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) [FirstOrder.Language.Structure L 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

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

Instances For

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

Set.Definable A L' s

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

Set.Definable A L s ↔ ∃ (φ : FirstOrder.Language.Formula L (↑A ⊕ α)), s = {v : α → M | FirstOrder.Language.Formula.Realize φ (Sum.elim Subtype.val v)}

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

Set.Definable ∅ L s ↔ ∃ (φ : FirstOrder.Language.Formula L α), s = setOf (FirstOrder.Language.Formula.Realize φ)

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

Set.Definable A L s ↔ Set.Definable ∅ (FirstOrder.Language.withConstants L ↑A) s

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

Set.Definable B L s

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@[simp]

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

Set.Definable A L ∅

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@[simp]

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

Set.Definable A L Set.univ

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@[simp]

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

Set.Definable A L (f ∩ g)

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@[simp]

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

Set.Definable A L (f ∪ g)

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

Set.Definable A L (Finset.inf s f)

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

Set.Definable A L (Finset.sup s f)

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

Set.Definable A L (⋂ i ∈ s, f i)

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

Set.Definable A L (⋃ i ∈ s, f i)

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@[simp]

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

Set.Definable A L sᶜ

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@[simp]

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

Set.Definable A L (s \ t)

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theorem Set.Definable.preimage_comp {M : Type w} {A : Set M} {L : FirstOrder.Language} [FirstOrder.Language.Structure L M] {α : Type u₁} {β : Type u_1} (f : α → β) {s : Set (α → M)} (h : Set.Definable A L s) :

Set.Definable A L ((fun (g : β → M) => g ∘ f) ⁻¹' s)

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

Set.Definable A L ((fun (g : β → M) => g ∘ ⇑f) '' s)

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

Set.Definable A L s ↔ ∃ (A0 : Finset M), ↑A0 ⊆ A ∧ Set.Definable (↑A0) L s

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theorem Set.Definable.image_comp_sum_inl_fin {M : Type w} {A : Set M} {L : FirstOrder.Language} [FirstOrder.Language.Structure L M] {α : Type u₁} (m : ℕ) {s : Set (α ⊕ Fin m → M)} (h : Set.Definable A L s) :

Set.Definable A 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} [FirstOrder.Language.Structure L M] {α : Type u₁} {β : Type u_1} {s : Set (β → M)} (h : Set.Definable A L s) (f : α ↪ β) [Finite β] :

Set.Definable A L ((fun (g : β → M) => g ∘ ⇑f) '' s)

Shows that definability is closed under finite projections.

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

Set.Definable A 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) [FirstOrder.Language.Structure L M] (s : Set M) :

Prop

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

Equations

Set.Definable₁ A L s = Set.Definable A 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) [FirstOrder.Language.Structure L M] (s : Set (M × M)) :

Prop

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

Equations

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

Instances For

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def FirstOrder.Language.DefinableSet (L : FirstOrder.Language) {M : Type w} [FirstOrder.Language.Structure L 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

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

Instances For

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

SetLike (FirstOrder.Language.DefinableSet L A α) (α → M)

Equations

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

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

Top (FirstOrder.Language.DefinableSet L A α)

Equations

FirstOrder.Language.DefinableSet.instTop = { top := { val := ⊤, property := ⋯ } }

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

Bot (FirstOrder.Language.DefinableSet L A α)

Equations

FirstOrder.Language.DefinableSet.instBot = { bot := { val := ⊥, property := ⋯ } }

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

Sup (FirstOrder.Language.DefinableSet L A α)

Equations

FirstOrder.Language.DefinableSet.instSup = { sup := fun (s t : FirstOrder.Language.DefinableSet L A α) => { val := ↑s ∪ ↑t, property := ⋯ } }

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

Inf (FirstOrder.Language.DefinableSet L A α)

Equations

FirstOrder.Language.DefinableSet.instInf = { inf := fun (s t : FirstOrder.Language.DefinableSet L A α) => { val := ↑s ∩ ↑t, property := ⋯ } }

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

HasCompl (FirstOrder.Language.DefinableSet L A α)

Equations

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

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

SDiff (FirstOrder.Language.DefinableSet L A α)

Equations

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

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

Inhabited (FirstOrder.Language.DefinableSet L A α)

Equations

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

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

s ≤ t ↔ ↑s ≤ ↑t

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@[simp]

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

x ∈ ⊤

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@[simp]

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

x ∉ ⊥

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@[simp]

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

x ∈ s ⊔ t ↔ x ∈ s ∨ x ∈ t

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@[simp]

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

x ∈ s ⊓ t ↔ x ∈ s ∧ x ∈ t

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@[simp]

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

x ∈ sᶜ ↔ x ∉ s

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@[simp]

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

x ∈ s \ t ↔ x ∈ s ∧ x ∉ t

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@[simp]

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

↑⊤ = Set.univ

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@[simp]

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

↑⊥ = ∅

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@[simp]

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

↑(s ⊔ t) = ↑s ∪ ↑t

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@[simp]

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

↑(s ⊓ t) = ↑s ∩ ↑t

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@[simp]

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

↑sᶜ = (↑s)ᶜ

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@[simp]

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

↑(s \ t) = ↑s \ ↑t

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

BooleanAlgebra (FirstOrder.Language.DefinableSet L A α)

Equations

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

Documentation

Mathlib.ModelTheory.Definability

Definable Sets #

Main Definitions #

Main Results #