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Lax678846.ExistentialSecondOrder

Existential second-order logic

concepts/Lax678846/ExistentialSecondOrder.lean · lax-678846

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    Definition and theorem

    An existential second-order sentence has the form ∃R1⋯∃Rs φ\exists R_1\cdots\exists R_s\,\varphi, where each relation has a fixed arity and the matrix φ\varphi is first-order. Witness relations range over all relations of their arities on the existing universe.

    The first-order basis consists of truth, equality, input-relation and witness-relation atoms, negation, conjunction, and existential element quantification. There is no built-in order or fixed-point operator. Element quantification binds variable zero. The witness vocabulary records the finite second-order quantifier prefix.

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    1import Lax678846.FiniteStructures
    2
    3/-!
    4---
    5title: Existential second-order logic
    6type: definition and theorem
    7---
    8An existential second-order sentence has the form
    9∃R1⋯∃Rs φ\exists R_1\cdots\exists R_s\,\varphi, where each relation has a fixed
    10arity and the matrix φ\varphi is first-order. Witness relations range over
    11all relations of their arities on the existing universe.
    12
    13The first-order basis consists of truth, equality, input-relation and
    14witness-relation atoms, negation, conjunction, and existential element
    15quantification. There is no built-in order or fixed-point operator.
    16Element quantification binds variable zero. The witness vocabulary records
    17the finite second-order quantifier prefix.
    18-/
    19
    20namespace Lax678846.ExistentialSecondOrder
    21
    22open Lax678846.FiniteStructures
    23
    24inductive FirstOrder (σ τ : Vocabulary) : Nat → Type
    25 | truth {m} : FirstOrder σ τ m
    26 | equal {m} (x y : Fin m) : FirstOrder σ τ m
    27 | relation {m} (r : Symbol σ) (args : Fin (σ.get r) → Fin m) : FirstOrder σ τ m
    28 | variable {m} (r : Symbol τ) (args : Fin (τ.get r) → Fin m) : FirstOrder σ τ m
    29 | neg {m} (φ : FirstOrder σ τ m) : FirstOrder σ τ m
    30 | conj {m} (φ ψ : FirstOrder σ τ m) : FirstOrder σ τ m
    31 | exists' {m} (φ : FirstOrder σ τ (m + 1)) : FirstOrder σ τ m
    32
    33def FirstOrder.eval {σ τ : Vocabulary} {m : Nat} (φ : FirstOrder σ τ m)
    34 (A : Structure σ) (R : Interpretation τ A.size) (v : Fin m → Fin A.size) : Prop :=
    35 match φ with
    36 | .truth => True
    37 | .equal x y => v x = v y
    38 | .relation r args => A.relation r (v ∘ args) = true
    39 | .variable r args => R r (v ∘ args) = true
    40 | .neg ψ => ¬ ψ.eval A R v
    41 | .conj ψ χ => ψ.eval A R v ∧ χ.eval A R v
    42 | .exists' ψ => ∃ a : Fin A.size, ψ.eval A R (Fin.cons a v)
    43
    44structure Sentence (σ : Vocabulary) where
    45 witnesses : Vocabulary
    46 matrix : FirstOrder σ witnesses 0
    47
    48def Satisfies {σ : Vocabulary} (A : Structure σ) (φ : Sentence σ) : Prop :=
    49 ∃ R : Interpretation φ.witnesses A.size, φ.matrix.eval A R Fin.elim0
    50
    51def Definable {σ : Vocabulary} (Q : Property σ) : Prop :=
    52 ∃ φ : Sentence σ, ∀ A : Structure σ, Q A ↔ Satisfies A φ
    53
    54axiom satisfiesInvariant {σ : Vocabulary} (φ : Sentence σ) :
    55 IsomorphismInvariant (fun A => Satisfies A φ)
    56
    57end Lax678846.ExistentialSecondOrder
    58
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