/export/starexec/sandbox2/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(terms(N)) -> s#(N) active#(terms(N)) -> terms#(s(N)) active#(terms(N)) -> sqr#(N) active#(terms(N)) -> recip#(sqr(N)) active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N))) active#(sqr(s(X))) -> dbl#(X) active#(sqr(s(X))) -> sqr#(X) active#(sqr(s(X))) -> add#(sqr(X),dbl(X)) active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X))) active#(dbl(s(X))) -> dbl#(X) active#(dbl(s(X))) -> s#(dbl(X)) active#(dbl(s(X))) -> s#(s(dbl(X))) active#(add(s(X),Y)) -> add#(X,Y) active#(add(s(X),Y)) -> s#(add(X,Y)) active#(first(s(X),cons(Y,Z))) -> first#(X,Z) active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(terms(X)) -> active#(X) active#(terms(X)) -> terms#(active(X)) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(recip(X)) -> active#(X) active#(recip(X)) -> recip#(active(X)) active#(sqr(X)) -> active#(X) active#(sqr(X)) -> sqr#(active(X)) active#(add(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> add#(active(X1),X2) active#(add(X1,X2)) -> active#(X2) active#(add(X1,X2)) -> add#(X1,active(X2)) active#(dbl(X)) -> active#(X) active#(dbl(X)) -> dbl#(active(X)) active#(first(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> first#(active(X1),X2) active#(first(X1,X2)) -> active#(X2) active#(first(X1,X2)) -> first#(X1,active(X2)) terms#(mark(X)) -> terms#(X) cons#(mark(X1),X2) -> cons#(X1,X2) recip#(mark(X)) -> recip#(X) sqr#(mark(X)) -> sqr#(X) add#(mark(X1),X2) -> add#(X1,X2) add#(X1,mark(X2)) -> add#(X1,X2) dbl#(mark(X)) -> dbl#(X) first#(mark(X1),X2) -> first#(X1,X2) first#(X1,mark(X2)) -> first#(X1,X2) proper#(terms(X)) -> proper#(X) proper#(terms(X)) -> terms#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(recip(X)) -> proper#(X) proper#(recip(X)) -> recip#(proper(X)) proper#(sqr(X)) -> proper#(X) proper#(sqr(X)) -> sqr#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(add(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(dbl(X)) -> proper#(X) proper#(dbl(X)) -> dbl#(proper(X)) proper#(first(X1,X2)) -> proper#(X2) proper#(first(X1,X2)) -> proper#(X1) proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) terms#(ok(X)) -> terms#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) recip#(ok(X)) -> recip#(X) sqr#(ok(X)) -> sqr#(X) s#(ok(X)) -> s#(X) add#(ok(X1),ok(X2)) -> add#(X1,X2) dbl#(ok(X)) -> dbl#(X) first#(ok(X1),ok(X2)) -> first#(X1,X2) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(terms(N)) -> s#(N) active#(terms(N)) -> terms#(s(N)) active#(terms(N)) -> sqr#(N) active#(terms(N)) -> recip#(sqr(N)) active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N))) active#(sqr(s(X))) -> dbl#(X) active#(sqr(s(X))) -> sqr#(X) active#(sqr(s(X))) -> add#(sqr(X),dbl(X)) active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X))) active#(dbl(s(X))) -> dbl#(X) active#(dbl(s(X))) -> s#(dbl(X)) active#(dbl(s(X))) -> s#(s(dbl(X))) active#(add(s(X),Y)) -> add#(X,Y) active#(add(s(X),Y)) -> s#(add(X,Y)) active#(first(s(X),cons(Y,Z))) -> first#(X,Z) active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(terms(X)) -> active#(X) active#(terms(X)) -> terms#(active(X)) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(recip(X)) -> active#(X) active#(recip(X)) -> recip#(active(X)) active#(sqr(X)) -> active#(X) active#(sqr(X)) -> sqr#(active(X)) active#(add(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> add#(active(X1),X2) active#(add(X1,X2)) -> active#(X2) active#(add(X1,X2)) -> add#(X1,active(X2)) active#(dbl(X)) -> active#(X) active#(dbl(X)) -> dbl#(active(X)) active#(first(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> first#(active(X1),X2) active#(first(X1,X2)) -> active#(X2) active#(first(X1,X2)) -> first#(X1,active(X2)) terms#(mark(X)) -> terms#(X) cons#(mark(X1),X2) -> cons#(X1,X2) recip#(mark(X)) -> recip#(X) sqr#(mark(X)) -> sqr#(X) add#(mark(X1),X2) -> add#(X1,X2) add#(X1,mark(X2)) -> add#(X1,X2) dbl#(mark(X)) -> dbl#(X) first#(mark(X1),X2) -> first