Lax Lean Archive

Invariance and characterization of 3-DNF-TAUT and 3-UNSAT

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

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    Natural Language Statement

    Lemma

    Being a DNF tautology of width at most three, and being an unsatisfiable CNF formula of width at most three, are invariant under isomorphism of instances; an instance is a yes-instance of 3-DNF-TAUT, respectively of 3-UNSAT, exactly when it has the corresponding property.

    Concept map
    23 concepts
    100%
    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 threeDnfTaut_iff proven

    2 threeDnfTautology_iso proven

    3 threeUnsat_iff proven

    4 threeUnsatisfiable_iso 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: Invariance and characterization of 3-DNF-TAUT and 3-UNSAT
    22type: lemma
    23---
    24Being a DNF tautology of width at most three, and being an unsatisfiable
    25CNF formula of width at most three, are invariant under isomorphism of
    26instances; an instance is a yes-instance of 3-DNF-TAUT, respectively of
    273-UNSAT, exactly when it has the corresponding property.
    28-/
    29
    30namespace Lax564036.ThreeDnfTautologyInvariance
    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/-- Being a tautology of width at most three is isomorphism-invariant. -/
    40axiom threeDnfTautology_iso : ∀ {A B : Type} [sat.Structure A] [sat.Structure B],
    41 (A ≃[sat] B) → (ThreeDnfTautology A ↔ ThreeDnfTautology B)
    42
    43/-- Being unsatisfiable of width at most three is isomorphism-invariant. -/
    44axiom threeUnsatisfiable_iso : ∀ {A B : Type} [sat.Structure A] [sat.Structure B],
    45 (A ≃[sat] B) → (ThreeUnsatisfiable A ↔ ThreeUnsatisfiable B)
    46
    47/-- The yes-instances of 3-DNF-TAUT are exactly the tautologies of width at
    48most three. -/
    49axiom threeDnfTaut_iff : ∀ (A : Type) [sat.Structure A],
    50 ThreeDnfTAUT A ↔ ThreeDnfTautology A
    51
    52/-- The yes-instances of 3-UNSAT are exactly the unsatisfiable instances of
    53width at most three. -/
    54axiom threeUnsat_iff : ∀ (A : Type) [sat.Structure A],
    55 ThreeUNSAT A ↔ ThreeUnsatisfiable A
    56
    57end Lax564036.ThreeDnfTautologyInvariance
    58
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