ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- FirstOrder.Language.BoundedFormula.realize_iExs
@[simp]
theorem realize_iExs [Finite γ] {f : α → β ⊕ γ}
    {φ : L.Formula α} {v : β → M} {v' : Fin 0 → M} :
    BoundedFormula.Realize (φ.iExs f) v v' ↔
      ∃ (i : γ → M), φ.Realize (fun a => Sum.elim v i (f a)) 
Arguments: {L : FirstOrder.Language} {M : Type w} [L.Structure M] {α : Type u'} {β : Type v'} {γ : Type u_3} [Finite γ] {f : α → β ⊕ γ} {φ : L.Formula α} {v : β → M} {v' : Fin 0 → M}
Goal: FirstOrder.Language.BoundedFormula.Realize (FirstOrder.Language.Formula.iExs f φ) v v' ↔
  ∃ i, φ.Realize fun a => Sum.elim v i (f a)
-- FirstOrder.Language.BoundedFormula.realize_toFormula
@[simp]
theorem realize_toFormula (φ : L.BoundedFormula α n) (v : α ⊕ (Fin n) → M) :
    φ.toFormula.Realize v ↔ φ.Realize (v ∘ Sum.inl) (v ∘ Sum.inr) 
Arguments: {L : FirstOrder.Language} {M : Type w} [L.Structure M] {α : Type u'} {n : ℕ} (φ : L.BoundedFormula α n) (v : α ⊕ Fin n → M)
Goal: φ.toFormula.Realize v ↔ φ.Realize (v ∘ Sum.inl) (v ∘ Sum.inr)
-- FirstOrder.Language.BoundedFormula.realize_iSup
@[simp]
theorem realize_iSup (s : Finset β) (f : β → L.BoundedFormula α n)
    (v : α → M) (v' : Fin n → M) :
    (iSup s f).Realize v v' ↔ ∃ b ∈ s, (f b).Realize v v' 
Arguments: {L : FirstOrder.Language} {M : Type w} [L.Structure M] {α : Type u'} {β : Type v'} {n : ℕ} (s : Finset β) (f : β → L.BoundedFormula α n) (v : α → M) (v' : Fin n → M)
Goal: (FirstOrder.Language.BoundedFormula.iSup s f).Realize v v' ↔ ∃ b ∈ s, (f b).Realize v v'
-- FirstOrder.Language.BoundedFormula.realize_iInf
@[simp]
theorem realize_iInf (s : Finset β) (f : β → L.BoundedFormula α n)
    (v : α → M) (v' : Fin n → M) :
    (iInf s f).Realize v v' ↔ ∀ b ∈ s, (f b).Realize v v' 
Arguments: {L : FirstOrder.Language} {M : Type w} [L.Structure M] {α : Type u'} {β : Type v'} {n : ℕ} (s : Finset β) (f : β → L.BoundedFormula α n) (v : α → M) (v' : Fin n → M)
Goal: (FirstOrder.Language.BoundedFormula.iInf s f).Realize v v' ↔ ∀ b ∈ s, (f b).Realize v v'
-- FirstOrder.Language.Equiv.realize_boundedFormula
@[simp]
theorem realize_boundedFormula (g : M ≃[L] N) (φ : L.BoundedFormula α n) {v : α → M}
    {xs : Fin n → M} : φ.Realize (g ∘ v) (g ∘ xs) ↔ φ.Realize v xs 
Arguments: {L : FirstOrder.Language} {M : Type w} {N : Type u_1} [L.Structure M] [L.Structure N] {α : Type u'} {n : ℕ} (g : L.Equiv M N) (φ : L.BoundedFormula α n) {v : α → M} {xs : Fin n → M}
Goal: φ.Realize (⇑g ∘ v) (⇑g ∘ xs) ↔ φ.Realize v xs
-- FirstOrder.Language.Equiv.realize_formula
@[simp]
theorem realize_formula (g : M ≃[L] N) (φ : L.Formula α) {v : α → M} :
    φ.Realize (g ∘ v) ↔ φ.Realize v 
Arguments: {L : FirstOrder.Language} {M : Type w} {N : Type u_1} [L.Structure M] [L.Structure N] {α : Type u'} (g : L.Equiv M N) (φ : L.Formula α) {v : α → M}
Goal: φ.Realize (⇑g ∘ v) ↔ φ.Realize v
-- FirstOrder.Language.Equiv.realize_sentence
theorem realize_sentence (g : M ≃[L] N) (φ : L.Sentence) : M ⊨ φ ↔ N ⊨ φ 
