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(.module:
[lux #*
[abstract
[monad (#+ do)]]
[data
[number
["n" nat]]]
[control
["." function]]
[math
["." random]]
["_" test (#+ Test)]]
{1
["." / (#+ Apply)]}
[//
[functor (#+ Injection Comparison)]])
(def: (identity injection comparison (^open "_@."))
(All [f] (-> (Injection f) (Comparison f) (Apply f) Test))
(do {@ random.monad}
[sample (:: @ map injection random.nat)]
(_.test "Identity."
((comparison n.=)
(_@apply (injection function.identity) sample)
sample))))
(def: (homomorphism injection comparison (^open "_@."))
(All [f] (-> (Injection f) (Comparison f) (Apply f) Test))
(do {@ random.monad}
[sample random.nat
increase (:: @ map n.+ random.nat)]
(_.test "Homomorphism."
((comparison n.=)
(_@apply (injection increase) (injection sample))
(injection (increase sample))))))
(def: (interchange injection comparison (^open "_@."))
(All [f] (-> (Injection f) (Comparison f) (Apply f) Test))
(do {@ random.monad}
[sample random.nat
increase (:: @ map n.+ random.nat)]
(_.test "Interchange."
((comparison n.=)
(_@apply (injection increase) (injection sample))
(_@apply (injection (function (_ f) (f sample))) (injection increase))))))
(def: (composition injection comparison (^open "_@."))
(All [f] (-> (Injection f) (Comparison f) (Apply f) Test))
(do {@ random.monad}
[sample random.nat
increase (:: @ map n.+ random.nat)
decrease (:: @ map n.- random.nat)]
(_.test "Composition."
((comparison n.=)
(_$ _@apply
(injection function.compose)
(injection increase)
(injection decrease)
(injection sample))
($_ _@apply
(injection increase)
(injection decrease)
(injection sample))))))
(def: #export (spec injection comparison apply)
(All [f] (-> (Injection f) (Comparison f) (Apply f) Test))
(_.with-cover [/.Apply]
($_ _.and
(..identity injection comparison apply)
(..homomorphism injection comparison apply)
(..interchange injection comparison apply)
(..composition injection comparison apply)
)))
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