Exponentiation

$\newcommand{\hyp}[2]{\stackrel{\tiny\text{hyp}}{#2}}$ $\newcommand{\let}[2]{\stackrel{\tiny\text{let}}{#2}}$ $\newcommand{\a}{a}$ $\newcommand{\b}{b}$ $\newcommand{\c}{c}$ $\newcommand{\p}{p}$ $\newcommand{\q}{q}$ $\newcommand{\r}{r}$ $\newcommand{\s}{s}$ $\newcommand{\x}{x}$ $\newcommand{\y}{y}$ $\newcommand{\z}{z}$ $\newcommand{\bool}{\mathbb{B}}$ $\newcommand{\btrue}{\mathsf{true}}$ $\newcommand{\bfalse}{\mathsf{false}}$ $\newcommand{\bnot}{\mathsf{not}}$ $\newcommand{\band}{\mathbin{\mathsf{and}}}$ $\newcommand{\bor}{\mathbin{\mathsf{or}}}$ $\newcommand{\bimpl}{\mathbin{\mathsf{impl}}}$ $\newcommand{\beq}{\mathsf{eq}}$ $\newcommand{\bif}[3]{\mathsf{if}\left(#1,#2,#3\right)}$ $\newcommand{\id}{\mathsf{id}}$ $\newcommand{\compose}{\mathsf{compose}}$ $\newcommand{\const}{\mathsf{const}}$ $\newcommand{\apply}{\mathsf{apply}}$ $\newcommand{\flip}{\mathsf{flip}}$ $\newcommand{\flipB}{\mathsf{flip2}}$ $\newcommand{\flipC}{\mathsf{flip3}}$ $\newcommand{\flipD}{\mathsf{flip4}}$ $\newcommand{\flipE}{\mathsf{flip5}}$ $\newcommand{\clone}{\mathsf{clone}}$ $\newcommand{\cloneC}{\mathsf{clone3}}$ $\newcommand{\composeAonB}{\mathsf{compose1on2}}$ $\newcommand{\composeAonC}{\mathsf{compose1on3}}$ $\newcommand{\composeAonD}{\mathsf{compose1on4}}$ $\newcommand{\composeBonA}{\mathsf{compose2on1}}$ $\newcommand{\composeBonB}{\mathsf{compose2on2}}$ $\newcommand{\composeConA}{\mathsf{compose3on1}}$ $\newcommand{\ptrue}{\mathsf{ptrue}}$ $\newcommand{\pfalse}{\mathsf{pfalse}}$ $\newcommand{\pnot}{\mathsf{pnot}}$ $\newcommand{\pand}{\mathbin{\mathsf{pand}}}$ $\newcommand{\por}{\mathbin{\mathsf{por}}}$ $\newcommand{\pimpl}{\mathbin{\mathsf{pimpl}}}$ $\newcommand{\fst}{\mathsf{first}}$ $\newcommand{\snd}{\mathsf{second}}$ $\newcommand{\dup}{\mathsf{dup}}$ $\newcommand{\tup}{\mathsf{tup}}$ $\newcommand{\tSwap}{\mathsf{swap}_{\times}}$ $\newcommand{\tPair}{\mathsf{pair}_{\times}}$ $\newcommand{\tAssocL}{\mathsf{assocL}_{\times}}$ $\newcommand{\tAssocR}{\mathsf{assocR}_{\times}}$ $\newcommand{\tupL}{\mathsf{tupL}}$ $\newcommand{\tupR}{\mathsf{tupR}}$ $\newcommand{\uncurry}{\mathsf{uncurry}}$ $\newcommand{\curry}{\mathsf{curry}}$ $\newcommand{\lft}{\mathsf{left}}$ $\newcommand{\rgt}{\mathsf{right}}$ $\newcommand{\either}{\mathsf{either}}$ $\newcommand{\uSwap}{\mathsf{swap}_{+}}$ $\newcommand{\uPair}{\mathsf{pair}_{+}}$ $\newcommand{\uAssocL}{\mathsf{assocL}_{+}}$ $\newcommand{\uAssocR}{\mathsf{assocR}_{+}}$ $\newcommand{\isLft}{\mathsf{isLeft}}$ $\newcommand{\isRgt}{\mathsf{isRight}}$ $\newcommand{\nats}{\mathbb{N}}$ $\newcommand{\zero}{\mathsf{0}}$ $\newcommand{\next}{\mathsf{next}}$ $\newcommand{\unnext}{\mathsf{unnext}}$ $\newcommand{\iszero}{\mathsf{isZero}}$ $\newcommand{\prev}{\mathsf{prev}}$ $\newcommand{\natrec}{\mathsf{natrec}}$ $\newcommand{\simprec}{\mathsf{simprec}}$ $\newcommand{\bailrec}{\mathsf{bailrec}}$ $\newcommand{\mutrec}{\mathsf{mutrec}}$ $\newcommand{\dnatrec}{\mathsf{dnatrec}}$ $\newcommand{\normrec}{\mathsf{normrec}}$ $\newcommand{\findsmallest}{\mathsf{findsmallest}}$ $\newcommand{\mnormrec}{\mathsf{mnormrec}}$ $\newcommand{\nequal}{\mathsf{eq}}$ $\newcommand{\nplus}{\mathsf{plus}}$ $\newcommand{\ntimes}{\mathsf{times}}$ $\newcommand{\nleq}{\mathsf{leq}}$ $\newcommand{\ngeq}{\mathsf{geq}}$ $\newcommand{\nminus}{\mathsf{minus}}$ $\newcommand{\nmax}{\mathsf{max}}$ $\newcommand{\nmin}{\mathsf{min}}$ $\newcommand{\ndivalg}{\mathsf{divalg}}$ $\newcommand{\nquo}{\mathsf{quo}}$ $\newcommand{\nrem}{\mathsf{rem}}$ $\newcommand{\ndiv}{\mathsf{div}}$ $\newcommand{\ngcd}{\mathsf{gcd}}$ $\newcommand{\ncoprime}{\mathsf{coprime}}$ $\newcommand{\nlcm}{\mathsf{lcm}}$ $\newcommand{\nmindiv}{\mathsf{mindiv}}$ $\newcommand{\nisprime}{\mathsf{isPrime}}$ $\newcommand{\npower}{\mathsf{power}}$ $\newcommand{\nchoose}{\mathsf{choose}}$ $\newcommand{\dom}{\mathsf{dom}}$ $\newcommand{\cod}{\mathsf{cod}}$ $\newcommand{\lists}[1]{\mathsf{List}(#1)}$ $\newcommand{\nil}{\mathsf{nil}}$ $\newcommand{\cons}{\mathsf{cons}}$ $\newcommand{\uncons}{\mathsf{uncons}}$ $\newcommand{\isnil}{\mathsf{isNil}}$ $\newcommand{\tail}{\mathsf{tail}}$ $\newcommand{\head}{\mathsf{head}}$ $\newcommand{\foldr}{\mathsf{foldr}}$ $\newcommand{\foldl}{\mathsf{foldl}}$ $\newcommand{\tacunfoldN}{\mathsf{tacunfoldN}}$ $\newcommand{\unfoldN}{\mathsf{unfoldN}}$ $\newcommand{\dfoldr}{\mathsf{dfoldr}}$ $\newcommand{\cfoldr}{\mathsf{cfoldr}}$ $\newcommand{\bfoldr}{\mathsf{bfoldr}}$ $\newcommand{\dbfoldr}{\mathsf{dbfoldr}}$ $\newcommand{\lfoldr}{\mathsf{lfoldr}}$ $\newcommand{\snoc}{\mathsf{snoc}}$ $\newcommand{\revcat}{\mathsf{revcat}}$ $\newcommand{\rev}{\mathsf{rev}}$ $\newcommand{\last}{\mathsf{last}}$ $\newcommand{\cat}{\mathsf{cat}}$ $\newcommand{\addlength}{\mathsf{addlength}}$ $\newcommand{\length}{\mathsf{length}}$ $\newcommand{\head}{\mathsf{head}}$ $\newcommand{\at}{\mathsf{at}}$ $\newcommand{\map}{\mathsf{map}}$ $\newcommand{\range}{\mathsf{range}}$ $\newcommand{\zip}{\mathsf{zip}}$ $\newcommand{\zipPad}{\mathsf{zipPad}}$ $\newcommand{\unzip}{\mathsf{unzip}}$ $\newcommand{\prefix}{\mathsf{prefix}}$ $\newcommand{\suffix}{\mathsf{suffix}}$ $\newcommand{\lcp}{\mathsf{lcp}}$ $\newcommand{\lcs}{\mathsf{lcs}}$ $\newcommand{\all}{\mathsf{all}}$ $\newcommand{\any}{\mathsf{any}}$ $\newcommand{\tails}{\mathsf{tails}}$ $\newcommand{\inits}{\mathsf{inits}}$ $\newcommand{\filter}{\mathsf{filter}}$ $\newcommand{\elt}{\mathsf{elt}}$ $\newcommand{\addcount}{\mathsf{addcount}}$ $\newcommand{\count}{\mathsf{count}}$ $\newcommand{\repeat}{\mathsf{repeat}}$ $\newcommand{\sublist}{\mathsf{sublist}}$ $\newcommand{\infix}{\mathsf{infix}}$ $\newcommand{\select}{\mathsf{select}}$ $\newcommand{\unique}{\mathsf{unique}}$ $\newcommand{\delete}{\mathsf{delete}}$ $\newcommand{\dedupeL}{\mathsf{dedupeL}}$ $\newcommand{\dedupeR}{\mathsf{dedupeR}}$ $\newcommand{\common}{\mathsf{common}}$ $\newcommand{\disjoint}{\mathsf{disjoint}}$ $\newcommand{\sublists}{\mathsf{sublists}}$ $\newcommand{\take}{\mathsf{take}}$ $\newcommand{\drop}{\mathsf{drop}}$ $\newcommand{\takeBut}{\mathsf{takeBut}}$ $\newcommand{\dropBut}{\mathsf{dropBut}}$ $\newcommand{\takeWhile}{\mathsf{takeWhile}}$ $\newcommand{\dropWhile}{\mathsf{dropWhile}}$