Arguments: {L : FirstOrder.Language} {M : Type w} {N : Type u_1} [L.Structure M] [L.Structure N] (g : L.Equiv M N) (φ : L.Sentence)
Goal: M ⊨ φ ↔ N ⊨ φ
-- FirstOrder.Language.Equiv.theory_model
theorem theory_model (g : M ≃[L] N) [M ⊨ T] : N ⊨ T 
Arguments: {L : FirstOrder.Language} {M : Type w} {N : Type u_1} [L.Structure M] [L.Structure N] {T : L.Theory} (g : L.Equiv M N) [M ⊨ T]
Goal: N ⊨ T
-- FirstOrder.Language.Equiv.elementarilyEquivalent
theorem elementarilyEquivalent (g : M ≃[L] N) : M ≅[L] N 
Arguments: {L : FirstOrder.Language} {M : Type w} {N : Type u_1} [L.Structure M] [L.Structure N] (g : L.Equiv M N)
Goal: L.ElementarilyEquivalent M N
-- FirstOrder.Language.Relations.realize_reflexive
@[simp]
theorem realize_reflexive : M ⊨ r.reflexive ↔ Reflexive fun x y : M => RelMap r ![x, y] 
Arguments: {L : FirstOrder.Language} {M : Type w} [L.Structure M] {r : L.Relations 2}
Goal: M ⊨ r.reflexive ↔ Reflexive fun x y => FirstOrder.Language.Structure.RelMap r ![x, y]
-- FirstOrder.Language.Relations.realize_irreflexive
@[simp]
theorem realize_irreflexive : M ⊨ r.irreflexive ↔ Irreflexive fun x y : M => RelMap r ![x, y] 
Arguments: {L : FirstOrder.Language} {M : Type w} [L.Structure M] {r : L.Relations 2}
Goal: M ⊨ r.irreflexive ↔ Irreflexive fun x y => FirstOrder.Language.Structure.RelMap r ![x, y]
-- FirstOrder.Language.Relations.realize_symmetric
@[simp]
theorem realize_symmetric : M ⊨ r.symmetric ↔ Symmetric fun x y : M => RelMap r ![x, y] 
Arguments: {L : FirstOrder.Language} {M : Type w} [L.Structure M] {r : L.Relations 2}
Goal: M ⊨ r.symmetric ↔ Symmetric fun x y => FirstOrder.Language.Structure.RelMap r ![x, y]
-- FirstOrder.Language.Relations.realize_antisymmetric
@[simp]
theorem realize_antisymmetric :
    M ⊨ r.antisymmetric ↔ AntiSymmetric fun x y : M => RelMap r ![x, y] 
Arguments: {L : FirstOrder.Language} {M : Type w} [L.Structure M] {r : L.Relations 2}
Goal: M ⊨ r.antisymmetric ↔ AntiSymmetric fun x y => FirstOrder.Language.Structure.RelMap r ![x, y]
-- FirstOrder.Language.Relations.realize_transitive
@[simp]
theorem realize_transitive : M ⊨ r.transitive ↔ Transitive fun x y : M => RelMap r ![x, y] 
Arguments: {L : FirstOrder.Language} {M : Type w} [L.Structure M] {r : L.Relations 2}
Goal: M ⊨ r.transitive ↔ Transitive fun x y => FirstOrder.Language.Structure.RelMap r ![x, y]
-- FirstOrder.Language.Relations.realize_total
@[simp]
theorem realize_total : M ⊨ r.total ↔ Total fun x y : M => RelMap r ![x, y] 
Arguments: {L : FirstOrder.Language} {M : Type w} [L.Structure M] {r : L.Relations 2}
Goal: M ⊨ r.total ↔ Total fun x y => FirstOrder.Language.Structure.RelMap r ![x, y]
-- FirstOrder.Language.Sentence.realize_cardGe
@[simp]
theorem Sentence.realize_cardGe (n) : M ⊨ Sentence.cardGe L n ↔ ↑n ≤ #M 
Arguments: (L : FirstOrder.Language) {M : Type w} [L.Structure M] (n : ℕ)
Goal: M ⊨ FirstOrder.Language.Sentence.cardGe L n ↔ ↑n ≤ Cardinal.mk M
-- FirstOrder.Language.model_infiniteTheory_iff
@[simp]
theorem model_infiniteTheory_iff : M ⊨ L.infiniteTheory ↔ Infinite M 
Arguments: (L : FirstOrder.Language) {M : Type w} [L.Structure M]
