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RelationBasics (E, R): trait
  % e1 \langle r \rangle e2 means e1 is related to e2 by r.
  introduces 
    __ \langle __ \rangle __: E, R, E -> Bool
    \bot, \top, I: -> R
    [__, __]: E, E -> R
    -__, __\inv: R -> R
    __ \U __: R, R -> R
  asserts
    R partitioned by __ \langle __ \rangle __
    forall e, e1, e2, e3, e4: E, r, r1, r2: R
      ~(e1 \langle \bot \rangle e2);
      e1 \langle \top \rangle e2;
      e1 \langle I \rangle e2 == e1 = e2;
      e1 \langle [e2, e3] \rangle e4 == e1 = e2 /\ e3 = e4;
      e1 \langle -r \rangle e2 == ~(e1 \langle r \rangle e2);
      e1 \langle r\inv \rangle e2 == e2 \langle r \rangle e1;
      e1 \langle r1 \U r2 \rangle e2 == e1 \langle r1 \rangle e2 \/ e1 \langle r2 \rangle e2
  implies
    AbelianMonoid (\bot for unit, \U for \circ, R for T),
    Involutive (__\inv, R),
    Involutive (-__, R)
    equations
      -\bot == \top;
      -\top == \bot;
      \bot\inv == \bot;
      \top\inv == \top
    converts \U, -__, __\inv
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