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⊕P and PP are closed under complement

Lax175070.CountClassComplements · concepts/Lax175070/CountClassComplements.lean · lax-175070

proven

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    Natural Language Statement

    Theorem

    ⊕P and PP are closed under complement. A first kernel is paired with a kernel that always holds, which adds one witness: a number is even exactly when its successor is odd, and c≤dc \le d exactly when c<d+1c < d + 1, with the two counts swapped. Whether C=_=P is closed under complement is open.

    Concept map
    23 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 compl_mem_parityP proven

    2 compl_mem_PP proven

    Lean source view on GitHub

    1import Lax904597.Problems
    2import Lax485149.Problems
    3import Lax485149.Complement
    4import Lax904597.Classes
    5import Lax904597.Sat
    6import Lax904597.Interpretations
    7import Lax904597.Machines
    8import Lax564036.Hierarchy
    9import Lax366625.CountingProblems
    10import Lax366625.CountingClasses
    11import Lax366625.WitnessCounting
    12import Lax366625.CountingSat
    13import Lax366625.CountingRuns
    14import Lax175070.CountDefinability
    15import Lax175070.SelectedSat
    16
    17/-!
    18---
    19title: ⊕P and PP are closed under complement
    20type: theorem
    21---
    22⊕P and PP are closed under complement. A first kernel is paired with a
    23kernel that always holds, which adds one witness: a number is even exactly
    24when its successor is odd, and c≤dc \le d exactly when c<d+1c < d + 1, with the
    25two counts swapped. Whether C=_=P is closed under complement is open.
    26-/
    27
    28namespace Lax175070.CountClassComplements
    29
    30open FirstOrder FirstOrder.Language
    31open Lax904597.Problems Lax904597.Classes Lax904597.Sat Lax904597.Interpretations
    32 Lax904597.Machines Lax564036.Hierarchy
    33open Lax366625.CountingProblems Lax366625.CountingClasses Lax366625.WitnessCounting
    34 Lax366625.CountingSat
    35open Lax366625.CountingRuns Lax175070.CountDefinability Lax175070.SelectedSat Lax485149.Problems
    36open Lax485149.Complement
    37
    38/-- ⊕P is closed under complement. -/
    39axiom compl_mem_parityP :
    40 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    41 ParityP.Mem P → ParityP.Mem (DecisionProblem.compl P)
    42
    43/-- PP is closed under complement. -/
    44axiom compl_mem_PP :
    45 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    46 PP.Mem P → PP.Mem (DecisionProblem.compl P)
    47
    48end Lax175070.CountClassComplements
    49
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