0.00/0.26 YES 0.00/0.26 0.00/0.26 Problem 1: 0.00/0.26 0.00/0.26 (VAR v_NonEmpty:S N:S X:S X1:S X2:S Y:S Z:S) 0.00/0.26 (RULES 0.00/0.26 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.26 active(add(0,X:S)) -> mark(X:S) 0.00/0.26 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.26 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.26 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.26 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.26 active(dbl(0)) -> mark(0) 0.00/0.26 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.26 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.26 active(first(0,X:S)) -> mark(nil) 0.00/0.26 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.26 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.26 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.26 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.26 active(sqr(0)) -> mark(0) 0.00/0.26 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.26 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.26 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.26 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.26 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.26 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.26 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.26 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.26 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.26 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.26 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.26 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.26 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.26 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.26 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.26 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.26 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.26 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.26 proper(0) -> ok(0) 0.00/0.26 proper(nil) -> ok(nil) 0.00/0.26 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.26 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.26 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.26 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.26 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.26 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.26 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.26 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.26 top(ok(X:S)) -> top(active(X:S)) 0.00/0.26 ) 0.00/0.26 (STRATEGY INNERMOST) 0.00/0.26 0.00/0.26 Problem 1: 0.00/0.26 0.00/0.26 Dependency Pairs Processor: 0.00/0.26 -> Pairs: 0.00/0.26 ACTIVE(add(s(X:S),Y:S)) -> ADD(X:S,Y:S) 0.00/0.26 ACTIVE(add(s(X:S),Y:S)) -> S(add(X:S,Y:S)) 0.00/0.26 ACTIVE(add(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.26 ACTIVE(add(X1:S,X2:S)) -> ACTIVE(X2:S) 0.00/0.26 ACTIVE(add(X1:S,X2:S)) -> ADD(active(X1:S),X2:S) 0.00/0.26 ACTIVE(add(X1:S,X2:S)) -> ADD(X1:S,active(X2:S)) 0.00/0.26 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.26 ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S) 0.00/0.26 ACTIVE(dbl(s(X:S))) -> DBL(X:S) 0.00/0.26 ACTIVE(dbl(s(X:S))) -> S(dbl(X:S)) 0.00/0.26 ACTIVE(dbl(s(X:S))) -> S(s(dbl(X:S))) 0.00/0.26 ACTIVE(dbl(X:S)) -> ACTIVE(X:S) 0.00/0.26 ACTIVE(dbl(X:S)) -> DBL(active(X:S)) 0.00/0.26 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> CONS(Y:S,first(X:S,Z:S)) 0.00/0.26 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> FIRST(X:S,Z:S) 0.00/0.26 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.26 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X2:S) 0.00/0.26 ACTIVE(first(X1:S,X2:S)) -> FIRST(active(X1:S),X2:S) 0.00/0.26 ACTIVE(first(X1:S,X2:S)) -> FIRST(X1:S,active(X2:S)) 0.00/0.26 ACTIVE(recip(X:S)) -> ACTIVE(X:S) 0.00/0.26 ACTIVE(recip(X:S)) -> RECIP(active(X:S)) 0.00/0.26 ACTIVE(sqr(s(X:S))) -> ADD(sqr(X:S),dbl(X:S)) 0.00/0.26 ACTIVE(sqr(s(X:S))) -> DBL(X:S) 0.00/0.26 ACTIVE(sqr(s(X:S))) -> S(add(sqr(X:S),dbl(X:S))) 0.00/0.26 ACTIVE(sqr(s(X:S))) -> SQR(X:S) 0.00/0.26 ACTIVE(sqr(X:S)) -> ACTIVE(X:S) 0.00/0.26 ACTIVE(sqr(X:S)) -> SQR(active(X:S)) 0.00/0.26 ACTIVE(terms(N:S)) -> CONS(recip(sqr(N:S)),terms(s(N:S))) 0.00/0.26 ACTIVE(terms(N:S)) -> RECIP(sqr(N:S)) 0.00/0.26 ACTIVE(terms(N:S)) -> S(N:S) 0.00/0.26 ACTIVE(terms(N:S)) -> SQR(N:S) 0.00/0.26 ACTIVE(terms(N:S)) -> TERMS(s(N:S)) 0.00/0.26 ACTIVE(terms(X:S)) -> ACTIVE(X:S) 0.00/0.26 ACTIVE(terms(X:S)) -> TERMS(active(X:S)) 0.00/0.26 ADD(mark(X1:S),X2:S) -> ADD(X1:S,X2:S) 0.00/0.26 ADD(ok(X1:S),ok(X2:S)) -> ADD(X1:S,X2:S) 0.00/0.26 ADD(X1:S,mark(X2:S)) -> ADD(X1:S,X2:S) 0.00/0.26 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.00/0.26 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.26 DBL(mark(X:S)) -> DBL(X:S) 0.00/0.26 DBL(ok(X:S)) -> DBL(X:S) 0.00/0.26 FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S) 0.00/0.26 FIRST(ok(X1:S),ok(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.26 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.26 PROPER(add(X1:S,X2:S)) -> ADD(proper(X1:S),proper(X2:S)) 0.00/0.26 PROPER(add(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.26 PROPER(add(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.26 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S)) 0.00/0.26 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.26 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.26 PROPER(dbl(X:S)) -> DBL(proper(X:S)) 0.00/0.26 PROPER(dbl(X:S)) -> PROPER(X:S) 0.00/0.26 PROPER(first(X1:S,X2:S)) -> FIRST(proper(X1:S),proper(X2:S)) 0.00/0.26 PROPER(first(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.26 PROPER(first(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.26 PROPER(recip(X:S)) -> PROPER(X:S) 0.00/0.26 PROPER(recip(X:S)) -> RECIP(proper(X:S)) 0.00/0.26 PROPER(s(X:S)) -> PROPER(X:S) 0.00/0.26 PROPER(s(X:S)) -> S(proper(X:S)) 0.00/0.26 PROPER(sqr(X:S)) -> PROPER(X:S) 0.00/0.26 PROPER(sqr(X:S)) -> SQR(proper(X:S)) 