Lax Lean Archive

Invariance and characterization of code halting

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

proven

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    Natural Language Statement

    Lemma

    Halting of the code drawn at the root is invariant under isomorphism of instances, and an instance is a yes-instance of CODEHALT exactly when its root draws a code that halts on zero.

    Concept map
    15 concepts
    100%
    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_iff proven

    2 codeHaltsOn_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 code halting
    18type: lemma
    19---
    20Halting of the code drawn at the root is invariant under isomorphism of
    21instances, and an instance is a yes-instance of CODEHALT exactly when its root
    22draws a code that halts on zero.
    23-/
    24
    25namespace Lax624099.CodeHaltingInvariance
    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/-- The property `CodeHaltsOn` is isomorphism-invariant. -/
    35axiom codeHaltsOn_iso : ∀ {A B : Type} [code.Structure A] [code.Structure B],
    36 (A ≃[code] B) → (CodeHaltsOn A ↔ CodeHaltsOn B)
    37
    38/-- The yes-instances of CODEHALT are exactly the instances whose root draws a
    39halting code. -/
    40axiom codehalt_iff : ∀ (A : Type) [code.Structure A], CODEHALT A ↔ CodeHaltsOn A
    41
    42end Lax624099.CodeHaltingInvariance
    43
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