modal/examples/multiply.modal

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

-- ( constants )
<> zero ((0 ()))
<> one ((1 ()))
<> ten ((0 (1 (0 (1 ())))))

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

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

-- reverse ()-terminated list
<> (reverse ?x) (reverse1 () ?x)
<> (reverse1 ?a ()) (?a)
<> (reverse1 ?a (?h ?t)) (reverse1 (?h ?a) ?t)

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

-- ( to binary str )
-- ( <> ((bstr ?x)) (emit force (0 (b  ?x))) )
-- ( <> ((bstr ?x)) ((bstr1 () ?x)) )
<> ((bstr ?x)) ((bstr1 force ?x ()))
<> ((bstr1 force/r () ?a)) (emit force/r (0 (b ?a)))
<> ((bstr1 force/r (?h ?t) ?a)) ((bstr1 force/r ?t (?h ?a)))

-- ( to string: TODO, need division for this one )
<> ((str ?x)) ((str1 ?x ()))
<> ((str1 (0 ()) ?a)) (emit force ?a)
<> ((str1 (?h ?t) ?a)) ((str2 (divmod (?h ?t) ten) ?a))
<> ((str2 (?q ?r) ?a)) ((str1 ?q ((decimal ?r) ?a)))

-- ( force a list to evaluate to digits/letters )
<> ((?h force/r ?t)) (force/r (?h ?t))
<> (force ()) (force/r ())
<> (force (0 ?t)) ((0 force ?t))
<> (force (1 ?t)) ((1 force ?t))
<> (force (2 ?t)) ((2 force ?t))
<> (force (3 ?t)) ((3 force ?t))
<> (force (4 ?t)) ((4 force ?t))
<> (force (5 ?t)) ((5 force ?t))
<> (force (6 ?t)) ((6 force ?t))
<> (force (7 ?t)) ((7 force ?t))
<> (force (8 ?t)) ((8 force ?t))
<> (force (9 ?t)) ((9 force ?t))
<> (force (a ?t)) ((a force ?t))
<> (force (b ?t)) ((b force ?t))
<> (force (c ?t)) ((c force ?t))
<> (force (d ?t)) ((d force ?t))
<> (force (e ?t)) ((e force ?t))
<> (force (f ?t)) ((f force ?t))
<> (force (x ?t)) ((x force ?t))

-- ( emit )
<> (emit force/r ?*) (?*)

-- ( comparison operartions )
<> ((cmp ?x ?y)) ((cmpc #eq ?x ?y))
<> ((cmpc ?e () ())) (?e)
<> ((cmpc ?e (1 ?x) ())) (#gt)
<> ((cmpc ?e (0 ?x) ())) ((cmpc ?e ?x ()))
<> ((cmpc ?e () (1 ?y))) (#lt)
<> ((cmpc ?e () (0 ?y))) ((cmpc ?e () ?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 () ())) (())
<> ((addc 1 () ())) ((1 ()))
-- ( <> ((addc ?c ?x ())) ((addc ?c ?x (0 ()))) )
-- ( <> ((addc ?c () ?y)) ((addc ?c (0 ()) ?y)) )
<> ((addc 0 ?x ())) (?x)
<> ((addc 0 () ?y)) (?y)
<> ((addc 1 ?x ())) ((addc 1 ?x (0 ())))
<> ((addc 1 () ?y)) ((addc 1 (0 ()) ?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 ())) ((0 ()))
<> ((sum (?a ()))) (?a)
<> ((sum (?a (?b ?c)))) ((sum ((add ?a ?b) ?c)))