Goal: M ⊨ L.infiniteTheory ↔ Infinite M
-- FirstOrder.Language.model_nonemptyTheory_iff
@[simp]
theorem model_nonemptyTheory_iff : M ⊨ L.nonemptyTheory ↔ Nonempty M 
Arguments: (L : FirstOrder.Language) {M : Type w} [L.Structure M]
Goal: M ⊨ L.nonemptyTheory ↔ Nonempty M
-- FirstOrder.Language.model_distinctConstantsTheory
theorem model_distinctConstantsTheory {M : Type w} [L[[α]].Structure M] (s : Set α) :
    M ⊨ L.distinctConstantsTheory s ↔ Set.InjOn (fun i : α => (L.con i : M)) s 
Arguments: (L : FirstOrder.Language) {α : Type u'} {M : Type w} [(L.withConstants α).Structure M] (s : Set α)
Goal: M ⊨ L.distinctConstantsTheory s ↔ Set.InjOn (fun i => ↑(L.con i)) s
-- FirstOrder.Language.card_le_of_model_distinctConstantsTheory
theorem card_le_of_model_distinctConstantsTheory (s : Set α) (M : Type w) [L[[α]].Structure M]
    [h : M ⊨ L.distinctConstantsTheory s] : Cardinal.lift.{w} #s ≤ Cardinal.lift.{u'} #M 
Arguments: (L : FirstOrder.Language) {α : Type u'} (s : Set α) (M : Type w) [(L.withConstants α).Structure M] [h : M ⊨ L.distinctConstantsTheory s]
Goal: Cardinal.lift.{w, u'} (Cardinal.mk ↑s) ≤ Cardinal.lift.{u', w} (Cardinal.mk M)
-- FirstOrder.Language.ElementarilyEquivalent.symm
@[symm]
nonrec theorem symm (h : M ≅[L] N) : N ≅[L] M 
Arguments: {L : FirstOrder.Language} {M : Type w} {N : Type u_1} [L.Structure M] [L.Structure N] (h : L.ElementarilyEquivalent M N)
Goal: L.ElementarilyEquivalent N M
-- FirstOrder.Language.ElementarilyEquivalent.trans
@[trans]
nonrec theorem trans (MN : M ≅[L] N) (NP : N ≅[L] P) : M ≅[L] P 
Arguments: {L : FirstOrder.Language} {M : Type w} {N : Type u_1} {P : Type u_2} [L.Structure M] [L.Structure N] [L.Structure P] (MN : L.ElementarilyEquivalent M N) (NP : L.ElementarilyEquivalent N P)
Goal: L.ElementarilyEquivalent M P
-- FirstOrder.Language.ElementarilyEquivalent.completeTheory_eq
theorem completeTheory_eq (h : M ≅[L] N) : L.completeTheory M = L.completeTheory N 
Arguments: {L : FirstOrder.Language} {M : Type w} {N : Type u_1} [L.Structure M] [L.Structure N] (h : L.ElementarilyEquivalent M N)
Goal: L.completeTheory M = L.completeTheory N
-- FirstOrder.Language.ElementarilyEquivalent.realize_sentence
theorem realize_sentence (h : M ≅[L] N) (φ : L.Sentence) : M ⊨ φ ↔ N ⊨ φ 
Arguments: {L : FirstOrder.Language} {M : Type w} {N : Type u_1} [L.Structure M] [L.Structure N] (h : L.ElementarilyEquivalent M N) (φ : L.Sentence)
Goal: M ⊨ φ ↔ N ⊨ φ
-- FirstOrder.Language.ElementarilyEquivalent.theory_model_iff
theorem theory_model_iff (h : M ≅[L] N) : M ⊨ T ↔ N ⊨ T 
Arguments: {L : FirstOrder.Language} {M : Type w} {N : Type u_1} [L.Structure M] [L.Structure N] {T : L.Theory} (h : L.ElementarilyEquivalent M N)
Goal: M ⊨ T ↔ N ⊨ T
-- FirstOrder.Language.ElementarilyEquivalent.theory_model
theorem theory_model [MT : M ⊨ T] (h : M ≅[L] N) : N ⊨ T 
Arguments: {L : FirstOrder.Language} {M : Type w} {N : Type u_1} [L.Structure M] [L.Structure N] {T : L.Theory} [MT : M ⊨ T] (h : L.ElementarilyEquivalent M N)
Goal: N ⊨ T
-- FirstOrder.Language.ElementarilyEquivalent.nonempty_iff
