Lax Lean Archive

3-DNF-TAUT and 3-UNSAT are coNP-complete

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

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    Natural Language Statement

    Theorem

    3-DNF-TAUT and 3-UNSAT are coNP-complete under first-order reductions. The clause-splitting reduction of SAT to 3SAT reduces the complement of SAT to 3-UNSAT, on the CNF side where the width bound is available, and the sign swap carries 3-UNSAT to 3-DNF-TAUT.

    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 threeDnfTaut_coNP_complete proven

    2 threeUnsat_coNP_complete 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: 3-DNF-TAUT and 3-UNSAT are coNP-complete
    22type: theorem
    23---
    243-DNF-TAUT and 3-UNSAT are coNP-complete under first-order reductions. The
    25clause-splitting reduction of SAT to 3SAT reduces the complement of SAT to
    263-UNSAT, on the CNF side where the width bound is available, and the sign
    27swap carries 3-UNSAT to 3-DNF-TAUT.
    28-/
    29
    30namespace Lax564036.ThreeDnfTautCoNPComplete
    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/-- 3-DNF-TAUT is coNP-complete. -/
    40axiom threeDnfTaut_coNP_complete : coNP.Complete ThreeDnfTAUT
    41
    42/-- 3-UNSAT is coNP-complete. -/
    43axiom threeUnsat_coNP_complete : coNP.Complete ThreeUNSAT
    44
    45end Lax564036.ThreeDnfTautCoNPComplete
    46
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