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Membership from two problems of #P

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

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    Natural Language Statement

    Theorem

    If two counting problems CC and DD are in #P, a decision problem PP such that, on nonempty finite structures, S(C(A),D(A))S(C(A), D(A)) holds and P(A)P(A) holds exactly when R(C(A),D(A))R(C(A), D(A)), is in the class of SS and RR. The witnesses of the two definitions of #P are the witnesses of the class, read over the ordered expansion.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

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    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: Membership from two problems of #P
    20type: theorem
    21---
    22If two counting problems CC and DD are in #P, a decision problem PP such
    23that, on nonempty finite structures, S(C(A),D(A))S(C(A), D(A)) holds and P(A)P(A) holds
    24exactly when R(C(A),D(A))R(C(A), D(A)), is in the class of SS and RR. The witnesses
    25of the two definitions of #P are the witnesses of the class, read over the
    26ordered expansion.
    27-/
    28
    29namespace Lax175070.CountClassMembership
    30
    31open FirstOrder FirstOrder.Language
    32open Lax904597.Problems Lax904597.Classes Lax904597.Sat Lax904597.Interpretations
    33 Lax904597.Machines Lax564036.Hierarchy
    34open Lax366625.CountingProblems Lax366625.CountingClasses Lax366625.WitnessCounting
    35 Lax366625.CountingSat
    36open Lax366625.CountingRuns Lax175070.CountDefinability Lax175070.SelectedSat Lax485149.Problems
    37open Lax485149.Complement
    38
    39/-- A problem of the class, from two counting problems of #P. -/
    40axiom mem_countClass_of_sharpP :
    41 ∀ {L : Language.{0, 0}} [L.IsRelational] {S R : ℕ → ℕ → Prop} {C D : CountingProblem L},
    42 SharpP.Mem C → SharpP.Mem D → ∀ {P : DecisionProblem L},
    43 (∀ (A : Type) [L.Structure A] [Finite A] [Nonempty A],
    44 S (C A) (D A) ∧ (P A ↔ R (C A) (D A))) → (countClass S R).Mem P
    45
    46end Lax175070.CountClassMembership
    47
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