0.00/0.26 PROPER(terms(X:S)) -> PROPER(X:S) 0.00/0.26 PROPER(terms(X:S)) -> TERMS(proper(X:S)) 0.00/0.26 RECIP(mark(X:S)) -> RECIP(X:S) 0.00/0.26 RECIP(ok(X:S)) -> RECIP(X:S) 0.00/0.26 S(ok(X:S)) -> S(X:S) 0.00/0.26 SQR(mark(X:S)) -> SQR(X:S) 0.00/0.26 SQR(ok(X:S)) -> SQR(X:S) 0.00/0.26 TERMS(mark(X:S)) -> TERMS(X:S) 0.00/0.26 TERMS(ok(X:S)) -> TERMS(X:S) 0.00/0.26 TOP(mark(X:S)) -> PROPER(X:S) 0.00/0.26 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.00/0.26 TOP(ok(X:S)) -> ACTIVE(X:S) 0.00/0.26 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.26 -> Rules: 0.00/0.26 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.26 active(add(0,X:S)) -> mark(X:S) 0.00/0.26 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.26 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.26 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.26 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.26 active(dbl(0)) -> mark(0) 0.00/0.26 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.26 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.26 active(first(0,X:S)) -> mark(nil) 0.00/0.26 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.26 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.26 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.26 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.26 active(sqr(0)) -> mark(0) 0.00/0.26 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.26 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.26 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.26 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.26 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.26 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.26 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.26 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.26 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.26 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.26 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.26 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.26 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.26 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.26 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.26 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.26 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.26 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.26 proper(0) -> ok(0) 0.00/0.26 proper(nil) -> ok(nil) 0.00/0.26 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.26 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.26 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.26 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.26 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.26 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.26 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.26 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.26 top(ok(X:S)) -> top(active(X:S)) 0.00/0.26 0.00/0.26 Problem 1: 0.00/0.26 0.00/0.26 SCC Processor: 0.00/0.26 -> Pairs: 0.00/0.26 ACTIVE(add(s(X:S),Y:S)) -> ADD(X:S,Y:S) 0.00/0.26 ACTIVE(add(s(X:S),Y:S)) -> S(add(X:S,Y:S)) 0.00/0.26 ACTIVE(add(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.26 ACTIVE(add(X1:S,X2:S)) -> ACTIVE(X2:S) 0.00/0.26 ACTIVE(add(X1:S,X2:S)) -> ADD(active(X1:S),X2:S) 0.00/0.26 ACTIVE(add(X1:S,X2:S)) -> ADD(X1:S,active(X2:S)) 0.00/0.26 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.26 ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S) 0.00/0.26 ACTIVE(dbl(s(X:S))) -> DBL(X:S) 0.00/0.26 ACTIVE(dbl(s(X:S))) -> S(dbl(X:S)) 0.00/0.26 ACTIVE(dbl(s(X:S))) -> S(s(dbl(X:S))) 0.00/0.26 ACTIVE(dbl(X:S)) -> ACTIVE(X:S) 0.00/0.26 ACTIVE(dbl(X:S)) -> DBL(active(X:S)) 0.00/0.26 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> CONS(Y:S,first(X:S,Z:S)) 0.00/0.26 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> FIRST(X:S,Z:S) 0.00/0.26 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.26 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X2:S) 0.00/0.26 ACTIVE(first(X1:S,X2:S)) -> FIRST(active(X1:S),X2:S) 0.00/0.26 ACTIVE(first(X1:S,X2:S)) -> FIRST(X1:S,active(X2:S)) 0.00/0.26 ACTIVE(recip(X:S)) -> ACTIVE(X:S) 0.00/0.26 ACTIVE(recip(X:S)) -> RECIP(active(X:S)) 0.00/0.26 ACTIVE(sqr(s(X:S))) -> ADD(sqr(X:S),dbl(X:S)) 0.00/0.26 ACTIVE(sqr(s(X:S))) -> DBL(X:S) 0.00/0.26 ACTIVE(sqr(s(X:S))) -> S(add(sqr(X:S),dbl(X:S))) 0.00/0.26 ACTIVE(sqr(s(X:S))) -> SQR(X:S) 0.00/0.26 ACTIVE(sqr(X:S)) -> ACTIVE(X:S) 0.00/0.26 ACTIVE(sqr(X:S)) -> SQR(active(X:S)) 0.00/0.26 ACTIVE(terms(N:S)) -> CONS(recip(sqr(N:S)),terms(s(N:S))) 0.00/0.26 ACTIVE(terms(N:S)) -> RECIP(sqr(N:S)) 0.00/0.26 ACTIVE(terms(N:S)) -> S(N:S) 0.00/0.26 ACTIVE(terms(N:S)) -> SQR(N:S) 0.00/0.26 ACTIVE(terms(N:S)) -> TERMS(s(N:S)) 0.00/0.26 ACTIVE(terms(X:S)) -> ACTIVE(X:S) 0.00/0.26 ACTIVE(terms(X:S)) -> TERMS(active(X:S)) 0.00/0.26 ADD(mark(X1:S),X2:S) -> ADD(X1:S,X2:S) 0.00/0.26 ADD(ok(X1:S),ok(X2:S)) -> ADD(X1:S,X2:S) 0.00/0.26 ADD(X1:S,mark(X2:S)) -> ADD(X1:S,X2:S) 0.00/0.26 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.00/0.26 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.26 DBL(mark(X:S)) -> DBL(X:S) 0.00/0.26 DBL(ok(X:S)) -> DBL(X:S) 0.00/0.26 FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S) 0.00/0.26 FIRST(ok(X1:S),ok(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.26 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.26 PROPER(add(X1:S,X2:S)) -> ADD(proper(X1:S),proper(X2:S)) 0.00/0.26 PROPER(add(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.26 PROPER(add(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.26 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S)) 0.00/0.26 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.26 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.26 PROPER(dbl(X:S)) -> DBL(proper(X:S)) 0.00/0.26 PROPER(dbl(X:S)) -> PROPER(X:S) 0.00/0.26 PROPER(first(X1:S,X2:S)) -> FIRST(proper(X1:S),proper(X2:S)) 