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Every RE-hard problem is undecidable

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

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    Natural Language Statement

    Theorem

    No RE-hard problem is decidable on concrete instances, over any presentation of its vocabulary: CODEHALT is not, since Mathlib's halting problem is not, and hardness carries undecidability along computable reductions.

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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: Every RE-hard problem is undecidable
    18type: theorem
    19---
    20No RE-hard problem is decidable on concrete instances, over any presentation
    21of its vocabulary: CODEHALT is not, since Mathlib's halting problem is not,
    22and hardness carries undecidability along computable reductions.
    23-/
    24
    25namespace Lax624099.REHardUndecidable
    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/-- Every RE-hard problem is undecidable. -/
    35axiom not_computablePred_of_RE_hard : ∀ {L : Language.{0, 0}} [L.IsRelational]
    36 {P : DecisionProblem L}, RE.Hard P → ∀ V : FinVocab L,
    37 ¬ComputablePred (DecisionProblem.toPred P V)
    38
    39end Lax624099.REHardUndecidable
    40
    Show Proof
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