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RE is closed under first-order reductions

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

proven

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    Natural Language Statement

    Theorem

    Membership in RE travels backward along first-order reductions and along ordered first-order reductions: if a problem reduces to a problem of RE, it is in RE.

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

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 RE_mem_of_foReduction proven

    2 RE_mem_of_orderedReduction proven

    Lean source view on GitHub

    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 closed under first-order reductions
    18type: theorem
    19---
    20Membership in RE travels backward along first-order reductions and along
    21ordered first-order reductions: if a problem reduces to a problem of RE, it is
    22in RE.
    23-/
    24
    25namespace Lax624099.REClosure
    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/-- Membership in RE travels backward along first-order reductions. -/
    35axiom RE_mem_of_foReduction : ∀ {L L' : Language.{0, 0}} [L.IsRelational] [L'.IsRelational]
    36 {P : DecisionProblem L} {Q : DecisionProblem L'}, FOReduction P Q → RE.Mem Q → RE.Mem P
    37
    38/-- Membership in RE travels backward along ordered first-order reductions. -/
    39axiom RE_mem_of_orderedReduction : ∀ {L L' : Language.{0, 0}} [L.IsRelational] [L'.IsRelational]
    40 {P : DecisionProblem L} {Q : DecisionProblem L'}, OrderedFOReduction P Q → RE.Mem Q → RE.Mem P
    41
    42end Lax624099.REClosure
    43
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