{-# LANGUAGE NoImplicitPrelude #-}
module Exponentiation
  ( power, _test_power, main_power
  ) where

import Testing
import Booleans
import NaturalNumbers
import SimpleRecursion
import Plus
import Times

power :: (Natural n) => n -> n -> n
power a b = simpleRec phi mu b a
  where
    phi _     = next zero
    mu  _ c d = times d c

_test_power_zero_right :: (Natural n, Equal n)
  => n -> Test (n -> Bool)
_test_power_zero_right _ =
  testName "power(a,zero) == next(zero)" $
  \a -> eq (power a zero) (next zero)


_test_power_next_right :: (Natural n, Equal n)
  => n -> Test (n -> n -> Bool)
_test_power_next_right _ =
  testName "power(a,next(b)) == times(power(a,b),a)" $
  \a b -> eq (power a (next b)) (times (power a b) a)

_test_power_one_right :: (Natural n, Equal n)
  => n -> Test (n -> Bool)
_test_power_one_right _ =
  testName "power(a,1) == a" $
  \a -> eq (power a (next zero)) a


_test_power_zero_left :: (Natural n, Equal n)
  => n -> Test (n -> Bool)
_test_power_zero_left _ =
  testName "power(zero,next(a)) == zero" $
  \a -> eq (power zero (next a)) zero


_test_power_one_left :: (Natural n, Equal n)
  => n -> Test (n -> Bool)
_test_power_one_left _ =
  testName "power(next(zero),a) == next(zero)" $
  \a -> eq (power (next zero) a) (next zero)

_test_power_plus_right :: (Natural n, Equal n)
  => n -> Test (n -> n -> n -> Bool)
_test_power_plus_right _ =
  testName "power(a,plus(b,c)) == times(power(a,b),power(a,c))" $
  \a b c -> eq (power a (plus b c)) (times (power a b) (power a c))


_test_power_times_right :: (Natural n, Equal n)
  => n -> Test (n -> n -> n -> Bool)
_test_power_times_right _ =
  testName "power(a,times(b,c)) == power(power(a,b),c)" $
  \a b c -> eq (power a (times b c)) (power (power a b) c)


_test_power_times_left :: (Natural n, Equal n)
  => n -> Test (n -> n -> n -> Bool)
_test_power_times_left _ =
  testName "power(times(a,b),c) == times(power(a,c),power(b,c))" $
  \a b c -> eq (power (times a b) c) (times (power a c) (power b c))

_test_power ::
  ( TypeName n, Natural n, Equal n, Arbitrary n, Show n
  ) => Int -> Int -> n -> IO ()
_test_power size cases n = do
  testLabel1 "power" n

  let args = testArgs size cases

  runTest args (_test_power_zero_right n)
  runTest args (_test_power_next_right n)
  runTest args (_test_power_one_right n)
  runTest args (_test_power_zero_left n)
  runTest args (_test_power_one_left n)
  runTest args (_test_power_plus_right n)
  runTest args (_test_power_times_right n)
  runTest args (_test_power_times_left n)

main_power :: IO ()
main_power = do
  _test_power 4 30 (zero :: Unary)