Lax Lean Archive

Invariance and characterization of 2SAT

Lax485149.TwoSatInvariance · concepts/Lax485149/TwoSatInvariance.lean · lax-485149

proven

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    Natural Language Statement

    Lemma

    Being a satisfiable CNF instance of width at most two is invariant under isomorphism of instances, and an instance is a yes-instance of 2SAT exactly when it has width at most two and is satisfiable.

    Concept map
    20 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 twoSat_iff proven

    2 twoSatisfiable_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 Lax485149.Problems
    8import Lax485149.Complement
    9import Lax485149.SecondOrderAtoms
    10import Lax485149.KromFragment
    11import Lax485149.TransitiveClosure
    12import Lax485149.DeterministicTransitiveClosure
    13import Lax485149.FirstOrderDefinability
    14import Lax485149.HeadAutomata
    15import Lax485149.Reachability
    16import Lax485149.DeterministicReachability
    17import Lax485149.TwoSat
    18import Lax485149.ClassNL
    19import Lax485149.ClassL
    20
    21/-!
    22---
    23title: Invariance and characterization of 2SAT
    24type: lemma
    25---
    26Being a satisfiable CNF instance of width at most two is invariant under
    27isomorphism of instances, and an instance is a yes-instance of 2SAT exactly
    28when it has width at most two and is satisfiable.
    29-/
    30
    31namespace Lax485149.TwoSatInvariance
    32
    33open FirstOrder FirstOrder.Language
    34open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    35open Lax904597.Classes Lax904597.Sat
    36open Lax485149.Problems Lax485149.Complement Lax485149.SecondOrderAtoms Lax485149.KromFragment
    37open Lax485149.TransitiveClosure Lax485149.DeterministicTransitiveClosure
    38open Lax485149.FirstOrderDefinability Lax485149.HeadAutomata Lax485149.Reachability
    39open Lax485149.DeterministicReachability Lax485149.TwoSat Lax485149.ClassNL Lax485149.ClassL
    40
    41/-- Being satisfiable of width at most two is isomorphism-invariant. -/
    42axiom twoSatisfiable_iso : ∀ {A B : Type} [sat.Structure A] [sat.Structure B],
    43 (A ≃[sat] B) → (TwoSatisfiable A ↔ TwoSatisfiable B)
    44
    45/-- The yes-instances of 2SAT are exactly the satisfiable instances of width at
    46most two. -/
    47axiom twoSat_iff : ∀ (A : Type) [sat.Structure A], TwoSAT A ↔ TwoSatisfiable A
    48
    49end Lax485149.TwoSatInvariance
    50
    Show ProofShow Proof
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    From Mathlib

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