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NP is contained in RE

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

proven

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    Natural Language Statement

    Theorem

    Every problem of NP is in RE: an existential second-order definition is an existential second-order definition with value invention that invents nothing.

    Concept map
    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.

    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: NP is contained in RE
    18type: theorem
    19---
    20Every problem of NP is in RE: an existential second-order definition is an
    21existential second-order definition with value invention that invents
    22nothing.
    23-/
    24
    25namespace Lax624099.NPSubsetRE
    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/-- NP ⊆ RE. -/
    35axiom NP_subset_RE : ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    36 NP.Mem P → RE.Mem P
    37
    38end Lax624099.NPSubsetRE
    39
    Show Proof
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