0.00/0.26 PROPER(first(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.26 PROPER(first(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.26 PROPER(recip(X:S)) -> PROPER(X:S) 0.00/0.26 PROPER(recip(X:S)) -> RECIP(proper(X:S)) 0.00/0.26 PROPER(s(X:S)) -> PROPER(X:S) 0.00/0.26 PROPER(s(X:S)) -> S(proper(X:S)) 0.00/0.26 PROPER(sqr(X:S)) -> PROPER(X:S) 0.00/0.26 PROPER(sqr(X:S)) -> SQR(proper(X:S)) 0.00/0.26 PROPER(terms(X:S)) -> PROPER(X:S) 0.00/0.26 PROPER(terms(X:S)) -> TERMS(proper(X:S)) 0.00/0.26 RECIP(mark(X:S)) -> RECIP(X:S) 0.00/0.26 RECIP(ok(X:S)) -> RECIP(X:S) 0.00/0.26 S(ok(X:S)) -> S(X:S) 0.00/0.26 SQR(mark(X:S)) -> SQR(X:S) 0.00/0.26 SQR(ok(X:S)) -> SQR(X:S) 0.00/0.26 TERMS(mark(X:S)) -> TERMS(X:S) 0.00/0.26 TERMS(ok(X:S)) -> TERMS(X:S) 0.00/0.26 TOP(mark(X:S)) -> PROPER(X:S) 0.00/0.26 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.00/0.26 TOP(ok(X:S)) -> ACTIVE(X:S) 0.00/0.26 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.26 -> Rules: 0.00/0.26 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.26 active(add(0,X:S)) -> mark(X:S) 0.00/0.26 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.26 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.26 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.26 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.26 active(dbl(0)) -> mark(0) 0.00/0.26 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.26 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.26 active(first(0,X:S)) -> mark(nil) 0.00/0.26 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.26 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.26 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.26 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.26 active(sqr(0)) -> mark(0) 0.00/0.26 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.26 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.26 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.26 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.26 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.26 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.26 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.26 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.26 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.26 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.26 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.26 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.26 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.26 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.26 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.26 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.26 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.26 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.26 proper(0) -> ok(0) 0.00/0.26 proper(nil) -> ok(nil) 0.00/0.26 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.26 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.26 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.26 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.26 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.26 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.26 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.26 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.26 top(ok(X:S)) -> top(active(X:S)) 0.00/0.26 ->Strongly Connected Components: 0.00/0.26 ->->Cycle: 0.00/0.26 ->->-> Pairs: 0.00/0.26 TERMS(mark(X:S)) -> TERMS(X:S) 0.00/0.26 TERMS(ok(X:S)) -> TERMS(X:S) 0.00/0.26 ->->-> Rules: 0.00/0.26 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.26 active(add(0,X:S)) -> mark(X:S) 0.00/0.26 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.26 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.26 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.26 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.26 active(dbl(0)) -> mark(0) 0.00/0.26 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.26 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.26 active(first(0,X:S)) -> mark(nil) 0.00/0.26 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.26 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.26 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.26 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.26 active(sqr(0)) -> mark(0) 0.00/0.26 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.26 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.26 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.26 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.26 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.26 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.26 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.26 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.26 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.26 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.26 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.26 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.26 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.26 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.26 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.26 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.26 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.26 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.26 proper(0) -> ok(0) 0.00/0.26 proper(nil) -> ok(nil) 0.00/0.26 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.26 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.26 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.26 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.26 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.26 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.26 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.26 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.26 top(ok(X:S)) -> top(active(X:S)) 