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The values of #3SAT, #1-in-SAT

Lax280166.SatVariantsValues · concepts/Lax280166/SatVariantsValues.lean · lax-280166

proven

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    Natural Language Statement

    Lemma

    The number counted by each of #3SAT, #1-in-SAT is invariant under isomorphism of instances, so the value of each problem on an instance is the number it counts.

    Concept map
    32 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 sharpOneInSat_count_iso proven

    2 sharpOneInSat_eq proven

    3 sharpThreeSat_count_iso proven

    4 sharpThreeSat_eq proven

    Lean source view on GitHub

    1import Lax904597.Problems
    2import Lax904597.Sat
    3import Lax799700.CliqueFamily
    4import Lax799700.DominatingSet
    5import Lax799700.Feedback
    6import Lax799700.Hamilton
    7import Lax799700.Knapsack
    8import Lax799700.OneInSat
    9import Lax799700.SetFamily
    10import Lax799700.Steiner
    11import Lax799700.ThreeSat
    12import Lax799700.ZeroOneIP
    13import Lax366625.CountingProblems
    14import Lax366625.CountingClasses
    15import Lax366625.WitnessCounting
    16import Lax366625.CountingSat
    17import Lax280166.CountingSatVariants
    18import Lax280166.CountingCliques
    19import Lax280166.CountingDominatingSets
    20import Lax280166.CountingFeedbackSets
    21import Lax280166.CountingHamiltonCircuits
    22import Lax280166.CountingSetFamilies
    23import Lax280166.CountingKnapsacks
    24import Lax280166.CountingSteinerTrees
    25
    26/-!
    27---
    28title: The values of #3SAT, #1-in-SAT
    29type: lemma
    30---
    31The number counted by each of #3SAT, #1-in-SAT is invariant under isomorphism of
    32instances, so the value of each problem on an instance is the number it
    33counts.
    34-/
    35
    36namespace Lax280166.SatVariantsValues
    37
    38open FirstOrder FirstOrder.Language
    39open Lax904597.Problems Lax904597.Sat
    40open Lax366625.CountingProblems Lax366625.CountingClasses Lax366625.WitnessCounting
    41 Lax366625.CountingSat
    42open Lax799700.ThreeSat Lax799700.SetFamily
    43open Lax280166.CountingSatVariants Lax280166.CountingCliques Lax280166.CountingDominatingSets
    44 Lax280166.CountingFeedbackSets Lax280166.CountingHamiltonCircuits Lax280166.CountingSetFamilies
    45 Lax280166.CountingKnapsacks Lax280166.CountingSteinerTrees
    46
    47/-- The number counted by #3SAT is isomorphism-invariant. -/
    48axiom sharpThreeSat_count_iso :
    49 ∀ {A B : Type} [Lax904597.Sat.sat.Structure A] [Lax904597.Sat.sat.Structure B],
    50 (A ≃[Lax904597.Sat.sat] B) → Nat.card {ν : A → Prop // WidthAtMostThree A ∧ SatModel A ν} =
    51 Nat.card {ν : B → Prop // WidthAtMostThree B ∧ SatModel B ν}
    52
    53/-- The value of #3SAT is the number it counts. -/
    54axiom sharpThreeSat_eq :
    55 ∀ (A : Type) [Lax904597.Sat.sat.Structure A], SharpThreeSAT A = Nat.card {ν : A → Prop //
    56 WidthAtMostThree A ∧ SatModel A ν}
    57
    58/-- The number counted by #1-in-SAT is isomorphism-invariant. -/
    59axiom sharpOneInSat_count_iso :
    60 ∀ {A B : Type} [Lax904597.Sat.sat.Structure A] [Lax904597.Sat.sat.Structure B],
    61 (A ≃[Lax904597.Sat.sat] B) → Nat.card {ν : A → Prop // OneInModel A ν} = Nat.card {ν : B → Prop
    62 // OneInModel B ν}
    63
    64/-- The value of #1-in-SAT is the number it counts. -/
    65axiom sharpOneInSat_eq :
    66 ∀ (A : Type) [Lax904597.Sat.sat.Structure A], SharpOneInSAT A = Nat.card {ν : A → Prop //
    67 OneInModel A ν}
    68
    69end Lax280166.SatVariantsValues
    70
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