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(.module:
[library
[lux #*
["_" test (#+ Test)]
[math
["." random]
[number
["n" nat]]]]]
[\\library
["." / (#+ do)]]
[//
[functor (#+ Injection Comparison)]])
(def: (left_identity injection comparison (^open "_//."))
(All [f] (-> (Injection f) (Comparison f) (/.Monad f) Test))
(do {! random.monad}
[sample random.nat
morphism (\ ! map (function (_ diff)
(|>> (n.+ diff) _//in))
random.nat)]
(_.test "Left identity."
((comparison n.=)
(|> (injection sample) (_//map morphism) _//join)
(morphism sample)))))
(def: (right_identity injection comparison (^open "_//."))
(All [f] (-> (Injection f) (Comparison f) (/.Monad f) Test))
(do random.monad
[sample random.nat]
(_.test "Right identity."
((comparison n.=)
(|> (injection sample) (_//map _//in) _//join)
(injection sample)))))
(def: (associativity injection comparison (^open "_//."))
(All [f] (-> (Injection f) (Comparison f) (/.Monad f) Test))
(do {! random.monad}
[sample random.nat
increase (\ ! map (function (_ diff)
(|>> (n.+ diff) _//in))
random.nat)
decrease (\ ! map (function (_ diff)
(|>> (n.- diff) _//in))
random.nat)]
(_.test "Associativity."
((comparison n.=)
(|> (injection sample) (_//map increase) _//join (_//map decrease) _//join)
(|> (injection sample) (_//map (|>> increase (_//map decrease) _//join)) _//join)))))
(def: #export (spec injection comparison monad)
(All [f] (-> (Injection f) (Comparison f) (/.Monad f) Test))
(<| (_.for [/.Monad])
($_ _.and
(..left_identity injection comparison monad)
(..right_identity injection comparison monad)
(..associativity injection comparison monad)
)))
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