Lax Lean Archive

Membership in RE depends only on the finite instances

Lax624099.REFinite · concepts/Lax624099/REFinite.lean · lax-624099

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    Natural Language Statement

    Theorem

    Two problems that agree on every finite structure are both in RE or neither.

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    15 concepts
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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    1import Lax904597.Problems
    2import Lax904597.Interpretations
    3import Lax904597.Relativized
    4import Lax904597.Classes
    5import Lax624099.Problems
    6import Lax624099.ValueInvention
    7import Lax624099.ClassRE
    8import Lax624099.FiniteSatisfiability
    9import Lax624099.Halting
    10import Lax624099.CodeHalting
    11import Lax624099.PostCorrespondence
    12import Lax624099.ConcreteInstances
    13import Lax904597.Machines
    14
    15/-!
    16---
    17title: Membership in RE depends only on the finite instances
    18type: theorem
    19---
    20Two problems that agree on every finite structure are both in RE or neither.
    21-/
    22
    23namespace Lax624099.REFinite
    24
    25open FirstOrder FirstOrder.Language
    26open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.Classes
    27open Lax904597.Machines Lax624099.Problems Lax624099.ValueInvention Lax624099.ClassRE
    28open Lax624099.FiniteSatisfiability
    29open Lax624099.Halting Lax624099.CodeHalting Lax624099.PostCorrespondence
    30open Lax624099.ConcreteInstances
    31
    32/-- Membership in RE depends only on the finite instances of a problem. -/
    33axiom RE_mem_congr_finite : ∀ {L : Language.{0, 0}} [L.IsRelational] {P Q : DecisionProblem L},
    34 (∀ (A : Type) [L.Structure A] [Finite A], P A ↔ Q A) → (RE.Mem P ↔ RE.Mem Q)
    35
    36end Lax624099.REFinite
    37
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