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FP and PTIME

Lax366625.FPAndPTIME · concepts/Lax366625/FPAndPTIME.lean · lax-366625

proven

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    Natural Language Statement

    Theorem

    The support of a problem of FP is in PTIME. Hence a problem of FP that is parsimoniously #P-hard gives NP ⊆ PTIME.

    Concept map
    28 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 NP_subset_PTIME_of_mem_FP_of_parsimoniousHard proven

    2 support_mem_PTIME_of_mem_FP proven

    Lean source view on GitHub

    1import Lax904597.Problems
    2import Lax904597.Interpretations
    3import Lax904597.Relativized
    4import Lax904597.SecondOrder
    5import Lax904597.Classes
    6import Lax904597.Sat
    7import Lax904597.Machines
    8import Lax535992.HornSat
    9import Lax535992.CircuitValue
    10import Lax535992.DeterministicMachines
    11import Lax535992.ClassPTIME
    12import Lax366625.CountingProblems
    13import Lax366625.CountingClasses
    14import Lax366625.WitnessCounting
    15import Lax366625.SecondOrderCounting
    16import Lax366625.QuantitativeLogic
    17import Lax366625.CountingSat
    18import Lax366625.MachineNumbers
    19import Lax366625.CountingRuns
    20import Lax366625.NumberedCircuits
    21import Lax366625.HornNumbers
    22
    23/-!
    24---
    25title: FP and PTIME
    26type: theorem
    27---
    28The support of a problem of FP is in PTIME. Hence a problem of FP that is
    29parsimoniously #P-hard gives NP ⊆ PTIME.
    30-/
    31
    32namespace Lax366625.FPAndPTIME
    33
    34open FirstOrder FirstOrder.Language
    35open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    36open Lax904597.Classes Lax904597.Sat Lax904597.Machines
    37open Lax535992.ClassPTIME
    38open Lax366625.CountingProblems Lax366625.CountingClasses Lax366625.WitnessCounting
    39open Lax366625.SecondOrderCounting
    40open Lax366625.QuantitativeLogic Lax366625.CountingSat Lax366625.MachineNumbers
    41open Lax366625.CountingRuns
    42open Lax366625.NumberedCircuits Lax366625.HornNumbers
    43
    44/-- The support of a problem of FP is in PTIME. -/
    45axiom support_mem_PTIME_of_mem_FP :
    46 ∀ {L : Language.{0, 0}} [L.IsRelational] {C : CountingProblem L},
    47 FP.Mem C → PTIME.Mem C.support
    48
    49/-- A parsimoniously #P-hard problem in FP gives NP ⊆ PTIME. -/
    50axiom NP_subset_PTIME_of_mem_FP_of_parsimoniousHard :
    51 ∀ {L : Language.{0, 0}} [L.IsRelational] {C : CountingProblem L},
    52 FP.Mem C → SharpP.ParsimoniousHard C →
    53 ∀ {L' : Language.{0, 0}} [L'.IsRelational] (P : DecisionProblem L'), NP.Mem P → PTIME.Mem P
    54
    55end Lax366625.FPAndPTIME
    56
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