theorem nonempty_iff (h : M ≅[L] N) : Nonempty M ↔ Nonempty N 
Arguments: {L : FirstOrder.Language} {M : Type w} {N : Type u_1} [L.Structure M] [L.Structure N] (h : L.ElementarilyEquivalent M N)
Goal: Nonempty M ↔ Nonempty N
-- FirstOrder.Language.ElementarilyEquivalent.nonempty
theorem nonempty [Mn : Nonempty M] (h : M ≅[L] N) : Nonempty N 
Arguments: {L : FirstOrder.Language} {M : Type w} {N : Type u_1} [L.Structure M] [L.Structure N] [Mn : Nonempty M] (h : L.ElementarilyEquivalent M N)
Goal: Nonempty N
-- FirstOrder.Language.ElementarilyEquivalent.infinite_iff
theorem infinite_iff (h : M ≅[L] N) : Infinite M ↔ Infinite N 
Arguments: {L : FirstOrder.Language} {M : Type w} {N : Type u_1} [L.Structure M] [L.Structure N] (h : L.ElementarilyEquivalent M N)
Goal: Infinite M ↔ Infinite N
-- FirstOrder.Language.ElementarilyEquivalent.infinite
theorem infinite [Mi : Infinite M] (h : M ≅[L] N) : Infinite N 
Arguments: {L : FirstOrder.Language} {M : Type w} {N : Type u_1} [L.Structure M] [L.Structure N] [Mi : Infinite M] (h : L.ElementarilyEquivalent M N)
Goal: Infinite N
-- Measurable.lmarginal
theorem _root_.Measurable.lmarginal (hf : Measurable f) : Measurable (∫⋯∫⁻_s, f ∂μ) 
Arguments: {δ : Type u_1} {π : δ → Type u_3} [(x : δ) → MeasurableSpace (π x)] (μ : (i : δ) → MeasureTheory.Measure (π i)) [∀ (i : δ), MeasureTheory.SigmaFinite (μ i)] [DecidableEq δ] {s : Finset δ} {f : ((i : δ) → π i) → ℝ≥0∞} (hf : Measurable f)
Goal: Measurable (∫⋯∫⁻_s, f ∂μ)
-- MeasureTheory.lmarginal_empty
@[simp] theorem lmarginal_empty (f : (∀ i, π i) → ℝ≥0∞) : ∫⋯∫⁻_∅, f ∂μ = f 
Arguments: {δ : Type u_1} {π : δ → Type u_3} [(x : δ) → MeasurableSpace (π x)] (μ : (i : δ) → MeasureTheory.Measure (π i)) [DecidableEq δ] (f : ((i : δ) → π i) → ℝ≥0∞)
Goal: ∫⋯∫⁻_∅, f ∂μ = f
-- MeasureTheory.lmarginal_congr
theorem lmarginal_congr {x y : ∀ i, π i} (f : (∀ i, π i) → ℝ≥0∞)
    (h : ∀ i ∉ s, x i = y i) :
    (∫⋯∫⁻_s, f ∂μ) x = (∫⋯∫⁻_s, f ∂μ) y 
Arguments: {δ : Type u_1} {π : δ → Type u_3} [(x : δ) → MeasurableSpace (π x)] (μ : (i : δ) → MeasureTheory.Measure (π i)) [DecidableEq δ] {s : Finset δ} {x y : (i : δ) → π i} (f : ((i : δ) → π i) → ℝ≥0∞) (h : ∀ i ∉ s, x i = y i)
Goal: (∫⋯∫⁻_s, f ∂μ) x = (∫⋯∫⁻_s, f ∂μ) y
-- MeasureTheory.lmarginal_update_of_mem
theorem lmarginal_update_of_mem {i : δ} (hi : i ∈ s)
    (f : (∀ i, π i) → ℝ≥0∞) (x : ∀ i, π i) (y : π i) :
    (∫⋯∫⁻_s, f ∂μ) (Function.update x i y) = (∫⋯∫⁻_s, f ∂μ) x 
Arguments: {δ : Type u_1} {π : δ → Type u_3} [(x : δ) → MeasurableSpace (π x)] (μ : (i : δ) → MeasureTheory.Measure (π i)) [DecidableEq δ] {s : Finset δ} {i : δ} (hi : i ∈ s) (f : ((i : δ) → π i) → ℝ≥0∞) (x : (i : δ) → π i) (y : π i)
Goal: (∫⋯∫⁻_s, f ∂μ) (Function.update x i y) = (∫⋯∫⁻_s, f ∂μ) x
-- MeasureTheory.lmarginal_union
theorem lmarginal_union (f : (∀ i, π i) → ℝ≥0∞) (hf : Measurable f)
    (hst : Disjoint s t) : ∫⋯∫⁻_s ∪ t, f ∂μ = ∫⋯∫⁻_s, ∫⋯∫⁻_t, f ∂μ ∂μ 
