modal/examples/arithmetic2.modal

<> (-- ?x) ()
-- ( little endian binary integers )

-- ( constants )
<> zero ((0 nil))
<> one ((1 nil))
<> two ((0 (1 nil)))
<> three ((1 (1 nil)))
<> ten ((0 (1 (0 (1 nil)))))

-- ( decimal digit to binary )
<> (binary 0) ((0 nil))
<> (binary 1) ((1 nil))
<> (binary 2) ((0 (1 nil)))
<> (binary 3) ((1 (1 nil)))
<> (binary 4) ((0 (0 (1 nil))))
<> (binary 5) ((1 (0 (1 nil))))
<> (binary 6) ((0 (1 (1 nil))))
<> (binary 7) ((1 (1 (1 nil))))
<> (binary 8) ((0 (0 (0 (1 nil)))))
<> (binary 9) ((1 (0 (0 (1 nil)))))

-- ( binary to decimal digit )
<> (decimal (0 nil)) (0)
<> (decimal (1 nil)) (1)
<> (decimal (0 (1 nil))) (2)
<> (decimal (1 (1 nil))) (3)
<> (decimal (0 (0 (1 nil)))) (4)
<> (decimal (1 (0 (1 nil)))) (5)
<> (decimal (0 (1 (1 nil)))) (6)
<> (decimal (1 (1 (1 nil)))) (7)
<> (decimal (0 (0 (0 (1 nil))))) (8)
<> (decimal (1 (0 (0 (1 nil))))) (9)

-- create nil-terminated list
<> (nilify (?h)) ((?h nil))
<> (nilify (?h ?t)) ((?h nilify ?t))

-- reverse nil-terminated list
<> (reverse ?x) (reverse' nil ?x)
<> (reverse' ?a nil) (?a)
<> (reverse' ?a (?h ?t)) (reverse' (?h ?a) ?t)

-- ( normalize, remove trailing zeros )
-- ( currently zero is (0 nil) though arguably it could be nil )
-- ( that change would require auditing our rules )
<> (normalize (?h ?t)) ((?h normalize' nil ?t))
<> (normalize' ?s nil) (nil)
<> (normalize' ?s (0 ?t)) (normalize' (0 ?s) ?t)
<> (normalize' nil (1 ?t)) ((1 normalize' nil ?t))
<> (normalize' (0 ?s) (1 ?t)) ((0 normalize' ?s (1 ?t)))

-- ( to integer )
<> ((int ?*)) ((sum f (one) g reverse nilify (?*)))
<> (g nil) (nil)
<> (g (?h ?t)) ((binary ?h g ?t))
<> (f (?u) nil) (nil) 
<> (f (?u) (?h ?t)) (((mul ?h ?u) f ((mul ?u ten)) ?t))

-- ( to string: TODO, need division for this one )

-- ( comparison operartions )
<> ((cmp ?x ?y)) ((cmpc #eq ?x ?y))
<> ((cmpc ?e nil nil)) (?e)
<> ((cmpc ?e (1 ?x) nil)) (#gt)
<> ((cmpc ?e (0 ?x) nil)) ((cmpc ?e ?x nil))
<> ((cmpc ?e nil (1 ?y))) (#lt)
<> ((cmpc ?e nil (0 ?y))) ((cmpc ?e nil ?y))
<> ((cmpc ?e (0 ?x) (0 ?y))) ((cmpc ?e ?x ?y))
<> ((cmpc ?e (1 ?x) (0 ?y))) ((cmpc #gt ?x ?y))
<> ((cmpc ?e (0 ?x) (1 ?y))) ((cmpc #lt ?x ?y))
<> ((cmpc ?e (1 ?x) (1 ?y))) ((cmpc ?e ?x ?y))

-- ( addition )
<> ((add ?x ?y)) (addc 0 ?x ?y)
<> (addc 0 nil nil) (nil)
<> (addc 1 nil nil) ((1 nil))
<> (addc ?c ?x nil) (addc ?c ?x (0 nil))
<> (addc ?c nil ?y) (addc ?c (0 nil) ?y)
<> (addc 0 (0 ?x) (0 ?y)) ((0 addc 0 ?x ?y))
<> (addc 0 (0 ?x) (1 ?y)) ((1 addc 0 ?x ?y))
<> (addc 0 (1 ?x) (0 ?y)) ((1 addc 0 ?x ?y))
<> (addc 0 (1 ?x) (1 ?y)) ((0 addc 1 ?x ?y))
<> (addc 1 (0 ?x) (0 ?y)) ((1 addc 0 ?x ?y))
<> (addc 1 (0 ?x) (1 ?y)) ((0 addc 1 ?x ?y))
<> (addc 1 (1 ?x) (0 ?y)) ((0 addc 1 ?x ?y))
<> (addc 1 (1 ?x) (1 ?y)) ((1 addc 1 ?x ?y))