0.00/0.26 ->->Cycle: 0.00/0.26 ->->-> Pairs: 0.00/0.26 SQR(mark(X:S)) -> SQR(X:S) 0.00/0.26 SQR(ok(X:S)) -> SQR(X:S) 0.00/0.26 ->->-> Rules: 0.00/0.26 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.26 active(add(0,X:S)) -> mark(X:S) 0.00/0.26 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.26 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.26 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.26 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.26 active(dbl(0)) -> mark(0) 0.00/0.26 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.26 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.26 active(first(0,X:S)) -> mark(nil) 0.00/0.26 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.26 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.26 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.26 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.26 active(sqr(0)) -> mark(0) 0.00/0.26 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.26 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.26 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.26 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.26 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.26 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.26 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.26 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.26 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.26 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.26 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.26 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.26 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.26 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.26 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.26 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.26 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.26 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.26 proper(0) -> ok(0) 0.00/0.26 proper(nil) -> ok(nil) 0.00/0.26 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.26 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.26 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.26 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.26 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.26 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.26 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.26 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.26 top(ok(X:S)) -> top(active(X:S)) 0.00/0.26 ->->Cycle: 0.00/0.26 ->->-> Pairs: 0.00/0.26 S(ok(X:S)) -> S(X:S) 0.00/0.26 ->->-> Rules: 0.00/0.26 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.26 active(add(0,X:S)) -> mark(X:S) 0.00/0.26 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.26 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.26 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.26 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.26 active(dbl(0)) -> mark(0) 0.00/0.26 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.26 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.26 active(first(0,X:S)) -> mark(nil) 0.00/0.26 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.26 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.26 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.26 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.26 active(sqr(0)) -> mark(0) 0.00/0.26 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.26 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.26 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.26 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.26 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.26 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.26 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.26 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.26 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.26 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.26 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.26 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.26 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.26 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.26 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.26 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.26 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.26 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.26 proper(0) -> ok(0) 0.00/0.26 proper(nil) -> ok(nil) 0.00/0.26 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.26 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.26 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.26 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.26 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.26 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.26 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.26 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.26 top(ok(X:S)) -> top(active(X:S)) 0.00/0.26 ->->Cycle: 0.00/0.26 ->->-> Pairs: 0.00/0.26 RECIP(mark(X:S)) -> RECIP(X:S) 0.00/0.26 RECIP(ok(X:S)) -> RECIP(X:S) 0.00/0.26 ->->-> Rules: 0.00/0.26 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.26 active(add(0,X:S)) -> mark(X:S) 0.00/0.26 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.26 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.26 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.26 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.26 active(dbl(0)) -> mark(0) 0.00/0.26 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.26 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.26 active(first(0,X:S)) -> mark(nil) 0.00/0.26 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.26 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.26 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.26 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.26 active(sqr(0)) -> mark(0) 0.00/0.26 