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Exactly one of four relations for graph parameters

Lax825442.RelationClassification · concepts/Lax825442/RelationClassification.lean · lax-825442

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    Natural Language Statement

    Theorem

    For every pair of graph parameters, exactly one of four relations holds: the first strictly bounds the second, the second strictly bounds the first, they are functionally equivalent, or they are incomparable.

    The statement asserts that at least one case holds and explicitly excludes all six pairs of simultaneous cases. Its proof uses classical excluded middle for functional boundedness in each direction; it does not provide an algorithm for deciding which case holds.

    Concept map
    7 concepts
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax825442.StrictlyBounds
    2import Lax825442.Equivalent
    3import Lax825442.Incomparable
    4
    5/-!
    6---
    7title: Exactly one of four relations for graph parameters
    8type: theorem
    9---
    10For every pair of graph parameters, exactly one of four relations holds:
    11the first strictly bounds the second, the second strictly bounds the first,
    12they are functionally equivalent, or they are incomparable.
    13
    14The statement asserts that at least one case holds and explicitly excludes
    15all six pairs of simultaneous cases. Its proof uses classical excluded
    16middle for functional boundedness in each direction; it does not provide an
    17algorithm for deciding which case holds.
    18-/
    19
    20namespace Lax825442.RelationClassification
    21open Lax153141.GraphParameters
    22open Lax825442.StrictlyBounds Lax825442.Equivalent Lax825442.Incomparable
    23
    24/-- Exactly one of the four cases holds: exhaustiveness and pairwise exclusiveness. -/
    25axiom relationClassification (p q : GraphParam) :
    26 (StrictlyBounds p q ∨ StrictlyBounds q p ∨ Equivalent p q ∨ Incomparable p q) ∧
    27 ¬ (StrictlyBounds p q ∧ StrictlyBounds q p) ∧
    28 ¬ (StrictlyBounds p q ∧ Equivalent p q) ∧
    29 ¬ (StrictlyBounds p q ∧ Incomparable p q) ∧
    30 ¬ (StrictlyBounds q p ∧ Equivalent p q) ∧
    31 ¬ (StrictlyBounds q p ∧ Incomparable p q) ∧
    32 ¬ (Equivalent p q ∧ Incomparable p q)
    33
    34end Lax825442.RelationClassification
    35
    36
    Show Proof
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