Lax Lean Archive

Invariance and characterization of SAT-UNSAT

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

proven

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    Natural Language Statement

    Lemma

    Satisfiability of either formula of a pair is invariant under isomorphism of instances, and an instance is a yes-instance of SAT-UNSAT exactly when its first formula is satisfiable and its second is not.

    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 satUnsat_iff proven

    2 satWith_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 SAT-UNSAT
    22type: lemma
    23---
    24Satisfiability of either formula of a pair is invariant under isomorphism
    25of instances, and an instance is a yes-instance of SAT-UNSAT exactly when
    26its first formula is satisfiable and its second is not.
    27-/
    28
    29namespace Lax564036.SatUnsatInvariance
    30
    31open FirstOrder FirstOrder.Language
    32open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    33open Lax904597.Classes Lax904597.Sat Lax904597.Machines
    34open Lax485149.Problems Lax485149.Complement Lax535992.ClassPTIME
    35open Lax564036.Hierarchy Lax564036.Difference Lax564036.Tautology Lax564036.ThreeDnfTautology
    36open Lax564036.SatUnsat Lax564036.QuantifiedBooleanFormulas Lax564036.AlternatingMachines
    37
    38/-- Satisfiability of a side of the pair is isomorphism-invariant. -/
    39axiom satWith_iso : ∀ {A B : Type} [satPair.Structure A] [satPair.Structure B],
    40 (A ≃[satPair] B) → ∀ (isCl : satPair.Relations 1) (pos neg : satPair.Relations 2),
    41 (SatWith A isCl pos neg ↔ SatWith B isCl pos neg)
    42
    43/-- The yes-instances of SAT-UNSAT are exactly the pairs whose first formula is
    44satisfiable and whose second is not. -/
    45axiom satUnsat_iff : ∀ (A : Type) [satPair.Structure A],
    46 SATUNSAT A ↔ (SatWith A spIsCl₁ spPos₁ spNeg₁ ∧ ¬SatWith A spIsCl₂ spPos₂ spNeg₂)
    47
    48end Lax564036.SatUnsatInvariance
    49
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