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.26 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.26 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.26 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.26 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.26 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.26 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.26 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.26 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.26 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.26 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.26 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.26 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.26 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.26 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.26 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.26 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.26 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.26 proper(0) -> ok(0) 0.00/0.26 proper(nil) -> ok(nil) 0.00/0.26 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.26 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.26 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.26 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.26 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.26 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.26 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.26 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.26 top(ok(X:S)) -> top(active(X:S)) 0.00/0.26 ->->Cycle: 0.00/0.26 ->->-> Pairs: 0.00/0.26 FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S) 0.00/0.26 FIRST(ok(X1:S),ok(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.26 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.26 ->->-> Rules: 0.00/0.26 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.26 active(add(0,X:S)) -> mark(X:S) 0.00/0.26 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.26 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.26 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.26 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.26 active(dbl(0)) -> mark(0) 0.00/0.26 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.26 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.26 active(first(0,X:S)) -> mark(nil) 0.00/0.26 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.26 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.26 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.26 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.26 active(sqr(0)) -> mark(0) 0.00/0.26 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.26 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.26 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.26 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.26 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.26 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.26 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.26 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.26 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.26 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.26 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.26 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.26 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.26 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.26 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.26 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.26 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.26 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.26 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.26 proper(0) -> ok(0) 0.00/0.26 proper(nil) -> ok(nil) 0.00/0.26 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.26 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.26 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.26 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.26 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.26 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.26 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->->Cycle: 0.00/0.27 ->->-> Pairs: 0.00/0.27 DBL(mark(X:S)) -> DBL(X:S) 0.00/0.27 DBL(ok(X:S)) -> DBL(X:S) 0.00/0.27 ->->-> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->->Cycle: 0.00/0.27 ->->-> Pairs: 0.00/0.27 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.00/0.27 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.27 ->->-> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->->Cycle: 0.00/0.27 ->->-> Pairs: 0.00/0.27 ADD(mark(X1:S),X2:S) -> ADD(X1:S,X2:S) 0.00/0.27 ADD(ok(X1:S),ok(X2:S)) -> ADD(X1:S,X2:S) 0.00/0.27 ADD(X1:S,mark(X2:S)) -> ADD(X1:S,X2:S) 0.00/0.27 ->->-> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->->Cycle: 0.00/0.27 ->->-> Pairs: 0.00/0.27 PROPER(add(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.27 PROPER(add(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.27 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.27 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.27 PROPER(dbl(X:S)) -> PROPER(X:S) 0.00/0.27 PROPER(first(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.27 PROPER(first(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.27 PROPER(recip(X:S)) -> PROPER(X:S) 0.00/0.27 PROPER(s(X:S)) -> PROPER(X:S) 0.00/0.27 PROPER(sqr(X:S)) -> PROPER(X:S) 0.00/0.27 PROPER(terms(X:S)) -> PROPER(X:S) 0.00/0.27 ->->-> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->->Cycle: 0.00/0.27 ->->-> Pairs: 0.00/0.27 ACTIVE(add(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.27 ACTIVE(add(X1:S,X2:S)) -> ACTIVE(X2:S) 0.00/0.27 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.27 ACTIVE(dbl(X:S)) -> ACTIVE(X:S) 0.00/0.27 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.27 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X2:S) 0.00/0.27 ACTIVE(recip(X:S)) -> ACTIVE(X:S) 0.00/0.27 ACTIVE(sqr(X:S)) -> ACTIVE(X:S) 0.00/0.27 ACTIVE(terms(X:S)) -> ACTIVE(X:S) 0.00/0.27 ->->-> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->->Cycle: 0.00/0.27 ->->-> Pairs: 0.00/0.27 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.00/0.27 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.27 ->->-> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 0.00/0.27 0.00/0.27 The problem is decomposed in 11 subproblems. 