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Post's correspondence problem is undecidable

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

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    Natural Language Statement

    Theorem

    Post's correspondence problem, Post (1946), is not decidable, over any presentation of its vocabulary.

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    15 concepts
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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: Post's correspondence problem is undecidable
    18type: theorem
    19---
    20Post's correspondence problem, Post (1946), is not decidable, over any
    21presentation of its vocabulary.
    22-/
    23
    24namespace Lax624099.PcpUndecidable
    25
    26open FirstOrder FirstOrder.Language
    27open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.Classes
    28open Lax904597.Machines Lax624099.Problems Lax624099.ValueInvention Lax624099.ClassRE
    29open Lax624099.FiniteSatisfiability
    30open Lax624099.Halting Lax624099.CodeHalting Lax624099.PostCorrespondence
    31open Lax624099.ConcreteInstances
    32
    33/-- PCP is undecidable. -/
    34axiom pcp_not_computable : ∀ V : FinVocab pcp, ¬ComputablePred (DecisionProblem.toPred PCP V)
    35
    36end Lax624099.PcpUndecidable
    37
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