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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 (\ ! each (function (_ diff)
(|>> (n.+ diff) _//in))
random.nat)]
(_.test "Left identity."
((comparison n.=)
(|> (injection sample) (_//each morphism) _//conjoint)
(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) (_//each _//in) _//conjoint)
(injection sample)))))
(def: (associativity injection comparison (^open "_//."))
(All (_ f) (-> (Injection f) (Comparison f) (/.Monad f) Test))
(do {! random.monad}
[sample random.nat
increase (\ ! each (function (_ diff)
(|>> (n.+ diff) _//in))
random.nat)
decrease (\ ! each (function (_ diff)
(|>> (n.- diff) _//in))
random.nat)]
(_.test "Associativity."
((comparison n.=)
(|> (injection sample) (_//each increase) _//conjoint (_//each decrease) _//conjoint)
(|> (injection sample) (_//each (|>> increase (_//each decrease) _//conjoint)) _//conjoint)))))
(def: .public (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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