.bundledInduced_α
@[simps]
def bundledInduced : CategoryTheory.Bundled.{w'} L.Structure 
Arguments: (L : FirstOrder.Language) {M : Type w} [L.Structure M] {N : Type w'} (g : M ≃ N)
Goal: ↑(Equiv.bundledInduced L g) = N
-- FirstOrder.Language.Theory.ModelType.is_model
structure ModelType where
  Carrier : Type w
  [struc : L.Structure Carrier]
  [is_model : T.Model Carrier]
  [nonempty' : Nonempty Carrier]
Arguments: {L : FirstOrder.Language} {T : L.Theory} (self : T.ModelType)
Goal: ↑self ⊨ T
-- FirstOrder.Language.Theory.ModelType.nonempty'
structure ModelType where
  Carrier : Type w
  [struc : L.Structure Carrier]
  [is_model : T.Model Carrier]
  [nonempty' : Nonempty Carrier]
Arguments: {L : FirstOrder.Language} {T : L.Theory} (self : T.ModelType)
Goal: Nonempty ↑self
-- FirstOrder.Language.Theory.ModelType.coe_of
@[simp]
theorem coe_of (M : Type w) [L.Structure M] [M ⊨ T] [Nonempty M] : (of T M : Type w) = M 
Arguments: {L : FirstOrder.Language} (T : L.Theory) (M : Type w) [L.Structure M] [M ⊨ T] [Nonempty M]
Goal: ↑(FirstOrder.Language.Theory.ModelType.of T M) = M
-- FirstOrder.Language.Theory.ModelType.reduct_struc
@[simps]
def reduct {L' : Language} (φ : L →ᴸ L') (M : (φ.onTheory T).ModelType) : T.ModelType 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {L' : FirstOrder.Language} (φ : L →ᴸ L') (M : (φ.onTheory T).ModelType)
Goal: (FirstOrder.Language.Theory.ModelType.reduct φ M).struc = φ.reduct ↑M
-- FirstOrder.Language.Theory.ModelType.reduct_Carrier
@[simps]
def reduct {L' : Language} (φ : L →ᴸ L') (M : (φ.onTheory T).ModelType) : T.ModelType 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {L' : FirstOrder.Language} (φ : L →ᴸ L') (M : (φ.onTheory T).ModelType)
Goal: ↑(FirstOrder.Language.Theory.ModelType.reduct φ M) = ↑M
-- FirstOrder.Language.Theory.ModelType.defaultExpansion_Carrier
@[simps]
noncomputable def defaultExpansion {L' : Language} {φ : L →ᴸ L'} (h : φ.Injective)
    [∀ (n) (f : L'.Functions n), Decidable (f ∈ Set.range fun f : L.Functions n => φ.onFunction f)]
    [∀ (n) (r : L'.Relations n), Decidable (r ∈ Set.range fun r : L.Relations n => φ.onRelation r)]
    (M : T.ModelType) [Inhabited M] : (φ.onTheory T).ModelType 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {L' : FirstOrder.Language} {φ : L →ᴸ L'} (h : φ.Injective) [(n : ℕ) → (f : L'.Functions n) → Decidable (f ∈ Set.range fun f => φ.onFunction f)] [(n : ℕ) → (r : L'.Relations n) → Decidable (r ∈ Set.range fun r => φ.onRelation r)] (M : T.ModelType) [Inhabited ↑M]
Goal: ↑(FirstOrder.Language.Theory.ModelType.defaultExpansion h M) = ↑M
-- FirstOrder.Language.Theory.ModelType.defaultExpansion_struc
@[simps]
noncomputable def defaultExpansion {L' : Language} {φ : L →ᴸ L'} (h : φ.Injective)
    [∀ (n) (f : L'.Functions n), Decidable (f ∈ Set.range fun f : L.Functions n => φ.onFunction f)]
    [∀ (n) (r : L'.Relations n), Decidable (r ∈ Set.range fun r : L.Relations n => φ.onRelation r)]
    (M : T.ModelType) [Inhabited M] : (φ.onTheory T).ModelType 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {L' : FirstOrder.Language} {φ : L →ᴸ L'} (h : φ.Injective) [(n : ℕ) → (f : L'.Functions n) → Decidable (f ∈ Set.range fun f => φ.onFunction f)] [(n : ℕ) → (r : L'.Relations n) → Decidable (r ∈ Set.range fun r => φ.onRelation r)] (M : T.ModelType) [Inhabited ↑M]
Goal: (FirstOrder.Language.Theory.ModelType.defaultExpansion h M).struc = φ.defaultExpansion ↑M
-- FirstOrder.Language.Theory.ModelType.subtheoryModel_struc
@[simps]
def subtheoryModel (M : T.ModelType) {T' : L.Theory} (h : T' ⊆ T) : T'.ModelType 
