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