/home/nowonder/forschung/aprove/TPDB05/TRS/TRCSR/ExIntrod_GM99_C.trs

The program

(VAR X Y Z X1 X2 X3)
(RULES 
active(primes) -> mark(sieve(from(s(s(0)))))
active(from(X)) -> mark(cons(X,from(s(X))))
active(head(cons(X,Y))) -> mark(X)
active(tail(cons(X,Y))) -> mark(Y)
active(if(true,X,Y)) -> mark(X)
active(if(false,X,Y)) -> mark(Y)
active(filter(s(s(X)),cons(Y,Z))) -> mark(if(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))))
active(sieve(cons(X,Y))) -> mark(cons(X,filter(X,sieve(Y))))
active(sieve(X)) -> sieve(active(X))
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(cons(X1,X2)) -> cons(active(X1),X2)
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
active(filter(X1,X2)) -> filter(active(X1),X2)
active(filter(X1,X2)) -> filter(X1,active(X2))
active(divides(X1,X2)) -> divides(active(X1),X2)
active(divides(X1,X2)) -> divides(X1,active(X2))
sieve(mark(X)) -> mark(sieve(X))
from(mark(X)) -> mark(from(X))
s(mark(X)) -> mark(s(X))
cons(mark(X1),X2) -> mark(cons(X1,X2))
head(mark(X)) -> mark(head(X))
tail(mark(X)) -> mark(tail(X))
if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
filter(mark(X1),X2) -> mark(filter(X1,X2))
filter(X1,mark(X2)) -> mark(filter(X1,X2))
divides(mark(X1),X2) -> mark(divides(X1,X2))
divides(X1,mark(X2)) -> mark(divides(X1,X2))
proper(primes) -> ok(primes)
proper(sieve(X)) -> sieve(proper(X))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(filter(X1,X2)) -> filter(proper(X1),proper(X2))
proper(divides(X1,X2)) -> divides(proper(X1),proper(X2))
sieve(ok(X)) -> ok(sieve(X))
from(ok(X)) -> ok(from(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
head(ok(X)) -> ok(head(X))
tail(ok(X)) -> ok(tail(X))
if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
filter(ok(X1),ok(X2)) -> ok(filter(X1,X2))
divides(ok(X1),ok(X2)) -> ok(divides(X1,X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
)

Submit to AProVE Web Frontend

Edit in AProVE Web Frontend