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binary_operation_name <is> equals(
'+', '-', '*', '/', '%', '|', '&', '**', '<<', '>>')
algebraic_class <is> (setof(Type.Tuple) &
attr('new', callable))
relation_class <is> (setof( setof(any*any) |
Type.Tuple))
relational_operator_name <is> equals(
'<', '<=', '>', '>=', '==', '!=', 'in', 'not in', 'is', 'is not')
s <is>s = lambda IS: N( R
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if 1:
abelian_group <in> setof(AA.group)
associative <in> setof(AA.binary_operation)
binary_operation <in> doc('''
A \emp{binary operation} $*$ on a set $S$ is a function $*: S \cross S \mapsto S$.
The element in $S$ assigned to $(x, y)$ is denoted $x*y$.
\citemh(p.21)
''') & LE.algebraic_class
boolean_algebra <in> LE.algebraic_class
commutative <in> LE.algebraic_class
distributive <in> setof(cprod(AA.binary_operation, AA.binary_operation))
distributive_1 <in> setof(cprod(AA.binary_operation, AA.binary_operation))
distributive_2 <in> setof(cprod(AA.binary_operation, AA.binary_operation))
field <in> LE.algebraic_class
group <in> (LE.algebraic_class & doc('''
''' ))
monoid <in> LE.algebraic_class
ring <in> (LE.algebraic_class,
attr('new', argnames('S', 'add', 'mul', 'neg', 'zero')))
semigroup <in> LE.algebraic_class
s <in>s = lambda IN:N( R
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reflexive <in> doc('x R x for every x in A',
AA.LE.relation_class)
symmetric <in> doc('x R y implies y R x, for all x, y in A',
AA.LE.relation_class)
transitive <in> doc('x R y, y R z implies x R z, for all x, y, z in A',
AA.LE.relation_class)
irreflexive <in> doc('not (x R y), for all x in A',
AA.LE.relation_class)
antisymmetric <in> doc('x R y, y R x implies x == y, for all x, y in A',
AA.LE.relation_class)
total_relation <in> doc('x R y or y R x, for all x, y in A',
AA.LE.relation_class)
equivalence_relation<in> doc('Reflexive, symmetric and transitive',
AA.LE.relation_class)
partial_order <in> doc('Reflexive, antisymmetric and transitive',
AA.LE.relation_class)
total_order <in> doc('Partial order and x R y or y R x, for all x, y in A',
AA.LE.relation_class)
lattice <in> attr('quadruple', doc('''Tuples (S, R, V, A), where:
S: set or convertible to set, i.e. 'setcastable'
R: relation operator on S
V: binary operator on S
A: binary operator on S
R, V and A are either operator symbols or functions.
(S, R) forms a partial order such that
every pair x, y of elements in S have a greatest
lower bound GLB and a least upper bound LUB.
The GLB is given by V(x, y) or x V y depending on if V is
a function or operator symbol. Likewise, ULB is given
by A(x, y) or x A y.
For example, these are lattice specifications:
(int, '<=', min, max)
(int, lambda x, y: x & y == x, '&', '|')
''', setof(tupleform(
('S', 'R', 'V', 'A'),
attr('S', SPLE.setcastable) &
expset('''attr('R', AA.LE.relational_operator_name | boolean<<(S, S)) &
attr('V', AA.LE.binary_operation_name | setcast(S)<<(S, S)) &
attr('A', AA.LE.binary_operation_name | setcast(S)<<(S, S))
''', 'S')
))))
s <in>s =lambda IN:N( R
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