-- ( multiplication )
<> ((mul ?x ?y)) ((mulc () ?x ?y))
<> ((mulc ?t () ?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)) (sub1 0 ?x ?y ())
<> (sub1 0 () () ?s) (())
<> (sub1 1 () () ?s) (#err)
<> (sub1 ?c ?x () ?s) (sub1 ?c ?x (0 ()) ?s)
<> (sub1 ?c () ?y ?s) (sub1 ?c (0 ()) ?y ?s)
<> (sub1 0 (0 ?x) (0 ?y) ?s) (sub1 0 ?x ?y (0 ?s))
<> (sub1 0 (0 ?x) (1 ?y) ?s) (sub2 1 ?x ?y ?s)
<> (sub1 0 (1 ?x) (0 ?y) ?s) (sub2 0 ?x ?y ?s)
<> (sub1 0 (1 ?x) (1 ?y) ?s) (sub1 0 ?x ?y (0 ?s))
<> (sub1 1 (0 ?x) (0 ?y) ?s) (sub2 1 ?x ?y ?s)
<> (sub1 1 (0 ?x) (1 ?y) ?s) (sub1 1 ?x ?y (0 ?s))
<> (sub1 1 (1 ?x) (0 ?y) ?s) (sub1 0 ?x ?y (0 ?s))
<> (sub1 1 (1 ?x) (1 ?y) ?s) (sub2 1 ?x ?y ?s)
<> (sub2 ?c ?x ?y ()) ((1 sub1 ?c ?x ?y ()))
<> (sub2 ?c ?x ?y (?h ?t)) ((0 sub2 ?c ?x ?y ?t))

<> (dec (0 ())) (#err)
<> (dec (1 ())) ((0 ()))
<> (dec (1 ?t)) ((0 ?t))
<> (dec (0 ?t)) (dec1 (0 ?t))
<> (dec1 (1 ())) (())
<> (dec1 (1 ?t)) ((0 ?t))
<> (dec1 (0 ?t)) ((1 dec1 ?t))

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

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

<> ((rshift1 (?a ()))) ((0 ()))
<> ((rshift1 (?a (?b ?c)))) ((?b ?c))

-- ( divmod, i.e. quotient and remainder )
-- ( x is the dividend, or what's left of it )
-- ( y is the divisor )
-- ( s is the number of bits to shift, so far )
-- ( o is the next valuet o add to the quotient )
-- ( m is the next multiple of y to work with )
-- ( d is the quotient, so far )
<> ((divmod ?x ?y)) ((divmod1 ?x ?y (cmp ?x ?y)))
<> ((divmod1 ?x ?y #lt)) ((zero ?x))
<> ((divmod1 ?x ?y #eq)) ((one zero))
<> ((divmod1 ?x ?y #gt)) ((divmod2 ?x ?y zero ?y))

<> ((divmod2 ?x ?y ?s ?m)) ((divmod3 ?x ?y ?s ?m (cmp ?x (0 ?m))))
<> ((divmod3 ?x ?y ?s ?m #gt)) ((divmod2 ?x ?y (inc ?s) (0 ?m)))
<> ((divmod3 ?x ?y ?s ?m #eq)) ((divmod4 ?x ?y (inc ?s) (0 ?m) zero))
<> ((divmod3 ?x ?y ?s ?m #lt)) ((divmod4 ?x ?y ?s ?m zero))

<> ((divmod4 ?x ?y (0 ()) ?m ?d)) (((add ?d one) (sub ?x ?y)))
<> ((divmod4 ?x ?y ?s ?m ?d)) ((divmod5 (sub ?x ?m) ?y dec ?s (rshift1 ?m) (add ?d (lshift one ?s))))

<> ((divmod5 (0 ()) ?y ?s ?m ?d)) ((?d (0 ())))
<> ((divmod5 ?x ?y ?s ?m ?d)) ((divmod6 ?x ?y ?s ?m ?d (cmp ?x ?m)))

<> ((divmod6 ?x ?y (0 ()) ?m ?d #lt)) ((?d ?x))
<> ((divmod6 ?x ?y ?s ?m ?d #lt)) ((divmod5 ?x ?y dec ?s (rshift1 ?m) ?d))
<> ((divmod6 ?x ?y ?s ?m ?d #eq)) ((divmod4 ?x ?y ?s ?m ?d))
<> ((divmod6 ?x ?y ?s ?m ?d #gt)) ((divmod4 ?x ?y ?s ?m ?d))

-- ( floor divison )
<> ((div ?x ?y)) ((div1 (divmod ?x ?y)))
<> ((div1 (?q ?r))) (?q)

-- ( remainder )
<> ((mod ?x ?y)) ((mod1 (divmod ?x ?y)))
<> ((mod1 (?q ?r))) (?r)

(bstr (mul (int 123456789) (int 987654321)))