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NP ∪ coNP ⊆ DP ⊆ Σ₂ ∩ Π₂

Lax564036.DPInclusions · concepts/Lax564036/DPInclusions.lean · lax-564036

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    Natural Language Statement

    Theorem

    NP and coNP are contained in DP, a condition of one kind being conjoined with a trivial condition of the other, and DP is contained in both Σ2p\Sigma_2^p and Π2p\Pi_2^p, the two kernels of a DP definition merging into one alternation in either order of the blocks.

    Concept map
    23 concepts
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 4 statements. Each proof establishes one of them relative to its assumptions.

    1 coNP_subset_DP proven

    2 DP_subset_piP_two proven

    3 DP_subset_sigmaP_two proven

    4 NP_subset_DP 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 Lax485149.Problems
    9import Lax485149.Complement
    10import Lax535992.ClassPTIME
    11import Lax564036.Hierarchy
    12import Lax564036.Difference
    13import Lax564036.Tautology
    14import Lax564036.ThreeDnfTautology
    15import Lax564036.SatUnsat
    16import Lax564036.QuantifiedBooleanFormulas
    17import Lax564036.AlternatingMachines
    18
    19/-!
    20---
    21title: NP ∪ coNP ⊆ DP ⊆ Σ₂ ∩ Π₂
    22type: theorem
    23---
    24NP and coNP are contained in DP, a condition of one kind being conjoined
    25with a trivial condition of the other, and DP is contained in both
    26Σ2p\Sigma_2^p and Π2p\Pi_2^p, the two kernels of a DP definition merging
    27into one alternation in either order of the blocks.
    28-/
    29
    30namespace Lax564036.DPInclusions
    31
    32open FirstOrder FirstOrder.Language
    33open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    34open Lax904597.Classes Lax904597.Sat Lax904597.Machines
    35open Lax485149.Problems Lax485149.Complement Lax535992.ClassPTIME
    36open Lax564036.Hierarchy Lax564036.Difference Lax564036.Tautology Lax564036.ThreeDnfTautology
    37open Lax564036.SatUnsat Lax564036.QuantifiedBooleanFormulas Lax564036.AlternatingMachines
    38
    39/-- NP is contained in DP. -/
    40axiom NP_subset_DP :
    41 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L), NP.Mem P → DP.Mem P
    42
    43/-- coNP is contained in DP. -/
    44axiom coNP_subset_DP :
    45 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L), coNP.Mem P → DP.Mem P
    46
    47/-- DP is contained in `Σ₂ᵖ`. -/
    48axiom DP_subset_sigmaP_two : ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    49 DP.Mem P → (SigmaP 2).Mem P
    50
    51/-- DP is contained in `Π₂ᵖ`. -/
    52axiom DP_subset_piP_two : ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    53 DP.Mem P → (PiP 2).Mem P
    54
    55end Lax564036.DPInclusions
    56
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