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