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Holes are symmetric, loopless, and triangle-free

Lax342547.HoleTriangleFree · concepts/Lax342547/HoleTriangleFree.lean · lax-342547

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    Natural Language Statement

    Theorem

    The hole relation of Definition 2.1 is symmetric, has no loops, and has no triangle. This is Lemma 2.2: the six functional identities around a triangle cancel in characteristic two, leaving 0=10=1.

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

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    Lean source view on GitHub

    1import Lax342547.HoleRelation
    2
    3/-!
    4---
    5title: Holes are symmetric, loopless, and triangle-free
    6type: theorem
    7---
    8The hole relation of Definition 2.1 is symmetric, has no loops, and has
    9no triangle. This is Lemma 2.2: the six functional identities around a
    10triangle cancel in characteristic two, leaving 0=10=1.
    11-/
    12
    13namespace Lax342547.HoleTriangleFree
    14
    15open HoleRelation
    16
    17universe u v w
    18
    19axiom hole_properties
    20 {Ω : Type u} {X : Type v} {V : Type w}
    21 [AddCommGroup X] [Module (ZMod 2) X]
    22 [AddCommGroup V] [Module (ZMod 2) V]
    23 (D : HoleData Ω X V) :
    24 (∀ ⦃i j⦄, Hole D i j → Hole D j i) ∧ (∀ i, ¬ Hole D i i) ∧
    25 ∀ i j k, Hole D i j → Hole D j k → ¬ Hole D k i
    26
    27end Lax342547.HoleTriangleFree
    28
    Show Proof
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