Lax Lean Archive

RE is recursive enumerability

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

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    Natural Language Statement

    Theorem

    A decision problem is in RE exactly when the set of its concrete instances, over any presentation of its vocabulary, is recursively enumerable in the sense of Mathlib's computability theory.

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

    Each proof establishes this claim relative to its assumptions.

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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: RE is recursive enumerability
    18type: theorem
    19---
    20A decision problem is in RE exactly when the set of its concrete instances,
    21over any presentation of its vocabulary, is recursively enumerable in the
    22sense of Mathlib's computability theory.
    23-/
    24
    25namespace Lax624099.REIsRecursivelyEnumerable
    26
    27open FirstOrder FirstOrder.Language
    28open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.Classes
    29open Lax904597.Machines Lax624099.Problems Lax624099.ValueInvention Lax624099.ClassRE
    30open Lax624099.FiniteSatisfiability
    31open Lax624099.Halting Lax624099.CodeHalting Lax624099.PostCorrespondence
    32open Lax624099.ConcreteInstances
    33
    34/-- RE is exactly the recursively enumerable properties of finite
    35structures. -/
    36axiom mem_RE_iff_rePred : ∀ {L : Language.{0, 0}} [L.IsRelational] (V : FinVocab L)
    37 (P : DecisionProblem L), RE.Mem P ↔ REPred (DecisionProblem.toPred P V)
    38
    39end Lax624099.REIsRecursivelyEnumerable
    40
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