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... https://en.wikipedia.org/wiki/Interval_(mathematics)
(.using
[library
[lux "*"]]
[//
[equivalence {"+" Equivalence}]
["[0]" order]
[enum {"+" Enum}]])
(type: .public (Interval a)
(Interface
(is (Enum a)
enum)
(is a
bottom)
(is a
top)))
(def: .public (between enum bottom top)
(All (_ a) (-> (Enum a) a a (Interval a)))
(implementation
(def: enum enum)
(def: bottom bottom)
(def: top top)))
(def: .public (singleton enum elem)
(All (_ a) (-> (Enum a) a (Interval a)))
(implementation
(def: enum enum)
(def: bottom elem)
(def: top elem)))
(template [<name> <comp>]
[(def: .public (<name> interval)
(All (_ a) (-> (Interval a) Bit))
(let [(open ",#[0]") interval]
(<comp> ,#bottom ,#top)))]
[inner? (order.> ,#order)]
[outer? ,#<]
[singleton? ,#=]
)
(def: .public (within? interval elem)
(All (_ a) (-> (Interval a) a Bit))
(let [(open ",#[0]") interval]
(cond (inner? interval)
(and (order.>= ,#order ,#bottom elem)
(order.<= ,#order ,#top elem))
(outer? interval)
(or (order.>= ,#order ,#bottom elem)
(order.<= ,#order ,#top elem))
... singleton
(and (,#= ,#bottom elem)
(,#= ,#top elem)))))
(template [<name> <limit>]
[(def: .public (<name> elem interval)
(All (_ a) (-> a (Interval a) Bit))
(let [(open "[0]") interval]
(= <limit> elem)))]
[starts_with? bottom]
[ends_with? top]
)
(def: .public (borders? interval elem)
(All (_ a) (-> (Interval a) a Bit))
(or (starts_with? elem interval)
(ends_with? elem interval)))
(implementation: .public (union left right)
(All (_ a) (-> (Interval a) (Interval a) (Interval a)))
(def: enum (the enum right))
(def: bottom (order.min (# right order) (# left bottom) (# right bottom)))
(def: top (order.max (# right order) (# left top) (# right top))))
(implementation: .public (intersection left right)
(All (_ a) (-> (Interval a) (Interval a) (Interval a)))
(def: enum (the enum right))
(def: bottom (order.max (# right order) (# left bottom) (# right bottom)))
(def: top (order.min (# right order) (# left top) (# right top))))
(implementation: .public (complement interval)
(All (_ a) (-> (Interval a) (Interval a)))
(def: enum (the enum interval))
(def: bottom (# interval succ (# interval top)))
(def: top (# interval pred (# interval bottom))))
(def: .public (precedes? reference sample)
(All (_ a) (-> (Interval a) (Interval a) Bit))
(let [(open "[0]") reference
limit (# reference bottom)]
(and (< limit (# sample bottom))
(< limit (# sample top)))))
(def: .public (succeeds? reference sample)
(All (_ a) (-> (Interval a) (Interval a) Bit))
(precedes? sample reference))
(template [<name> <comp>]
[(def: .public (<name> reference sample)
(All (_ a) (-> a (Interval a) Bit))
(let [(open ",#[0]") sample]
(and (<comp> reference ,#bottom)
(<comp> reference ,#top))))]
[before? ,#<]
[after? (order.> ,#order)]
)
(def: .public (meets? reference sample)
(All (_ a) (-> (Interval a) (Interval a) Bit))
(let [(open ",#[0]") reference
limit (# reference bottom)]
(and (,#= limit (# sample top))
(order.<= ,#order limit (# sample bottom)))))
(def: .public (touches? reference sample)
(All (_ a) (-> (Interval a) (Interval a) Bit))
(or (meets? reference sample)
(meets? sample reference)))
(template [<name> <eq_side> <ineq> <ineq_side>]
[(def: .public (<name> reference sample)
(All (_ a) (-> (Interval a) (Interval a) Bit))
(let [(open ",#[0]") reference]
(and (,#= (# reference <eq_side>)
(# sample <eq_side>))
(<ineq> ,#order
(# reference <ineq_side>)
(# sample <ineq_side>)))))]
[starts? ,#bottom order.<= ,#top]
[finishes? ,#top order.>= ,#bottom]
)
(implementation: .public equivalence
(All (_ a) (Equivalence (Interval a)))
(def: (= reference sample)
(let [(open ",#[0]") reference]
(and (,#= ,#bottom (# sample bottom))
(,#= ,#top (# sample top))))))
(def: .public (nested? reference sample)
(All (_ a) (-> (Interval a) (Interval a) Bit))
(cond (or (singleton? sample)
(and (inner? reference) (inner? sample))
(and (outer? reference) (outer? sample)))
(let [(open ",#[0]") reference]
(and (order.>= ,#order (# reference bottom) (# sample bottom))
(order.<= ,#order (# reference top) (# sample top))))
(or (singleton? reference)
(and (inner? reference) (outer? sample)))
#0
... (and (outer? reference) (inner? sample))
(let [(open ",#[0]") reference]
(or (and (order.>= ,#order (# reference bottom) (# sample bottom))
(order.> ,#order (# reference bottom) (# sample top)))
(and (,#< (# reference top) (# sample bottom))
(order.<= ,#order (# reference top) (# sample top)))))
))
(def: .public (overlaps? reference sample)
(All (_ a) (-> (Interval a) (Interval a) Bit))
(let [(open ",#[0]") reference]
(and (not (# ..equivalence = reference sample))
(cond (singleton? sample)
#0
(singleton? reference)
(nested? sample reference)
(or (and (inner? sample) (outer? reference))
(and (outer? sample) (inner? reference)))
(or (order.>= ,#order (# reference bottom) (# sample top))
(order.<= ,#order (# reference top) (# sample bottom)))
... both inner
(inner? sample)
(inner? (intersection reference sample))
... both outer
(not (nested? reference sample))
))))
|
