Lax Lean Archive

Invariance and characterization of the halting problem

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

proven

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    Natural Language Statement

    Lemma

    Acceptance on an unbounded tape by a well-formed machine instance is invariant under isomorphism of instances, and an instance is a yes-instance of HALT exactly when it is well-formed and accepts its input.

    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 halt_iff proven

    2 haltsOn_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 the halting problem
    18type: lemma
    19---
    20Acceptance on an unbounded tape by a well-formed machine instance is invariant
    21under isomorphism of instances, and an instance is a yes-instance of HALT
    22exactly when it is well-formed and accepts its input.
    23-/
    24
    25namespace Lax624099.HaltingInvariance
    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 `HaltsOn` is isomorphism-invariant. -/
    35axiom haltsOn_iso : ∀ {A B : Type} [turing.Structure A] [turing.Structure B],
    36 (A ≃[turing] B) → (HaltsOn A ↔ HaltsOn B)
    37
    38/-- The yes-instances of HALT are exactly the well-formed accepting machine
    39instances. -/
    40axiom halt_iff : ∀ (A : Type) [turing.Structure A], HALT A ↔ HaltsOn A
    41
    42end Lax624099.HaltingInvariance
    43
    Show ProofShow Proof
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