Arguments: {δ : Type u_1} {π : δ → Type u_3} [(x : δ) → MeasurableSpace (π x)] (μ : (i : δ) → MeasureTheory.Measure (π i)) [∀ (i : δ), MeasureTheory.SigmaFinite (μ i)] [DecidableEq δ] {s t : Finset δ} (f : ((i : δ) → π i) → ℝ≥0∞) (hf : Measurable f) (hst : Disjoint s t)
Goal: ∫⋯∫⁻_s ∪ t, f ∂μ = ∫⋯∫⁻_s, ∫⋯∫⁻_t, f ∂μ ∂μ
-- MeasureTheory.lmarginal_union'
theorem lmarginal_union' (f : (∀ i, π i) → ℝ≥0∞) (hf : Measurable f) {s t : Finset δ}
    (hst : Disjoint s t) : ∫⋯∫⁻_s ∪ t, f ∂μ = ∫⋯∫⁻_t, ∫⋯∫⁻_s, f ∂μ ∂μ 
Arguments: {δ : Type u_1} {π : δ → Type u_3} [(x : δ) → MeasurableSpace (π x)] (μ : (i : δ) → MeasureTheory.Measure (π i)) [∀ (i : δ), MeasureTheory.SigmaFinite (μ i)] [DecidableEq δ] (f : ((i : δ) → π i) → ℝ≥0∞) (hf : Measurable f) {s t : Finset δ} (hst : Disjoint s t)
Goal: ∫⋯∫⁻_s ∪ t, f ∂μ = ∫⋯∫⁻_t, ∫⋯∫⁻_s, f ∂μ ∂μ
-- MeasureTheory.lmarginal_singleton
theorem lmarginal_singleton (f : (∀ i, π i) → ℝ≥0∞) (i : δ) :
    ∫⋯∫⁻_{i}, f ∂μ = fun x => ∫⁻ xᵢ, f (Function.update x i xᵢ) ∂μ i 
Arguments: {δ : Type u_1} {π : δ → Type u_3} [(x : δ) → MeasurableSpace (π x)] {μ : (i : δ) → MeasureTheory.Measure (π i)} [DecidableEq δ] (f : ((i : δ) → π i) → ℝ≥0∞) (i : δ)
Goal: ∫⋯∫⁻_{i}, f ∂μ = fun x => ∫⁻ (xᵢ : π i), f (Function.update x i xᵢ) ∂μ i
-- MeasureTheory.lmarginal_insert
theorem lmarginal_insert (f : (∀ i, π i) → ℝ≥0∞) (hf : Measurable f) {i : δ}
    (hi : i ∉ s) (x : ∀ i, π i) :
    (∫⋯∫⁻_insert i s, f ∂μ) x = ∫⁻ xᵢ, (∫⋯∫⁻_s, f ∂μ) (Function.update x i xᵢ) ∂μ i 
Arguments: {δ : Type u_1} {π : δ → Type u_3} [(x : δ) → MeasurableSpace (π x)] {μ : (i : δ) → MeasureTheory.Measure (π i)} [∀ (i : δ), MeasureTheory.SigmaFinite (μ i)] [DecidableEq δ] {s : Finset δ} (f : ((i : δ) → π i) → ℝ≥0∞) (hf : Measurable f) {i : δ} (hi : i ∉ s) (x : (i : δ) → π i)
Goal: (∫⋯∫⁻_insert i s, f ∂μ) x = ∫⁻ (xᵢ : π i), (∫⋯∫⁻_s, f ∂μ) (Function.update x i xᵢ) ∂μ i
-- MeasureTheory.lmarginal_erase
theorem lmarginal_erase (f : (∀ i, π i) → ℝ≥0∞) (hf : Measurable f) {i : δ}
    (hi : i ∈ s) (x : ∀ i, π i) :
    (∫⋯∫⁻_s, f ∂μ) x = ∫⁻ xᵢ, (∫⋯∫⁻_(erase s i), f ∂μ) (Function.update x i xᵢ) ∂μ i 
Arguments: {δ : Type u_1} {π : δ → Type u_3} [(x : δ) → MeasurableSpace (π x)] {μ : (i : δ) → MeasureTheory.Measure (π i)} [∀ (i : δ), MeasureTheory.SigmaFinite (μ i)] [DecidableEq δ] {s : Finset δ} (f : ((i : δ) → π i) → ℝ≥0∞) (hf : Measurable f) {i : δ} (hi : i ∈ s) (x : (i : δ) → π i)
Goal: (∫⋯∫⁻_s, f ∂μ) x = ∫⁻ (xᵢ : π i), (∫⋯∫⁻_s.erase i, f ∂μ) (Function.update x i xᵢ) ∂μ i
-- MeasureTheory.lmarginal_insert'
theorem lmarginal_insert' (f : (∀ i, π i) → ℝ≥0∞) (hf : Measurable f) {i : δ}
    (hi : i ∉ s) :
    ∫⋯∫⁻_insert i s, f ∂μ = ∫⋯∫⁻_s, (fun x ↦ ∫⁻ xᵢ, f (Function.update x i xᵢ) ∂μ i) ∂μ 
Arguments: {δ : Type u_1} {π : δ → Type u_3} [(x : δ) → MeasurableSpace (π x)] {μ : (i : δ) → MeasureTheory.Measure (π i)} [∀ (i : δ), MeasureTheory.SigmaFinite (μ i)] [DecidableEq δ] {s : Finset δ} (f : ((i : δ) → π i) → ℝ≥0∞) (hf : Measurable f) {i : δ} (hi : i ∉ s)
Goal: ∫⋯∫⁻_insert i s, f ∂μ = ∫⋯∫⁻_s, fun x => ∫⁻ (xᵢ : π i), f (Function.update x i xᵢ) ∂μ i ∂μ
-- MeasureTheory.lmarginal_erase'
theorem lmarginal_erase' (f : (∀ i, π i) → ℝ≥0∞) (hf : Measurable f) {i : δ}
