Definable Sets

This file defines what it means for a set over a first-order structure to be definable.

Main Definitions

• Set.Definable is defined so that A.Definable L s indicates that the set s of a finite cartesian power of M is definable with parameters in A.
• Set.Definable₁ is defined so that A.Definable₁ L s indicates that (s : Set M) is definable with parameters in A.
• Set.Definable₂ is defined so that A.Definable₂ L s indicates that (s : Set (M × M)) is definable with parameters in A.
• A FirstOrder.Language.DefinableSet is defined so that L.DefinableSet A α is the boolean algebra of subsets of α → M defined by formulas with parameters in A.

Main Results

• L.DefinableSet A α forms a BooleanAlgebra
• Set.Definable.image_comp shows that definability is closed under projections in finite dimensions.

def Set.Definable {M : Type w} (A : Set M) (L : FirstOrder.Language) [L.Structure M] {α : Type u₁} (s : Set (α → M)) : Prop

A subset of a finite Cartesian product of a structure is definable over a set A when membership in the set is given by a first-order formula with parameters from A.

Equations

• A.Definable L s = ∃ (φ : (L.withConstants ↑A).Formula α), s = setOf φ.Realize

theorem Set.Definable.map_expansion {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {s : Set (α → M)} {L' : FirstOrder.Language} [L'.Structure M] (h : A.Definable L s) (φ : L →ᴸ L') [φ.IsExpansionOn M] : A.Definable L' s

theorem Set.definable_iff_exists_formula_sum {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {s : Set (α → M)} : A.Definable L s ↔ ∃ (φ : L.Formula (↑A ⊕ α)), s = {v : α → M | φ.Realize (Sum.elim Subtype.val v)}

theorem Set.empty_definable_iff {M : Type w} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {s : Set (α → M)} : ∅.Definable L s ↔ ∃ (φ : L.Formula α), s = setOf φ.Realize

theorem Set.definable_iff_empty_definable_with_params {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {s : Set (α → M)} : A.Definable L s ↔ ∅.Definable (L.withConstants ↑A) s

theorem Set.Definable.mono {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {B : Set M} {s : Set (α → M)} (hAs : A.Definable L s) (hAB : A ⊆ B) : B.Definable L s

@[simp]

theorem Set.definable_empty {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} : A.Definable L ∅

@[simp]

theorem Set.definable_univ {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} : A.Definable L univ

@[simp]

theorem Set.Definable.inter {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {f g : Set (α → M)} (hf : A.Definable L f) (hg : A.Definable L g) : A.Definable L (f ∩ g)

@[simp]

theorem Set.Definable.union {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {f g : Set (α → M)} (hf : A.Definable L f) (hg : A.Definable L g) : A.Definable L (f ∪ g)

theorem Set.definable_finset_inf {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {ι : Type u_2} {f : ι → Set (α → M)} (hf : ∀ (i : ι), A.Definable L (f i)) (s : Finset ι) : A.Definable L (s.inf f)

theorem Set.definable_finset_sup {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {ι : Type u_2} {f : ι → Set (α → M)} (hf : ∀ (i : ι), A.Definable L (f i)) (s : Finset ι) : A.Definable L (s.sup f)

theorem Set.definable_finset_biInter {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {ι : Type u_2} {f : ι → Set (α → M)} (hf : ∀ (i : ι), A.Definable L (f i)) (s : Finset ι) : A.Definable L (⋂ i ∈ s, f i)

theorem Set.definable_finset_biUnion {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {ι : Type u_2} {f : ι → Set (α → M)} (hf : ∀ (i : ι), A.Definable L (f i)) (s : Finset ι) : A.Definable L (⋃ i ∈ s, f i)

@[simp]

theorem Set.Definable.compl {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {s : Set (α → M)} (hf : A.Definable L s) : A.Definable L sᶜ

@[simp]

theorem Set.Definable.sdiff {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {s t : Set (α → M)} (hs : A.Definable L s) (ht : A.Definable L t) : A.Definable L (s \ t)