Arguments: {L : FirstOrder.Language} {T : L.Theory} (M : T.ModelType) {T' : L.Theory} (h : T' ⊆ T)
Goal: (M.subtheoryModel h).struc = M.struc
-- FirstOrder.Language.Theory.ModelType.subtheoryModel_Carrier
@[simps]
def subtheoryModel (M : T.ModelType) {T' : L.Theory} (h : T' ⊆ T) : T'.ModelType 
Arguments: {L : FirstOrder.Language} {T : L.Theory} (M : T.ModelType) {T' : L.Theory} (h : T' ⊆ T)
Goal: ↑(M.subtheoryModel h) = ↑M
-- FirstOrder.Language.Theory.coe_of
@[simp]
theorem coe_of {M : Type w} [L.Structure M] [Nonempty M] (h : M ⊨ T) : (h.bundled : Type w) = M 
Arguments: {L : FirstOrder.Language} {T : L.Theory} {M : Type w} [L.Structure M] [Nonempty M] (h : M ⊨ T)
Goal: ↑h.bundled = M
-- FirstOrder.Language.Term.relabel_id
theorem relabel_id (t : L.Term α) : t.relabel id = t 
Arguments: {L : FirstOrder.Language} {α : Type u'} (t : L.Term α)
Goal: FirstOrder.Language.Term.relabel id t = t
-- FirstOrder.Language.Term.relabel_id_eq_id
@[simp]
theorem relabel_id_eq_id : (Term.relabel id : L.Term α → L.Term α) = id 
Arguments: {L : FirstOrder.Language} {α : Type u'}
Goal: FirstOrder.Language.Term.relabel id = id
-- FirstOrder.Language.Term.relabel_relabel
@[simp]
theorem relabel_relabel (f : α → β) (g : β → γ) (t : L.Term α) :
    (t.relabel f).relabel g = t.relabel (g ∘ f) 
Arguments: {L : FirstOrder.Language} {α : Type u'} {β : Type v'} {γ : Type u_3} (f : α → β) (g : β → γ) (t : L.Term α)
Goal: FirstOrder.Language.Term.relabel g (FirstOrder.Language.Term.relabel f t) = FirstOrder.Language.Term.relabel (g ∘ f) t
-- FirstOrder.Language.Term.relabel_comp_relabel
@[simp]
theorem relabel_comp_relabel (f : α → β) (g : β → γ) :
    (Term.relabel g ∘ Term.relabel f : L.Term α → L.Term γ) = Term.relabel (g ∘ f) 
Arguments: {L : FirstOrder.Language} {α : Type u'} {β : Type v'} {γ : Type u_3} (f : α → β) (g : β → γ)
Goal: FirstOrder.Language.Term.relabel g ∘ FirstOrder.Language.Term.relabel f = FirstOrder.Language.Term.relabel (g ∘ f)
-- FirstOrder.Language.Term.relabelEquiv_apply
@[simps]
def relabelEquiv (g : α ≃ β) : L.Term α ≃ L.Term β 
Arguments: {L : FirstOrder.Language} {α : Type u'} {β : Type v'} (g : α ≃ β)
Goal: ∀ (a : L.Term α), (FirstOrder.Language.Term.relabelEquiv g) a = FirstOrder.Language.Term.relabel (⇑g) a
-- FirstOrder.Language.Term.relabelEquiv_symm_apply
@[simps]
def relabelEquiv (g : α ≃ β) : L.Term α ≃ L.Term β 
Arguments: {L : FirstOrder.Language} {α : Type u'} {β : Type v'} (g : α ≃ β)
Goal: ∀ (a : L.Term β), (FirstOrder.Language.Term.relabelEquiv g).symm a = FirstOrder.Language.Term.relabel (⇑g.symm) a
-- FirstOrder.Language.Term.constantsVarsEquiv_symm_apply
@[simps]
def constantsVarsEquiv : L[[γ]].Term α ≃ L.Term (γ ⊕ α) 
Arguments: {L : FirstOrder.Language} {α : Type u'} {γ : Type u_3}
Goal: ∀ (a : L.Term (γ ⊕ α)), FirstOrder.Language.Term.constantsVarsEquiv.symm a = a.varsToConstants
-- FirstOrder.Language.Term.constantsVarsEquiv_apply
@[simps]
def constantsVarsEquiv : L[[γ]].Term α ≃ L.Term (γ ⊕ α) 
Arguments: {L : FirstOrder.Language} {α : Type u'} {γ : Type u_3}
Goal: ∀ (a : (L.withConstants γ).Term α), FirstOrder.Language.Term.constantsVarsEquiv a = a.constantsToVars
-- FirstOrder.Language.Term.constantsVarsEquivLeft_apply
@[simp]
theorem constantsVarsEquivLeft_apply (t : L[[γ]].Term (α ⊕ β)) :
    constantsVarsEquivLeft t = (constantsToVars t).relabel (Equiv.sumAssoc _ _ _).symm 
Arguments: {L : FirstOrder.Language} {α : Type u'} {β : Type v'} {γ : Type u_3} (t : (L.withConstants γ).Term (α ⊕ β))
Goal: FirstOrder.Language.Term.constantsVarsEquivLeft t =
  FirstOrder.Language.Term.relabel (⇑(Equiv.sumAssoc γ α β).symm) t.constantsToVars