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.
Set.TermDefinable functions are those equivalent to some term expressible in the language.
Set.TermDefinable₁ specialize this to case of unary functions.

Main Results #

L.DefinableSet A α forms a BooleanAlgebra
Set.Definable.image_comp shows that definability is closed under projections in finite dimensions.
The Set.TermDefinable property is transitive, and TermDefinable functions are closed under composition.

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

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

Instances For

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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

source

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)}

source

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

source

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

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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

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@[simp]

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

A.Definable L ∅

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@[simp]

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

A.Definable L univ

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@[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)

source

@[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)

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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)

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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)

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theorem Set.definable_biInter_finset {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)

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theorem Set.definable_biUnion_finset {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)

source

@[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

source

@[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)

source

@[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)

source

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)

source

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)

source

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

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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.

source

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.

source

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.

source

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}

Instances For

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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}

Instances For

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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 }

Instances For

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@[implicit_reducible]

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' := ⋯ }

source

@[implicit_reducible]

instance FirstOrder.Language.DefinableSet.instPartialOrder {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} :

PartialOrder (L.DefinableSet A α)

Equations

FirstOrder.Language.DefinableSet.instPartialOrder = PartialOrder.ofSetLike (L.DefinableSet A α) (α → M)

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@[implicit_reducible]

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 := ⟨⊤, ⋯⟩ }

source

@[implicit_reducible]

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 := ⟨⊥, ⋯⟩ }

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@[implicit_reducible]

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, ⋯⟩ }

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@[implicit_reducible]

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, ⋯⟩ }

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@[implicit_reducible]

instance FirstOrder.Language.DefinableSet.instCompl {L : Language} {M : Type w} [L.Structure M] {A : Set M} {α : Type u₁} :

Compl (L.DefinableSet A α)

Equations

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

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@[implicit_reducible]

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, ⋯⟩ }

source

@[implicit_reducible]

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, ⋯⟩ }

source

@[implicit_reducible]

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 := ⊥ }

source

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

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@[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 ∈ ⊤

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@[simp]

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

x ∉ ⊥

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@[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

source

@[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

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@[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

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@[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

source

@[simp]

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

↑⊤ = Set.univ

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@[simp]

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

↑⊥ = ∅

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@[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

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@[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

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@[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

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@[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

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@[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

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@[implicit_reducible]

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) ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

source

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

Prop

A function from a Cartesian power of a structure to that structure is term-definable over a set A when the value of the function is given by a term with constants A.

Equations

A.TermDefinable L f = ∃ (φ : (L.withConstants ↑A).Term α), f = fun (v : α → M) => FirstOrder.Language.Term.realize v φ

Instances For

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theorem Set.TermDefinable.definable_tupleGraph {M : Type w} (A : Set M) (L : FirstOrder.Language) [L.Structure M] {α : Type u₁} {f : (α → M) → M} (h : A.TermDefinable L f) :

A.Definable L (Function.tupleGraph f)

Every TermDefinable function has a tupleGraph that is definable.

source

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

A.TermDefinable L' f

source

theorem Set.termDefinable_empty_iff {M : Type w} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {f : (α → M) → M} :

∅.TermDefinable L f ↔ ∃ (φ : L.Term α), f = fun (v : α → M) => FirstOrder.Language.Term.realize v φ

source

theorem Set.termDefinable_empty_withConstants_iff {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {f : (α → M) → M} :

∅.TermDefinable (L.withConstants ↑A) f ↔ A.TermDefinable L f

source

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

B.TermDefinable L f

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theorem Set.TermDefinable.trans {M : Type w} {A : Set M} {L : FirstOrder.Language} {L' : FirstOrder.Language} [L.Structure M] [L'.Structure M] {β : Type u_1} {f : (β → M) → M} (h₁ : A.TermDefinable L f) (h₂ : ∀ {n : ℕ} (g : (L.withConstants ↑A).Functions n), A.TermDefinable L' fun (v : Fin n → M) => FirstOrder.Language.Term.realize v g.term) :

A.TermDefinable L' f

TermDefinable is transitive. If f is TermDefinable in a structure S on L, and all of the functions' realizations on S are TermDefinable on a structure T on L', then f is TermDefinable on T in L'.

source

def Set.TermDefinable₁ {M : Type w} {A : Set M} (L : FirstOrder.Language) [L.Structure M] (f : M → M) :

Prop

A function from a structure to itself is term-definable over a set A when the value of the function is given by a term with constants A. Like TermDefinable but specialized for unary functions in order to write M → M instead of (Unit → M) → M.