@[simp]

theorem Set.Definable.himp {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {s t : Set (α → M)} (hs : A.Definable L s) (ht : A.Definable L t) : A.Definable L (s ⇨ t)

theorem Set.Definable.preimage_comp {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {β : Type u_1} (f : α → β) {s : Set (α → M)} (h : A.Definable L s) : A.Definable L ((fun (g : β → M) => g ∘ f) ⁻¹' s)

theorem Set.Definable.image_comp_equiv {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {β : Type u_1} {s : Set (β → M)} (h : A.Definable L s) (f : α ≃ β) : A.Definable L ((fun (g : β → M) => g ∘ ⇑f) '' s)

theorem Set.definable_iff_finitely_definable {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {s : Set (α → M)} : A.Definable L s ↔ ∃ (A0 : Finset M), ↑A0 ⊆ A ∧ (↑A0).Definable L s

theorem Set.Definable.image_comp_sumInl_fin {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} (m : ℕ) {s : Set (α ⊕ Fin m → M)} (h : A.Definable L s) : A.Definable L ((fun (g : α ⊕ Fin m → M) => g ∘ Sum.inl) '' s)

This lemma is only intended as a helper for Definable.image_comp.

@[deprecated Set.Definable.image_comp_sumInl_fin (since := "2025-02-21")]

theorem Set.Definable.image_comp_sum_inl_fin {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} (m : ℕ) {s : Set (α ⊕ Fin m → M)} (h : A.Definable L s) : A.Definable L ((fun (g : α ⊕ Fin m → M) => g ∘ Sum.inl) '' s)

Alias of Set.Definable.image_comp_sumInl_fin.

---

This lemma is only intended as a helper for Definable.image_comp.

theorem Set.Definable.image_comp_embedding {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {β : Type u_1} {s : Set (β → M)} (h : A.Definable L s) (f : α ↪ β) [Finite β] : A.Definable L ((fun (g : β → M) => g ∘ ⇑f) '' s)

Shows that definability is closed under finite projections.

theorem Set.Definable.image_comp {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {β : Type u_1} {s : Set (β → M)} (h : A.Definable L s) (f : α → β) [Finite α] [Finite β] : A.Definable L ((fun (g : β → M) => g ∘ f) '' s)

Shows that definability is closed under finite projections.

def Set.Definable₁ {M : Type w} (A : Set M) (L : FirstOrder.Language) [L.Structure M] (s : Set M) : Prop

A 1-dimensional version of Definable, for Set M.

Equations

• A.Definable₁ L s = A.Definable L {x : Fin 1 → M | x 0 ∈ s}

def Set.Definable₂ {M : Type w} (A : Set M) (L : FirstOrder.Language) [L.Structure M] (s : Set (M × M)) : Prop

A 2-dimensional version of Definable, for Set (M × M).

Equations

• A.Definable₂ L s = A.Definable L {x : Fin 2 → M | (x 0, x 1) ∈ s}

def FirstOrder.Language.DefinableSet (L : Language) {M : Type w} [L.Structure M] (A : Set M) (α : Type u₁) : Type (max 0 u₁ w)

Definable sets are subsets of finite Cartesian products of a structure such that membership is given by a first-order formula.

Equations

• L.DefinableSet A α = { s : Set (α → M) // A.Definable L s }

instance FirstOrder.Language.DefinableSet.instSetLike {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} : SetLike (L.DefinableSet A α) (α → M)

Equations

• FirstOrder.Language.DefinableSet.instSetLike = { coe := Subtype.val, coe_injective' := ⋯ }

instance FirstOrder.Language.DefinableSet.instTop {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} : Top (L.DefinableSet A α)

Equations

• FirstOrder.Language.DefinableSet.instTop = { top := ⟨⊤, ⋯⟩ }

instance FirstOrder.Language.DefinableSet.instBot {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} : Bot (L.DefinableSet A α)

Equations

• FirstOrder.Language.DefinableSet.instBot = { bot := ⟨⊥, ⋯⟩ }

instance FirstOrder.Language.DefinableSet.instSup {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} : Max (L.DefinableSet A α)

Equations

• FirstOrder.Language.DefinableSet.instSup = { max := fun (s t : L.DefinableSet A α) => ⟨↑s ∪ ↑t, ⋯⟩ }

instance FirstOrder.Language.DefinableSet.instInf {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} : Min (L.DefinableSet A α)