0.00/0.27 0.00/0.27 Problem 1.1: 0.00/0.27 0.00/0.27 Subterm Processor: 0.00/0.27 -> Pairs: 0.00/0.27 TERMS(mark(X:S)) -> TERMS(X:S) 0.00/0.27 TERMS(ok(X:S)) -> TERMS(X:S) 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Projection: 0.00/0.27 pi(TERMS) = 1 0.00/0.27 0.00/0.27 Problem 1.1: 0.00/0.27 0.00/0.27 SCC Processor: 0.00/0.27 -> Pairs: 0.00/0.27 Empty 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Strongly Connected Components: 0.00/0.27 There is no strongly connected component 0.00/0.27 0.00/0.27 The problem is finite. 0.00/0.27 0.00/0.27 Problem 1.2: 0.00/0.27 0.00/0.27 Subterm Processor: 0.00/0.27 -> Pairs: 0.00/0.27 SQR(mark(X:S)) -> SQR(X:S) 0.00/0.27 SQR(ok(X:S)) -> SQR(X:S) 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Projection: 0.00/0.27 pi(SQR) = 1 0.00/0.27 0.00/0.27 Problem 1.2: 0.00/0.27 0.00/0.27 SCC Processor: 0.00/0.27 -> Pairs: 0.00/0.27 Empty 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Strongly Connected Components: 0.00/0.27 There is no strongly connected component 0.00/0.27 0.00/0.27 The problem is finite. 0.00/0.27 0.00/0.27 Problem 1.3: 0.00/0.27 0.00/0.27 Subterm Processor: 0.00/0.27 -> Pairs: 0.00/0.27 S(ok(X:S)) -> S(X:S) 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Projection: 0.00/0.27 pi(S) = 1 0.00/0.27 0.00/0.27 Problem 1.3: 0.00/0.27 0.00/0.27 SCC Processor: 0.00/0.27 -> Pairs: 0.00/0.27 Empty 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Strongly Connected Components: 0.00/0.27 There is no strongly connected component 0.00/0.27 0.00/0.27 The problem is finite. 0.00/0.27 0.00/0.27 Problem 1.4: 0.00/0.27 0.00/0.27 Subterm Processor: 0.00/0.27 -> Pairs: 0.00/0.27 RECIP(mark(X:S)) -> RECIP(X:S) 0.00/0.27 RECIP(ok(X:S)) -> RECIP(X:S) 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Projection: 0.00/0.27 pi(RECIP) = 1 0.00/0.27 0.00/0.27 Problem 1.4: 0.00/0.27 0.00/0.27 SCC Processor: 0.00/0.27 -> Pairs: 0.00/0.27 Empty 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Strongly Connected Components: 0.00/0.27 There is no strongly connected component 0.00/0.27 0.00/0.27 The problem is finite. 0.00/0.27 0.00/0.27 Problem 1.5: 0.00/0.27 0.00/0.27 Subterm Processor: 0.00/0.27 -> Pairs: 0.00/0.27 FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S) 0.00/0.27 FIRST(ok(X1:S),ok(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.27 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Projection: 0.00/0.27 pi(FIRST) = 1 0.00/0.27 0.00/0.27 Problem 1.5: 0.00/0.27 0.00/0.27 SCC Processor: 0.00/0.27 -> Pairs: 0.00/0.27 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Strongly Connected Components: 0.00/0.27 ->->Cycle: 0.00/0.27 ->->-> Pairs: 0.00/0.27 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.27 ->->-> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 0.00/0.27 Problem 1.5: 0.00/0.27 0.00/0.27 Subterm Processor: 0.00/0.27 -> Pairs: 0.00/0.27 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Projection: 0.00/0.27 pi(FIRST) = 2 0.00/0.27 0.00/0.27 Problem 1.5: 0.00/0.27 0.00/0.27 SCC Processor: 0.00/0.27 -> Pairs: 0.00/0.27 Empty 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Strongly Connected Components: 0.00/0.27 There is no strongly connected component 0.00/0.27 0.00/0.27 The problem is finite. 0.00/0.27 0.00/0.27 Problem 1.6: 0.00/0.27 0.00/0.27 Subterm Processor: 0.00/0.27 -> Pairs: 0.00/0.27 DBL(mark(X:S)) -> DBL(X:S) 0.00/0.27 DBL(ok(X:S)) -> DBL(X:S) 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Projection: 0.00/0.27 pi(DBL) = 1 0.00/0.27 0.00/0.27 Problem 1.6: 0.00/0.27 0.00/0.27 SCC Processor: 0.00/0.27 -> Pairs: 0.00/0.27 Empty 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Strongly Connected Components: 0.00/0.27 There is no strongly connected component 0.00/0.27 0.00/0.27 The problem is finite. 0.00/0.27 0.00/0.27 Problem 1.7: 0.00/0.27 0.00/0.27 Subterm Processor: 0.00/0.27 -> Pairs: 0.00/0.27 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.00/0.27 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Projection: 0.00/0.27 pi(CONS) = 1 0.00/0.27 0.00/0.27 Problem 1.7: 0.00/0.27 0.00/0.27 SCC Processor: 0.00/0.27 -> Pairs: 0.00/0.27 Empty 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Strongly Connected Components: 0.00/0.27 There is no strongly connected component 0.00/0.27 0.00/0.27 The problem is finite. 0.00/0.27 0.00/0.27 Problem 1.8: 0.00/0.27 0.00/0.27 Subterm Processor: 0.00/0.27 -> Pairs: 0.00/0.27 ADD(mark(X1:S),X2:S) -> ADD(X1:S,X2:S) 0.00/0.27 ADD(ok(X1:S),ok(X2:S)) -> ADD(X1:S,X2:S) 0.00/0.27 ADD(X1:S,mark(X2:S)) -> ADD(X1:S,X2:S) 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Projection: 0.00/0.27 pi(ADD) = 1 0.00/0.27 0.00/0.27 Problem 1.8: 0.00/0.27 0.00/0.27 SCC Processor: 0.00/0.27 -> Pairs: 0.00/0.27 ADD(X1:S,mark(X2:S)) -> ADD(X1:S,X2:S) 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Strongly Connected Components: 0.00/0.27 ->->Cycle: 0.00/0.27 ->->-> Pairs: 0.00/0.27 ADD(X1:S,mark(X2:S)) -> ADD(X1:S,X2:S) 0.00/0.27 ->->-> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 0.00/0.27 Problem 1.8: 0.00/0.27 0.00/0.27 Subterm Processor: 0.00/0.27 -> Pairs: 0.00/0.27 ADD(X1:S,mark(X2:S)) -> ADD(X1:S,X2:S) 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Projection: 0.00/0.27 pi(ADD) = 2 0.00/0.27 0.00/0.27 Problem 1.8: 0.00/0.27 0.00/0.27 SCC Processor: 0.00/0.27 -> Pairs: 0.00/0.27 Empty 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Strongly Connected Components: 0.00/0.27 There is no strongly connected component 0.00/0.27 0.00/0.27 The problem is finite. 