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(.module:
[lux #*
[control
[monad (#+ do)]]
data/text/format
["." function]
[math
["r" random]]
["_" test (#+ Test)]]
{1
["." / (#+ Apply)]}
[//
[functor (#+ Injection Comparison)]])
(def: (identity injection comparison (^open "_/."))
(All [f] (-> (Injection f) (Comparison f) (Apply f) Test))
(do r.monad
[sample (:: @ map injection r.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 r.monad
[sample r.nat
increase (:: @ map n/+ r.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 r.monad
[sample r.nat
increase (:: @ map n/+ r.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 r.monad
[sample r.nat
increase (:: @ map n/+ r.nat)
decrease (:: @ map n/- r.nat)]
(_.test "Composition."
((comparison n/=)
(_$ _/apply
(injection function.compose)
(injection increase)
(injection decrease)
(injection sample))
($_ _/apply
(injection increase)
(injection decrease)
(injection sample))))))
(def: #export (laws injection comparison apply)
(All [f] (-> (Injection f) (Comparison f) (Apply f) Test))
(_.context (%name (name-of /.Apply))
($_ _.and
(..identity injection comparison apply)
(..homomorphism injection comparison apply)
(..interchange injection comparison apply)
(..composition injection comparison apply)
)))
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