    (hi : i ∈ s) :
    ∫⋯∫⁻_s, f ∂μ = ∫⋯∫⁻_(erase s i), (fun x ↦ ∫⁻ xᵢ, f (Function.update x i xᵢ) ∂μ i) ∂μ 
Arguments: {δ : Type u_1} {π : δ → Type u_3} [(x : δ) → MeasurableSpace (π x)] {μ : (i : δ) → MeasureTheory.Measure (π i)} [∀ (i : δ), MeasureTheory.SigmaFinite (μ i)] [DecidableEq δ] {s : Finset δ} (f : ((i : δ) → π i) → ℝ≥0∞) (hf : Measurable f) {i : δ} (hi : i ∈ s)
Goal: ∫⋯∫⁻_s, f ∂μ = ∫⋯∫⁻_s.erase i, fun x => ∫⁻ (xᵢ : π i), f (Function.update x i xᵢ) ∂μ i ∂μ
-- MeasureTheory.lmarginal_mono
@[gcongr]
theorem lmarginal_mono {f g : (∀ i, π i) → ℝ≥0∞} (hfg : f ≤ g) : ∫⋯∫⁻_s, f ∂μ ≤ ∫⋯∫⁻_s, g ∂μ 
Arguments: {δ : Type u_1} {π : δ → Type u_3} [(x : δ) → MeasurableSpace (π x)] {μ : (i : δ) → MeasureTheory.Measure (π i)} [DecidableEq δ] {s : Finset δ} {f g : ((i : δ) → π i) → ℝ≥0∞} (hfg : f ≤ g)
Goal: ∫⋯∫⁻_s, f ∂μ ≤ ∫⋯∫⁻_s, g ∂μ
-- MeasureTheory.lmarginal_univ
@[simp] theorem lmarginal_univ [Fintype δ] {f : (∀ i, π i) → ℝ≥0∞} :
    ∫⋯∫⁻_univ, f ∂μ = fun _ => ∫⁻ x, f x ∂Measure.pi μ 
Arguments: {δ : Type u_1} {π : δ → Type u_3} [(x : δ) → MeasurableSpace (π x)] {μ : (i : δ) → MeasureTheory.Measure (π i)} [∀ (i : δ), MeasureTheory.SigmaFinite (μ i)] [DecidableEq δ] [Fintype δ] {f : ((i : δ) → π i) → ℝ≥0∞}
Goal: ∫⋯∫⁻_Finset.univ, f ∂μ = fun x => ∫⁻ (x : (i : δ) → π i), f x ∂MeasureTheory.Measure.pi μ
-- MeasureTheory.lintegral_eq_lmarginal_univ
theorem lintegral_eq_lmarginal_univ [Fintype δ] {f : (∀ i, π i) → ℝ≥0∞} (x : ∀ i, π i) :
    ∫⁻ x, f x ∂Measure.pi μ = (∫⋯∫⁻_univ, f ∂μ) x 
Arguments: {δ : Type u_1} {π : δ → Type u_3} [(x : δ) → MeasurableSpace (π x)] {μ : (i : δ) → MeasureTheory.Measure (π i)} [∀ (i : δ), MeasureTheory.SigmaFinite (μ i)] [DecidableEq δ] [Fintype δ] {f : ((i : δ) → π i) → ℝ≥0∞} (x : (i : δ) → π i)
Goal: ∫⁻ (x : (i : δ) → π i), f x ∂MeasureTheory.Measure.pi μ = (∫⋯∫⁻_Finset.univ, f ∂μ) x
-- MeasureTheory.lmarginal_image
theorem lmarginal_image [DecidableEq δ'] {e : δ' → δ} (he : Injective e) (s : Finset δ')
    {f : (∀ i, π (e i)) → ℝ≥0∞} (hf : Measurable f) (x : ∀ i, π i) :
      (∫⋯∫⁻_s.image e, f ∘ (· ∘' e) ∂μ) x = (∫⋯∫⁻_s, f ∂μ ∘' e) (x ∘' e) 
Arguments: {δ : Type u_1} {δ' : Type u_2} {π : δ → Type u_3} [(x : δ) → MeasurableSpace (π x)] {μ : (i : δ) → MeasureTheory.Measure (π i)} [∀ (i : δ), MeasureTheory.SigmaFinite (μ i)] [DecidableEq δ] [DecidableEq δ'] {e : δ' → δ} (he : Function.Injective e) (s : Finset δ') {f : ((i : δ') → π (e i)) → ℝ≥0∞} (hf : Measurable f) (x : (i : δ) → π i)
Goal: (∫⋯∫⁻_Finset.image e s, f ∘ fun x => (fun {x_1} => x) ∘' e ∂μ) x =
  (∫⋯∫⁻_s, f ∂(fun {x} => μ) ∘' e) ((fun {x_1} => x) ∘' e)
-- MeasureTheory.lmarginal_update_of_not_mem
theorem lmarginal_update_of_not_mem {i : δ}
    {f : (∀ i, π i) → ℝ≥0∞} (hf : Measurable f) (hi : i ∉ s) (x : ∀ i, π i) (y : π i) :
    (∫⋯∫⁻_s, f ∂μ) (Function.update x i y) = (∫⋯∫⁻_s, f ∘ (Function.update · i y) ∂μ) x 
Arguments: {δ : Type u_1} {π : δ → Type u_3} [(x : δ) → MeasurableSpace (π x)] {μ : (i : δ) → MeasureTheory.Measure (π i)} [∀ (i : δ), MeasureTheory.SigmaFinite (μ i)] [DecidableEq δ] {s : Finset δ} {i : δ} {f : ((i : δ) → π i) → ℝ≥0∞} (hf : Measurable f) (hi : i ∉ s) (x : (i : δ) → π i) (y : π i)
