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The values of #Steiner Tree

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

proven

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    Natural Language Statement

    Lemma

    The number counted by #Steiner Tree is invariant under isomorphism of instances, so the value of the 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 2 statements. Each proof establishes one of them relative to its assumptions.

    1 sharpSteinerTree_count_iso proven

    2 sharpSteinerTree_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 #Steiner Tree
    29type: lemma
    30---
    31The number counted by #Steiner Tree is invariant under isomorphism of instances,
    32so the value of the problem on an instance is the number it counts.
    33-/
    34
    35namespace Lax280166.SteinerTreesValues
    36
    37open FirstOrder FirstOrder.Language
    38open Lax904597.Problems Lax904597.Sat
    39open Lax366625.CountingProblems Lax366625.CountingClasses Lax366625.WitnessCounting
    40 Lax366625.CountingSat
    41open Lax799700.ThreeSat Lax799700.SetFamily
    42open Lax280166.CountingSatVariants Lax280166.CountingCliques Lax280166.CountingDominatingSets
    43 Lax280166.CountingFeedbackSets Lax280166.CountingHamiltonCircuits Lax280166.CountingSetFamilies
    44 Lax280166.CountingKnapsacks Lax280166.CountingSteinerTrees
    45
    46/-- The number counted by #Steiner Tree is isomorphism-invariant. -/
    47axiom sharpSteinerTree_count_iso :
    48 ∀ {A B : Type} [Lax799700.Steiner.steinerGraph.Structure A]
    49 [Lax799700.Steiner.steinerGraph.Structure B],
    50 (A ≃[Lax799700.Steiner.steinerGraph] B) → Nat.card {S : A → Prop // SteinerOfSize A S} =
    51 Nat.card {S : B → Prop // SteinerOfSize B S}
    52
    53/-- The value of #Steiner Tree is the number it counts. -/
    54axiom sharpSteinerTree_eq :
    55 ∀ (A : Type) [Lax799700.Steiner.steinerGraph.Structure A], SharpSteinerTree A = Nat.card {S : A →
    56 Prop // SteinerOfSize A S}
    57
    58end Lax280166.SteinerTreesValues
    59
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