(VAR x y z dummy dummy2 dummy3 dummy4 dummy5 dummy6)
(RULES
function(iszero,0,dummy,dummy2) -> true
function(iszero,s(x),dummy,dummy2) -> false

function(p,0,dummy,dummy2) -> 0
function(p,s(0),dummy,dummy2) -> 0
function(p,s(s(x)), dummy, dummy2) -> s(function(p,s(x),x,x))


function(plus,dummy,x,y) -> function(if,function(iszero,x,x,x),x,y)
function(if,true,x,y) -> y
function(if,false,x,y) ->
function(plus,function(third,x,y,y),function(p,x,x,y),s(y))

function(third,x,y,z) -> z
)

(COMMENT

basic system:

iszero(0) -> true
iszero(s(x)) -> false

p(0) -> 0
p(s(0)) -> 0
p(s(s(x))) -> s(p(s(x)))

plus(x,y) -> if(iszero(x),x,y)
if(true,x,y) -> y
if(false,x,y) -> plus(p(x),s(y))

third(x,y,z) -> z



alternative 1: function only for defined symbols

function(iszero,0,dummy,dummy2) -> true
function(iszero,s(x),dummy,dummy2) -> false

function(p,0,dummy,dummy2) -> 0
function(p,s(0),dummy,dummy2) -> 0
function(p,s(s(x)), dummy, dummy2) -> s(function(p,s(x),x,x))


function(plus,dummy,x,y) -> function(if,function(iszero,x,x,x),x,y)
function(if,true,x,y) -> y
function(if,false,x,y) ->
function(plus,function(third,x,y,y),function(p,x,x,y),s(y))

function(third,x,y,z) -> z




alternative 2: function also for constructors, easier pred

function(iszero,0,dummy,dummy2) -> true
function(iszero,function(s,x,dummy3,dummy4),dummy,dummy2) -> false

function(p,0,dummy,dummy2) -> 0
function(p,function(s,x,dummy3,dummy4),dummy,dummy2) -> x

function(plus,dummy,x,y) -> function(if,function(iszero,x,x,x),x,y)
function(if,true,x,y) -> y
function(if,false,x,y) ->
function(plus,function(third,x,y,y),function(p,x,x,y),function(s,y,x,y))

function(third,x,y,z) -> z
)