0.00/0.27 0.00/0.27 Problem 1.9: 0.00/0.27 0.00/0.27 Subterm Processor: 0.00/0.27 -> Pairs: 0.00/0.27 PROPER(add(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.27 PROPER(add(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.27 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.27 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.27 PROPER(dbl(X:S)) -> PROPER(X:S) 0.00/0.27 PROPER(first(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.27 PROPER(first(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.27 PROPER(recip(X:S)) -> PROPER(X:S) 0.00/0.27 PROPER(s(X:S)) -> PROPER(X:S) 0.00/0.27 PROPER(sqr(X:S)) -> PROPER(X:S) 0.00/0.27 PROPER(terms(X:S)) -> PROPER(X:S) 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Projection: 0.00/0.27 pi(PROPER) = 1 0.00/0.27 0.00/0.27 Problem 1.9: 0.00/0.27 0.00/0.27 SCC Processor: 0.00/0.27 -> Pairs: 0.00/0.27 Empty 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Strongly Connected Components: 0.00/0.27 There is no strongly connected component 0.00/0.27 0.00/0.27 The problem is finite. 0.00/0.27 0.00/0.27 Problem 1.10: 0.00/0.27 0.00/0.27 Subterm Processor: 0.00/0.27 -> Pairs: 0.00/0.27 ACTIVE(add(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.27 ACTIVE(add(X1:S,X2:S)) -> ACTIVE(X2:S) 0.00/0.27 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.27 ACTIVE(dbl(X:S)) -> ACTIVE(X:S) 0.00/0.27 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.27 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X2:S) 0.00/0.27 ACTIVE(recip(X:S)) -> ACTIVE(X:S) 0.00/0.27 ACTIVE(sqr(X:S)) -> ACTIVE(X:S) 0.00/0.27 ACTIVE(terms(X:S)) -> ACTIVE(X:S) 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Projection: 0.00/0.27 pi(ACTIVE) = 1 0.00/0.27 0.00/0.27 Problem 1.10: 0.00/0.27 0.00/0.27 SCC Processor: 0.00/0.27 -> Pairs: 0.00/0.27 Empty 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Strongly Connected Components: 0.00/0.27 There is no strongly connected component 0.00/0.27 0.00/0.27 The problem is finite. 0.00/0.27 0.00/0.27 Problem 1.11: 0.00/0.27 0.00/0.27 Reduction Pairs Processor: 0.00/0.27 -> Pairs: 0.00/0.27 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.00/0.27 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 -> Usable rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 ->Interpretation type: 0.00/0.27 Linear 0.00/0.27 ->Coefficients: 0.00/0.27 Natural Numbers 0.00/0.27 ->Dimension: 0.00/0.27 1 0.00/0.27 ->Bound: 0.00/0.27 2 0.00/0.27 ->Interpretation: 0.00/0.27 0.00/0.27 [active](X) = X 0.00/0.27 [add](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.27 [cons](X1,X2) = X1 0.00/0.27 [dbl](X) = 2.X + 2 0.00/0.27 [first](X1,X2) = 2.X1 + X2 + 2 0.00/0.27 [proper](X) = X 0.00/0.27 [recip](X) = X 0.00/0.27 [s](X) = 2 0.00/0.27 [sqr](X) = 2.X 0.00/0.27 [terms](X) = 2.X + 2 0.00/0.27 [top](X) = 0 0.00/0.27 [0] = 2 0.00/0.27 [fSNonEmpty] = 0 0.00/0.27 [mark](X) = X + 2 0.00/0.27 [nil] = 2 0.00/0.27 [ok](X) = X 0.00/0.27 [ACTIVE](X) = 0 0.00/0.27 [ADD](X1,X2) = 0 0.00/0.27 [CONS](X1,X2) = 0 0.00/0.27 [DBL](X) = 0 0.00/0.27 [FIRST](X1,X2) = 0 0.00/0.27 [PROPER](X) = 0 0.00/0.27 [RECIP](X) = 0 0.00/0.27 [S](X) = 0 0.00/0.27 [SQR](X) = 0 0.00/0.27 [TERMS](X) = 0 0.00/0.27 [TOP](X) = 2.X 0.00/0.27 0.00/0.27 Problem 1.11: 0.00/0.27 0.00/0.27 SCC Processor: 0.00/0.27 -> Pairs: 0.00/0.27 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ->Strongly Connected Components: 0.00/0.27 ->->Cycle: 0.00/0.27 ->->-> Pairs: 0.00/0.27 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.27 ->->-> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.27 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.27 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.27 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.27 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.27 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.27 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(nil) -> ok(nil) 0.00/0.27 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.27 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.27 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.27 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.27 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 0.00/0.27 Problem 1.11: 0.00/0.27 0.00/0.27 Reduction Pairs Processor: 0.00/0.27 -> Pairs: 0.00/0.27 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.27 -> Rules: 0.00/0.27 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.27 active(add(0,X:S)) -> mark(X:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.27 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.27 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.27 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.27 active(dbl(0)) -> mark(0) 0.00/0.27 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.27 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.27 active(first(0,X:S)) -> mark(nil) 0.00/0.27 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.27 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.27 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.27 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.27 active(sqr(0)) -> mark(0) 0.00/0.27 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.27 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.27 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.27 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.27 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.27 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.27 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.27 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.27 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.27 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.28 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.28 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.28 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.28 