Goal: (∫⋯∫⁻_s, f ∂μ) (Function.update x i y) = (∫⋯∫⁻_s, f ∘ fun x => Function.update x i y ∂μ) x
-- MeasureTheory.lmarginal_eq_of_subset
theorem lmarginal_eq_of_subset {f g : (∀ i, π i) → ℝ≥0∞} (hst : s ⊆ t)
    (hf : Measurable f) (hg : Measurable g) (hfg : ∫⋯∫⁻_s, f ∂μ = ∫⋯∫⁻_s, g ∂μ) :
    ∫⋯∫⁻_t, f ∂μ = ∫⋯∫⁻_t, g ∂μ 
Arguments: {δ : Type u_1} {π : δ → Type u_3} [(x : δ) → MeasurableSpace (π x)] {μ : (i : δ) → MeasureTheory.Measure (π i)} [∀ (i : δ), MeasureTheory.SigmaFinite (μ i)] [DecidableEq δ] {s t : Finset δ} {f g : ((i : δ) → π i) → ℝ≥0∞} (hst : s ⊆ t) (hf : Measurable f) (hg : Measurable g) (hfg : ∫⋯∫⁻_s, f ∂μ = ∫⋯∫⁻_s, g ∂μ)
Goal: ∫⋯∫⁻_t, f ∂μ = ∫⋯∫⁻_t, g ∂μ
-- MeasureTheory.lmarginal_le_of_subset
theorem lmarginal_le_of_subset {f g : (∀ i, π i) → ℝ≥0∞} (hst : s ⊆ t)
    (hf : Measurable f) (hg : Measurable g) (hfg : ∫⋯∫⁻_s, f ∂μ ≤ ∫⋯∫⁻_s, g ∂μ) :
    ∫⋯∫⁻_t, f ∂μ ≤ ∫⋯∫⁻_t, g ∂μ 
Arguments: {δ : Type u_1} {π : δ → Type u_3} [(x : δ) → MeasurableSpace (π x)] {μ : (i : δ) → MeasureTheory.Measure (π i)} [∀ (i : δ), MeasureTheory.SigmaFinite (μ i)] [DecidableEq δ] {s t : Finset δ} {f g : ((i : δ) → π i) → ℝ≥0∞} (hst : s ⊆ t) (hf : Measurable f) (hg : Measurable g) (hfg : ∫⋯∫⁻_s, f ∂μ ≤ ∫⋯∫⁻_s, g ∂μ)
Goal: ∫⋯∫⁻_t, f ∂μ ≤ ∫⋯∫⁻_t, g ∂μ
-- MeasureTheory.lintegral_eq_of_lmarginal_eq
theorem lintegral_eq_of_lmarginal_eq [Fintype δ] (s : Finset δ) {f g : (∀ i, π i) → ℝ≥0∞}
    (hf : Measurable f) (hg : Measurable g) (hfg : ∫⋯∫⁻_s, f ∂μ = ∫⋯∫⁻_s, g ∂μ) :
    ∫⁻ x, f x ∂Measure.pi μ = ∫⁻ x, g x ∂Measure.pi μ 
Arguments: {δ : Type u_1} {π : δ → Type u_3} [(x : δ) → MeasurableSpace (π x)] {μ : (i : δ) → MeasureTheory.Measure (π i)} [∀ (i : δ), MeasureTheory.SigmaFinite (μ i)] [DecidableEq δ] [Fintype δ] (s : Finset δ) {f g : ((i : δ) → π i) → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) (hfg : ∫⋯∫⁻_s, f ∂μ = ∫⋯∫⁻_s, g ∂μ)
Goal: ∫⁻ (x : (i : δ) → π i), f x ∂MeasureTheory.Measure.pi μ = ∫⁻ (x : (i : δ) → π i), g x ∂MeasureTheory.Measure.pi μ
-- MeasureTheory.lintegral_le_of_lmarginal_le
theorem lintegral_le_of_lmarginal_le [Fintype δ] (s : Finset δ) {f g : (∀ i, π i) → ℝ≥0∞}
    (hf : Measurable f) (hg : Measurable g) (hfg : ∫⋯∫⁻_s, f ∂μ ≤ ∫⋯∫⁻_s, g ∂μ) :
    ∫⁻ x, f x ∂Measure.pi μ ≤ ∫⁻ x, g x ∂Measure.pi μ 
Arguments: {δ : Type u_1} {π : δ → Type u_3} [(x : δ) → MeasurableSpace (π x)] {μ : (i : δ) → MeasureTheory.Measure (π i)} [∀ (i : δ), MeasureTheory.SigmaFinite (μ i)] [DecidableEq δ] [Fintype δ] (s : Finset δ) {f g : ((i : δ) → π i) → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) (hfg : ∫⋯∫⁻_s, f ∂μ ≤ ∫⋯∫⁻_s, g ∂μ)
Goal: ∫⁻ (x : (i : δ) → π i), f x ∂MeasureTheory.Measure.pi μ ≤ ∫⁻ (x : (i : δ) → π i), g x ∂MeasureTheory.Measure.pi μ
-- FirstOrder.Language.Term.listDecode_encode_list
theorem listDecode_encode_list (l : List (L.Term α)) :
    listDecode (l.bind listEncode) = l 
Arguments: {L : FirstOrder.Language} {α : Type u'} (l : List (L.Term α))
