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The exponential of classes is monotone

Lax480241.ExponentialMonotone · concepts/Lax480241/ExponentialMonotone.lean · lax-480241

proven

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    Natural Language Statement

    Theorem

    If every problem of a class CC is in a class DD, every problem of Cexp⁡C^{\exp} is in Dexp⁡D^{\exp}: the same expansion serves.

    Concept map
    25 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.Classes
    3import Lax904597.Machines
    4import Lax485149.Problems
    5import Lax485149.Complement
    6import Lax485149.ClassNL
    7import Lax535992.ClassPTIME
    8import Lax564036.Hierarchy
    9import Lax564036.AlternatingMachines
    10import Lax134656.ClassPSPACE
    11import Lax480241.Expansions
    12import Lax480241.SecondOrderFixedPoints
    13import Lax480241.AlternatingSpace
    14import Lax480241.ExponentialClasses
    15
    16/-!
    17---
    18title: The exponential of classes is monotone
    19type: theorem
    20---
    21If every problem of a class CC is in a class DD, every problem of
    22Cexp⁡C^{\exp} is in Dexp⁡D^{\exp}: the same expansion serves.
    23-/
    24
    25namespace Lax480241.ExponentialMonotone
    26
    27open FirstOrder FirstOrder.Language
    28open Lax904597.Problems Lax904597.Classes Lax904597.Machines Lax485149.Problems
    29 Lax485149.Complement Lax485149.ClassNL
    30open Lax535992.ClassPTIME Lax564036.Hierarchy Lax564036.AlternatingMachines Lax134656.ClassPSPACE
    31open Lax480241.Expansions Lax480241.SecondOrderFixedPoints Lax480241.AlternatingSpace
    32 Lax480241.ExponentialClasses
    33
    34/-- The exponential of classes is monotone. -/
    35axiom expClass_mono :
    36 ∀ {C D : ComplexityClass},
    37 (∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L), C.Mem P → D.Mem P) →
    38 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L), (expClass C).Mem P →
    39 (expClass D).Mem P
    40
    41end Lax480241.ExponentialMonotone
    42
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