-- ( summation )
<> ((sum nil)) ((0 nil))
<> ((sum (?a nil))) (?a)
<> ((sum (?a (?b ?c)))) ((sum ((add ?a ?b) ?c)))

-- ( multiplication )
<> ((mul ?x ?y)) (mulc nil ?x ?y)
<> (mulc ?t nil ?y) ((sum ?t))
<> (mulc ?t (0 ?x) ?y) (mulc ?t ?x (0 ?y))
<> (mulc ?t (1 ?x) ?y) (mulc (?y ?t) ?x (0 ?y))

-- ( subtraction )
<> ((sub ?x ?y)) (normalize subc 0 ?x ?y)
<> (subc 0 nil nil) (nil)
<> (subc 1 nil nil) (#err)
<> (subc 0 ?x nil) (?x)
<> (subc 1 ?x nil) (subc 1 ?x (0 nil))
<> (subc ?c nil ?y) (subc ?c (0 nil) ?y)
<> (subc 0 (0 ?x) (0 ?y)) ((0 subc 0 ?x ?y))
<> (subc 0 (0 ?x) (1 ?y)) ((1 subc 1 ?x ?y))
<> (subc 0 (1 ?x) (0 ?y)) ((1 subc 0 ?x ?y))
<> (subc 0 (1 ?x) (1 ?y)) ((0 subc 0 ?x ?y))
<> (subc 1 (0 ?x) (0 ?y)) ((1 subc 1 ?x ?y))
<> (subc 1 (0 ?x) (1 ?y)) ((0 subc 1 ?x ?y))
<> (subc 1 (1 ?x) (0 ?y)) ((0 subc 0 ?x ?y))
<> (subc 1 (1 ?x) (1 ?y)) ((1 subc 1 ?x ?y))

-- ( dec )
<> (dec (0 nil)) (#err)
<> (dec ?x) (normalize dec' ?x)
<> (dec' (0 ?t)) ((1 dec' ?t))
<> (dec' (1 ?t)) ((0 ?t))

-- ( inc )
<> ((inc nil)) ((1 nil))
<> ((inc (0 ?t))) ((1 ?t))
<> ((inc (1 ?t))) ((0 (inc ?t)))

-- ( left shift; lshift x b means x<<b )
<> ((lshift ?x (0 nil))) (?x)
<> ((lshift ?x (1 nil))) ((0 ?x))
<> ((lshift ?x (?h (?a ?b)))) ((lshift (0 ?x) dec (?h (?a ?b))))

-- ( divmod, i.e. quotient and remainder )
<> ((divmod ?x ?y)) ((divmod1 ?x ?y (cmp ?x ?y)))
<> ((divmod1 ?x ?y #lt)) (zero)
<> ((divmod1 ?x ?y #eq)) (one)
<> ((divmod1 ?x ?y #gt)) ((divmod2 ?x ?y zero (0 ?y)))
<> ((divmod2 ?x ?y ?s ?m)) ((divmod3 ?x ?y ?s ?m (cmp ?x ?m)))
<> ((divmod3 ?x ?y ?s ?m #lt)) ((divmod4 ?x ?y ?s zero))
<> ((divmod3 ?x ?y ?s ?m #eq)) ((divmod4 ?x ?y (inc ?s) zero))
<> ((divmod3 ?x ?y ?s ?m #gt)) ((divmod2 ?x ?y (inc ?s) (0 ?m)))
<> ((divmod4 ?x ?y (0 nil) ?d)) (((add ?d one) (sub ?x ?y)))
<> ((divmod4 ?x ?y ?s ?d)) ((divmod5 (sub ?x (lshift ?y ?s)) ?y dec ?s (add ?d (lshift one ?s))))
<> ((divmod5 (0 nil) ?y ?s ?d)) ((?d (0 nil)))
<> ((divmod5 ?x ?y ?s ?d)) ((divmod6 ?x ?y ?s ?d (cmp ?x (lshift ?y ?s))))
<> ((divmod6 ?x ?y (0 nil) ?d #lt)) ((?d ?x))
<> ((divmod6 ?x ?y ?s ?d #lt)) ((divmod5 ?x ?y dec ?s ?d))
<> ((divmod6 ?x ?y ?s ?d #eq)) ((divmod4 ?x ?y ?s ?d))
<> ((divmod6 ?x ?y ?s ?d #gt)) ((divmod4 ?x ?y ?s ?d))

-- ( floor divison )
<> ((div ?x ?y)) ((div' (divmod ?x ?y)))
<> ((div' (?q ?r))) (?q)

-- ( remainder )
<> ((mod ?x ?y)) ((mod' (divmod ?x ?y)))
<> ((mod' (?q ?r))) (?r)

(divmod (int 1234567) (int 1357))