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Splitting by a transversal gives a partition homogeneous modulo the pair relation

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

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    Natural Language Statement

    Lemma

    Fix a graph G and a partition 𝒫 of V(G), let T be a transversal of 𝒫, and R ≔ R(G, 𝒫). Then 𝒬 ≔ 𝒫 ∧ T is homogeneous modulo R, that is, for all X, Y ∈ 𝒬 either

    XY ∖ R ⊆ E(G) or XY ∖ R ⊆ E(Ḡ).

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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 34 of this submission's paper

    Lean source view on GitHub

    1import Lax185574.PairRelation
    2import Lax185574.TypeSplitting
    3import Lax185574.MergeSequences
    4
    5/-!
    6---
    7title: Splitting by a transversal gives a partition homogeneous modulo the pair relation
    8type: lemma
    9---
    10Fix a graph G and a partition 𝒫 of V(G), let T be a transversal of 𝒫,
    11and R ≔ R(G, 𝒫). Then 𝒬 ≔ 𝒫 ∧ T is *homogeneous modulo R*, that is,
    12for all X, Y ∈ 𝒬 either
    13
    14XY ∖ R ⊆ E(G) or XY ∖ R ⊆ E(Ḡ).
    15
    16# Formalization notes
    17
    18Homogeneity modulo R is `MergeSequences.HomogeneousModulo`, the fourth
    19condition of a merge sequence. The graph may be infinite.
    20-/
    21
    22namespace Lax185574.HomogeneousSplitting
    23
    24open Lax185574.Partitions Lax185574.PairRelation Lax185574.TypeSplitting
    25open Lax185574.MergeSequences
    26
    27universe z
    28
    29/-- **Homogeneity modulo the pair relation.** For a transversal `T` of
    30`𝒫`, the partition `𝒫 ∧ T` is homogeneous modulo `R(G, 𝒫)`. -/
    31axiom homogeneousModulo_split {V : Type z} (G : SimpleGraph V) (P : Setoid V) (T : Set V)
    32 (hT : IsTransversal P T) :
    33 HomogeneousModulo G (split G P T) (pairRelation G P)
    34
    35end Lax185574.HomogeneousSplitting
    36
    Show Proof
    Formalization notes

    Homogeneity modulo R is MergeSequences.HomogeneousModuloMergeSequences.HomogeneousModulo, the fourth condition of a merge sequence. The graph may be infinite.

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