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.28 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.28 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.28 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.28 proper(0) -> ok(0) 0.00/0.28 proper(nil) -> ok(nil) 0.00/0.28 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.28 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.28 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.28 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.28 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.28 top(ok(X:S)) -> top(active(X:S)) 0.00/0.28 -> Usable rules: 0.00/0.28 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.28 active(add(0,X:S)) -> mark(X:S) 0.00/0.28 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.28 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.28 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.28 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.28 active(dbl(0)) -> mark(0) 0.00/0.28 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.28 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.28 active(first(0,X:S)) -> mark(nil) 0.00/0.28 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.28 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.28 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.28 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.28 active(sqr(0)) -> mark(0) 0.00/0.28 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.28 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.28 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.28 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.28 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.28 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.28 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.28 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.28 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.28 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.28 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.28 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.28 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.28 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.28 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.28 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.28 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.28 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.28 ->Interpretation type: 0.00/0.28 Linear 0.00/0.28 ->Coefficients: 0.00/0.28 Natural Numbers 0.00/0.28 ->Dimension: 0.00/0.28 1 0.00/0.28 ->Bound: 0.00/0.28 2 0.00/0.28 ->Interpretation: 0.00/0.28 0.00/0.28 [active](X) = 2.X + 1 0.00/0.28 [add](X1,X2) = X2 0.00/0.28 [cons](X1,X2) = 2.X1 + 1 0.00/0.28 [dbl](X) = X 0.00/0.28 [first](X1,X2) = 2.X1 + X2 + 1 0.00/0.28 [proper](X) = 0 0.00/0.28 [recip](X) = X 0.00/0.28 [s](X) = 2.X 0.00/0.28 [sqr](X) = X 0.00/0.28 [terms](X) = X 0.00/0.28 [top](X) = 0 0.00/0.28 [0] = 2 0.00/0.28 [fSNonEmpty] = 0 0.00/0.28 [mark](X) = X 0.00/0.28 [nil] = 2 0.00/0.28 [ok](X) = 2.X + 2 0.00/0.28 [ACTIVE](X) = 0 0.00/0.28 [ADD](X1,X2) = 0 0.00/0.28 [CONS](X1,X2) = 0 0.00/0.28 [DBL](X) = 0 0.00/0.28 [FIRST](X1,X2) = 0 0.00/0.28 [PROPER](X) = 0 0.00/0.28 [RECIP](X) = 0 0.00/0.28 [S](X) = 0 0.00/0.28 [SQR](X) = 0 0.00/0.28 [TERMS](X) = 0 0.00/0.28 [TOP](X) = X 0.00/0.28 0.00/0.28 Problem 1.11: 0.00/0.28 0.00/0.28 SCC Processor: 0.00/0.28 -> Pairs: 0.00/0.28 Empty 0.00/0.28 -> Rules: 0.00/0.28 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.28 active(add(0,X:S)) -> mark(X:S) 0.00/0.28 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.28 active(add(X1:S,X2:S)) -> add(X1:S,active(X2:S)) 0.00/0.28 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.00/0.28 active(dbl(s(X:S))) -> mark(s(s(dbl(X:S)))) 0.00/0.28 active(dbl(0)) -> mark(0) 0.00/0.28 active(dbl(X:S)) -> dbl(active(X:S)) 0.00/0.28 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.28 active(first(0,X:S)) -> mark(nil) 0.00/0.28 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.28 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.28 active(recip(X:S)) -> recip(active(X:S)) 0.00/0.28 active(sqr(s(X:S))) -> mark(s(add(sqr(X:S),dbl(X:S)))) 0.00/0.28 active(sqr(0)) -> mark(0) 0.00/0.28 active(sqr(X:S)) -> sqr(active(X:S)) 0.00/0.28 active(terms(N:S)) -> mark(cons(recip(sqr(N:S)),terms(s(N:S)))) 0.00/0.28 active(terms(X:S)) -> terms(active(X:S)) 0.00/0.28 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.28 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.28 add(X1:S,mark(X2:S)) -> mark(add(X1:S,X2:S)) 0.00/0.28 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.00/0.28 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.28 dbl(mark(X:S)) -> mark(dbl(X:S)) 0.00/0.28 dbl(ok(X:S)) -> ok(dbl(X:S)) 0.00/0.28 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.28 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.28 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.28 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(dbl(X:S)) -> dbl(proper(X:S)) 0.00/0.28 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(recip(X:S)) -> recip(proper(X:S)) 0.00/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.28 proper(sqr(X:S)) -> sqr(proper(X:S)) 0.00/0.28 proper(terms(X:S)) -> terms(proper(X:S)) 0.00/0.28 proper(0) -> ok(0) 0.00/0.28 proper(nil) -> ok(nil) 0.00/0.28 recip(mark(X:S)) -> mark(recip(X:S)) 0.00/0.28 recip(ok(X:S)) -> ok(recip(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 sqr(mark(X:S)) -> mark(sqr(X:S)) 0.00/0.28 sqr(ok(X:S)) -> ok(sqr(X:S)) 0.00/0.28 terms(mark(X:S)) -> mark(terms(X:S)) 0.00/0.28 terms(ok(X:S)) -> ok(terms(X:S)) 0.00/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.28 top(ok(X:S)) -> top(active(X:S)) 0.00/0.28 ->Strongly Connected Components: 0.00/0.28 There is no strongly connected component 0.00/0.28 0.00/0.28 The problem is finite. 0.00/0.28 EOF