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RE is not closed under complement

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

proven

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    Natural Language Statement

    Theorem

    The complement of CODEHALT is not in RE, so CODEHALT is not in co-RE and the classes RE and co-RE differ: a problem and its complement both recursively enumerable would be decidable, by Post's theorem.

    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 codehalt_not_mem_coRE proven

    2 RE_ne_coRE 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 not closed under complement
    18type: theorem
    19---
    20The complement of CODEHALT is not in RE, so CODEHALT is not in co-RE and the
    21classes RE and co-RE differ: a problem and its complement both recursively
    22enumerable would be decidable, by Post's theorem.
    23-/
    24
    25namespace Lax624099.RENeCoRE
    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/-- CODEHALT is not in co-RE. -/
    35axiom codehalt_not_mem_coRE : ¬coRE.Mem CODEHALT
    36
    37/-- RE differs from co-RE. -/
    38axiom RE_ne_coRE : RE ≠ coRE
    39
    40end Lax624099.RENeCoRE
    41
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