Equations

• FirstOrder.Language.DefinableSet.instInf = { min := fun (s t : L.DefinableSet A α) => ⟨↑s ∩ ↑t, ⋯⟩ }

instance FirstOrder.Language.DefinableSet.instHasCompl {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} : HasCompl (L.DefinableSet A α)

Equations

• FirstOrder.Language.DefinableSet.instHasCompl = { compl := fun (s : L.DefinableSet A α) => ⟨(↑s)ᶜ, ⋯⟩ }

instance FirstOrder.Language.DefinableSet.instSDiff {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} : SDiff (L.DefinableSet A α)

Equations

• FirstOrder.Language.DefinableSet.instSDiff = { sdiff := fun (s t : L.DefinableSet A α) => ⟨↑s \ ↑t, ⋯⟩ }

noncomputable instance FirstOrder.Language.DefinableSet.instHImp {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} : HImp (L.DefinableSet A α)

Equations

• FirstOrder.Language.DefinableSet.instHImp = { himp := fun (s t : L.DefinableSet A α) => ⟨↑s ⇨ ↑t, ⋯⟩ }

instance FirstOrder.Language.DefinableSet.instInhabited {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} : Inhabited (L.DefinableSet A α)

Equations

• FirstOrder.Language.DefinableSet.instInhabited = { default := ⊥ }

theorem FirstOrder.Language.DefinableSet.le_iff {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} {s t : L.DefinableSet A α} : s ≤ t ↔ ↑s ≤ ↑t

@[simp]

theorem FirstOrder.Language.DefinableSet.mem_top {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} {x : α → M} : x ∈ ⊤

@[simp]

theorem FirstOrder.Language.DefinableSet.not_mem_bot {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} {x : α → M} : x ∉ ⊥

@[simp]

theorem FirstOrder.Language.DefinableSet.mem_sup {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} {s t : L.DefinableSet A α} {x : α → M} : x ∈ s ⊔ t ↔ x ∈ s ∨ x ∈ t

@[simp]

theorem FirstOrder.Language.DefinableSet.mem_inf {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} {s t : L.DefinableSet A α} {x : α → M} : x ∈ s ⊓ t ↔ x ∈ s ∧ x ∈ t

@[simp]

theorem FirstOrder.Language.DefinableSet.mem_compl {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} {s : L.DefinableSet A α} {x : α → M} : x ∈ sᶜ ↔ x ∉ s

@[simp]

theorem FirstOrder.Language.DefinableSet.mem_sdiff {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} {s t : L.DefinableSet A α} {x : α → M} : x ∈ s \ t ↔ x ∈ s ∧ x ∉ t

@[simp]

theorem FirstOrder.Language.DefinableSet.coe_top {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} : ↑⊤ = Set.univ

@[simp]

theorem FirstOrder.Language.DefinableSet.coe_bot {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} : ↑⊥ = ∅

@[simp]

theorem FirstOrder.Language.DefinableSet.coe_sup {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} (s t : L.DefinableSet A α) : ↑(s ⊔ t) = ↑s ∪ ↑t

@[simp]

theorem FirstOrder.Language.DefinableSet.coe_inf {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} (s t : L.DefinableSet A α) : ↑(s ⊓ t) = ↑s ∩ ↑t

@[simp]

theorem FirstOrder.Language.DefinableSet.coe_compl {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} (s : L.DefinableSet A α) : ↑sᶜ = (↑s)ᶜ

@[simp]

theorem FirstOrder.Language.DefinableSet.coe_sdiff {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} (s t : L.DefinableSet A α) : ↑(s \ t) = ↑s \ ↑t

@[simp]

theorem FirstOrder.Language.DefinableSet.coe_himp {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} (s t : L.DefinableSet A α) : ↑(s ⇨ t) = ↑s ⇨ ↑t

noncomputable instance FirstOrder.Language.DefinableSet.instBooleanAlgebra {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} : BooleanAlgebra (L.DefinableSet A α)

Equations

• FirstOrder.Language.DefinableSet.instBooleanAlgebra = Function.Injective.booleanAlgebra (fun (a : { s : Set (α → M) // A.Definable L s }) => ↑a) ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