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Invariance and characterization of Post's correspondence problem

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

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    Natural Language Statement

    Lemma

    Having a match is invariant under isomorphism of Post correspondence systems, and a system is a yes-instance of PCP exactly when it has a match.

    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 pcp_iff proven

    2 pcpOn_iso 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: Invariance and characterization of Post's correspondence problem
    18type: lemma
    19---
    20Having a match is invariant under isomorphism of Post correspondence systems,
    21and a system is a yes-instance of PCP exactly when it has a match.
    22-/
    23
    24namespace Lax624099.PostCorrespondenceInvariance
    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/-- The property `Pcp.PcpOn` is isomorphism-invariant. -/
    34axiom pcpOn_iso : ∀ {A B : Type} [pcp.Structure A] [pcp.Structure B],
    35 (A ≃[pcp] B) → (Pcp.PcpOn A ↔ Pcp.PcpOn B)
    36
    37/-- The yes-instances of PCP are exactly the systems with a match. -/
    38axiom pcp_iff : ∀ (A : Type) [pcp.Structure A], PCP A ↔ Pcp.PcpOn A
    39
    40end Lax624099.PostCorrespondenceInvariance
    41
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