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Definable flips preserve monadic dependence

Lax185574.FlipsMonadicallyDependent · concepts/Lax185574/FlipsMonadicallyDependent.lean · lax-185574

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    Natural Language Statement

    Observation

    Let 𝒞 be a monadically dependent graph class and s ∈ ℕ. The class consisting of all s-definable flips of graphs from 𝒞 is monadically dependent.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

    In the paper

    • page 20 of this submission's paper

    Lean source view on GitHub

    1import Lax185574.DefinableFlips
    2import Lax185574.MonadicDependence
    3
    4/-!
    5---
    6title: Definable flips preserve monadic dependence
    7type: observation
    8---
    9Let 𝒞 be a monadically dependent graph class and s ∈ ℕ. The class
    10consisting of all s-definable flips of graphs from 𝒞 is monadically
    11dependent.
    12
    13# Formalization notes
    14
    15The paper allows infinite graphs in this part. Here the graphs are
    16those of a `GraphClass`, which are finite, as in every use.
    17
    18A flip H of a graph G ∈ 𝒞 is a `SimpleGraph` on the vertices of G, as
    19are all flips in this submission. The class contains it as a graph,
    20`Graph.ofSimpleGraph H`, which has the same vertices and adjacency.
    21Monadic dependence is that of `Lax185574.MonadicDependence`, by
    22transductions.
    23-/
    24
    25namespace Lax185574.FlipsMonadicallyDependent
    26
    27open Lax185574.Graphs Lax185574.MonadicDependence Lax185574.DefinableFlips
    28
    29/-- The class of all `s`-definable flips of graphs from `C`. -/
    30def definableFlipClass (C : GraphClass) (s : ℕ) : GraphClass :=
    31 {H | ∃ G ∈ C, ∃ F : SimpleGraph (Fin G.n),
    32 IsDefinableFlip G.toSimpleGraph s F ∧ H = Graph.ofSimpleGraph F}
    33
    34/-- **Definable flips preserve monadic dependence.** The `s`-definable
    35flips of the graphs of a monadically dependent class form a monadically
    36dependent class. -/
    37axiom monadicallyDependent_definableFlipClass (C : GraphClass)
    38 (hC : MonadicallyDependent C) (s : ℕ) :
    39 MonadicallyDependent (definableFlipClass C s)
    40
    41end Lax185574.FlipsMonadicallyDependent
    42
    Show Proof
    Formalization notes

    The paper allows infinite graphs in this part. Here the graphs are those of a GraphClassGraphClass, which are finite, as in every use.

    A flip H of a graph G ∈ 𝒞 is a SimpleGraphSimpleGraph on the vertices of G, as are all flips in this submission. The class contains it as a graph, Graph.ofSimpleGraphHGraph.ofSimpleGraph H, which has the same vertices and adjacency. Monadic dependence is that of Lax185574.MonadicDependenceLax185574.MonadicDependence, by transductions.

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