Goal: FirstOrder.Language.Term.listDecode (l.bind FirstOrder.Language.Term.listEncode) = l
-- FirstOrder.Language.Term.encoding_Γ
@[simps]
protected def encoding : Encoding (L.Term α) 
Arguments: {L : FirstOrder.Language} {α : Type u'}
Goal: FirstOrder.Language.Term.encoding.Γ = (α ⊕ (i : ℕ) × L.Functions i)
-- FirstOrder.Language.Term.encoding_encode
@[simps]
protected def encoding : Encoding (L.Term α) 
Arguments: {L : FirstOrder.Language} {α : Type u'}
Goal: ∀ (a : L.Term α), FirstOrder.Language.Term.encoding.encode a = a.listEncode
-- FirstOrder.Language.Term.encoding_decode
@[simps]
protected def encoding : Encoding (L.Term α) 
Arguments: {L : FirstOrder.Language} {α : Type u'} (l : List (α ⊕ (i : ℕ) × L.Functions i))
Goal: FirstOrder.Language.Term.encoding.decode l =
  (do
      let a ← (FirstOrder.Language.Term.listDecode l).head?
      pure (some a)).join
-- FirstOrder.Language.Term.listEncode_injective
theorem listEncode_injective :
    Function.Injective (listEncode : L.Term α → List (α ⊕ (Σi, L.Functions i))) 
Arguments: {L : FirstOrder.Language} {α : Type u'}
Goal: Function.Injective FirstOrder.Language.Term.listEncode
-- FirstOrder.Language.Term.card_le
theorem card_le : #(L.Term α) ≤ max ℵ₀ #(α ⊕ (Σi, L.Functions i)) 
Arguments: {L : FirstOrder.Language} {α : Type u'}
Goal: Cardinal.mk (L.Term α) ≤ max Cardinal.aleph0 (Cardinal.mk (α ⊕ (i : ℕ) × L.Functions i))
-- FirstOrder.Language.Term.card_sigma
theorem card_sigma : #(Σn, L.Term (α ⊕ (Fin n))) = max ℵ₀ #(α ⊕ (Σi, L.Functions i)) 
Arguments: {L : FirstOrder.Language} {α : Type u'}
Goal: Cardinal.mk ((n : ℕ) × L.Term (α ⊕ Fin n)) = max Cardinal.aleph0 (Cardinal.mk (α ⊕ (i : ℕ) × L.Functions i))
-- FirstOrder.Language.BoundedFormula.sigmaAll_apply
@[simp]
lemma sigmaAll_apply {n} {φ : L.BoundedFormula α (n + 1)} :
    sigmaAll ⟨n + 1, φ⟩ = ⟨n, φ.all⟩ 
Arguments: {L : FirstOrder.Language} {α : Type u'} {n : ℕ} {φ : L.BoundedFormula α (n + 1)}
Goal: FirstOrder.Language.BoundedFormula.sigmaAll ⟨n + 1, φ⟩ = ⟨n, φ.all⟩
-- FirstOrder.Language.BoundedFormula.sigmaImp_apply
@[simp]
lemma sigmaImp_apply {n} {φ ψ : L.BoundedFormula α n} :
    sigmaImp ⟨n, φ⟩ ⟨n, ψ⟩ = ⟨n, φ.imp ψ⟩ 
Arguments: {L : FirstOrder.Language} {α : Type u'} {n : ℕ} {φ ψ : L.BoundedFormula α n}
Goal: FirstOrder.Language.BoundedFormula.sigmaImp ⟨n, φ⟩ ⟨n, ψ⟩ = ⟨n, φ.imp ψ⟩
-- FirstOrder.Language.BoundedFormula.listDecode_encode_list
@[simp]
theorem listDecode_encode_list (l : List (Σn, L.BoundedFormula α n)) :
    listDecode (l.bind (fun φ => φ.2.listEncode)) = l 
Arguments: {L : FirstOrder.Language} {α : Type u'} (l : List ((n : ℕ) × L.BoundedFormula α n))
Goal: FirstOrder.Language.BoundedFormula.listDecode (l.bind fun φ => φ.snd.listEncode) = l
-- FirstOrder.Language.BoundedFormula.encoding_Γ
@[simps]
protected def encoding : Encoding (Σn, L.BoundedFormula α n) 
Arguments: {L : FirstOrder.Language} {α : Type u'}
Goal: FirstOrder.Language.BoundedFormula.encoding.Γ = ((k : ℕ) × L.Term (α ⊕ Fin k) ⊕ (n : ℕ) × L.Relations n ⊕ ℕ)