Equations

Set.TermDefinable₁ L f = A.TermDefinable L fun (x : Unit → M) => f (x ())

Instances For

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theorem Set.termDefinable₁_iff_termDefinable {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] (f : M → M) :

TermDefinable₁ L f ↔ A.TermDefinable L fun (v : Unit → M) => f (v ())

TermDefinable₁ is defined as TermDefinable on the Unit index type.

source

theorem Set.TermDefinable.termDefinable₁ {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] (f : M → M) :

(A.TermDefinable L fun (v : Unit → M) => f (v ())) → TermDefinable₁ L f

Alias of the reverse direction of Set.termDefinable₁_iff_termDefinable.

TermDefinable₁ is defined as TermDefinable on the Unit index type.

source

theorem Set.TermDefinable₁.termDefinable {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] (f : M → M) :

TermDefinable₁ L f → A.TermDefinable L fun (v : Unit → M) => f (v ())

Alias of the forward direction of Set.termDefinable₁_iff_termDefinable.

TermDefinable₁ is defined as TermDefinable on the Unit index type.

source

theorem Set.termDefinable₁_iff_exists_term {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {f : M → M} :

TermDefinable₁ L f ↔ ∃ (φ : (L.withConstants ↑A).Term Unit), f = (fun (v : Unit → M) => FirstOrder.Language.Term.realize v φ) ∘ Function.const Unit

source

theorem Set.TermDefinable₁.definable₂_graph {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {f : M → M} (h : TermDefinable₁ L f) :

A.Definable₂ L (Function.graph f)

A TermDefinable₁ function has a graph that's Definable₂.

source

theorem Set.TermDefinable₁.id {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] :

TermDefinable₁ L _root_.id

The identity function is TermDefinable₁

source

theorem Set.TermDefinable.const {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} (C : (L.withConstants ↑A).Constants) :

A.TermDefinable L (Function.const (α → M) ↑C)

Constant functions are TermDefinable, assuming the constant value is a language constant.

source

theorem Set.TermDefinable₁.const {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] (C : (L.withConstants ↑A).Constants) :

TermDefinable₁ L (Function.const M ↑C)

Constant functions are TermDefinable₁, assuming the constant value is a language constant.

source

theorem Set.TermDefinable.comp {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {β : Type u_1} {f : (α → M) → M} {g : α → (β → M) → M} (hf : A.TermDefinable L f) (hg : ∀ (i : α), A.TermDefinable L (g i)) :

A.TermDefinable L fun (b : β → M) => f fun (x : α) => g x b

A k-ary TermDefinable function composed with k TermDefinable others is TermDefinable.

source

theorem Set.TermDefinable₁.comp {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {f g : M → M} (hf : TermDefinable₁ L f) (hg : TermDefinable₁ L g) :

TermDefinable₁ L (f ∘ g)

TermDefinable₁ functions are closed under composition.

source

theorem Set.TermDefinable₁.comp_termDefinable {M : Type w} {A : Set M} {L : FirstOrder.Language} [L.Structure M] {α : Type u₁} {f : M → M} {g : (α → M) → M} (hf : TermDefinable₁ L f) (hg : A.TermDefinable L g) :

A.TermDefinable L (f ∘ g)

A TermDefinable function postcomposed with TermDefinable₁ is TermDefinable.

Documentation

Mathlib.ModelTheory.Definability

Definable Sets #

Main Definitions #

Main Results #