Lax Lean Archive

Spanning 2-uniform hypergraph

Lax683916.SpanningTwoUniformHypergraphs · concepts/Lax683916/SpanningTwoUniformHypergraphs.lean · lax-683916

definition

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Definition

    A spanning 2-uniform hypergraph on VV has all elements of VV as vertices and every hyperedge contains exactly two vertices.

    Concept map
    1 concept; 1 descendant hidden
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A

    Lean source view on GitHub

    1import Mathlib.Combinatorics.Hypergraph.Basic
    2
    3/-!
    4---
    5title: Spanning 2-uniform hypergraph
    6type: definition
    7---
    8A spanning 2-uniform hypergraph on `V` has all elements of `V` as vertices
    9and every hyperedge contains exactly two vertices.
    10
    11# Formalization notes
    12
    13Spanning is stated by equality with `Set.univ`, because `Hypergraph V` keeps
    14an explicit vertex set inside the ambient type. A separate looplessness field
    15would be derivable: every edge has cardinality two and therefore cannot be a
    16singleton.
    17-/
    18
    19namespace Lax683916.SpanningTwoUniformHypergraphs
    20
    21/-- A hypergraph with full vertex set and two-element edges. -/
    22structure SpanningTwoUniformHypergraph (V : Type*) where
    23 /-- The underlying hypergraph. -/
    24 hypergraph : Hypergraph V
    25 /-- Every element of the ambient type is a vertex. -/
    26 spanning : hypergraph.vertexSet = Set.univ
    27 /-- Every hyperedge contains exactly two vertices. -/
    28 twoUniform : ∀ e ∈ hypergraph.edgeSet, Set.ncard e = 2
    29
    30end Lax683916.SpanningTwoUniformHypergraphs
    31
    Formalization notes

    Spanning is stated by equality with Set.univSet.univ, because HypergraphVHypergraph V keeps an explicit vertex set inside the ambient type. A separate looplessness field would be derivable: every edge has cardinality two and therefore cannot be a singleton.

    Builds on

    none

    Used by
    From Mathlib

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…