#(X1,X2) first#(X1,mark(X2)) -> first#(X1,X2) proper#(terms(X)) -> proper#(X) proper#(terms(X)) -> terms#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(recip(X)) -> proper#(X) proper#(recip(X)) -> recip#(proper(X)) proper#(sqr(X)) -> proper#(X) proper#(sqr(X)) -> sqr#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(add(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(dbl(X)) -> proper#(X) proper#(dbl(X)) -> dbl#(proper(X)) proper#(first(X1,X2)) -> proper#(X2) proper#(first(X1,X2)) -> proper#(X1) proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) terms#(ok(X)) -> terms#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) recip#(ok(X)) -> recip#(X) sqr#(ok(X)) -> sqr#(X) s#(ok(X)) -> s#(X) add#(ok(X1),ok(X2)) -> add#(X1,X2) dbl#(ok(X)) -> dbl#(X) first#(ok(X1),ok(X2)) -> first#(X1,X2) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X)) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X) top#(ok(X)) -> active#(X) -> active#(first(X1,X2)) -> first#(X1,active(X2)) top#(ok(X)) -> active#(X) -> active#(first(X1,X2)) -> active#(X2) top#(ok(X)) -> active#(X) -> active#(first(X1,X2)) -> first#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(first(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(dbl(X)) -> dbl#(active(X)) top#(ok(X)) -> active#(X) -> active#(dbl(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(add(X1,X2)) -> add#(X1,active(X2)) top#(ok(X)) -> active#(X) -> active#(add(X1,X2)) -> active#(X2) top#(ok(X)) -> active#(X) -> active#(add(X1,X2)) -> add#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(add(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(sqr(X)) -> sqr#(active(X)) top#(ok(X)) -> active#(X) -> active#(sqr(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(recip(X)) -> recip#(active(X)) top#(ok(X)) -> active#(X) -> active#(recip(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(terms(X)) -> terms#(active(X)) top#(ok(X)) -> active#(X) -> active#(terms(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) top#(ok(X)) -> active#(X) -> active#(first(s(X),cons(Y,Z))) -> first#(X,Z) top#(ok(X)) -> active#(X) -> active#(add(s(X),Y)) -> s#(add(X,Y)) top#(ok(X)) -> active#(X) -> active#(add(s(X),Y)) -> add#(X,Y) top#(ok(X)) -> active#(X) -> active#(dbl(s(X))) -> s#(s(dbl(X))) top#(ok(X)) -> active#(X) -> active#(dbl(s(X))) -> s#(dbl(X)) top#(ok(X)) -> active#(X) -> active#(dbl(s(X))) -> dbl#(X) top#(ok(X)) -> active#(X) -> active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X))) top#(ok(X)) -> active#(X) -> active#(sqr(s(X))) -> add#(sqr(X),dbl(X)) top#(ok(X)) -> active#(X) -> active#(sqr(s(X))) -> sqr#(X) top#(ok(X)) -> active#(X) -> active#(sqr(s(X))) -> dbl#(X) top#(ok(X)) -> active#(X) -> active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N))) top#(ok(X)) -> active#(X) -> active#(terms(N)) -> recip#(sqr(N)) top#(ok(X)) -> active#(X) -> active#(terms(N)) -> sqr#(N) top#(ok(X)) -> active#(X) -> active#(terms(N)) -> terms#(s(N)) top#(ok(X)) -> active#(X) -> active#(terms(N)) -> s#(N) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> top#(proper(X)) top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(dbl(X)) -> dbl#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(dbl(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(sqr(X)) -> sqr#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(sqr(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(recip(X)) -> recip#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(recip(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(terms(X)) -> terms#(proper(X)) top#(mark(X)) -> proper#(X) -> proper#(terms(X)) -> proper#(X) proper#(first(X1,X2)) -> proper#(X2) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(first(X1,X2)) -> proper#(X2) -> proper#(first(X1,X2)) -> proper#(X1) proper#(first(X1,X2)) -> proper#(X2) -> proper#(first(X1,X2)) -> proper#(X2) proper#(first(X1,X2)) -> proper#(X2) -> proper#(dbl(X)) -> dbl#(proper(X)) proper#(first(X1,X2)) -> proper#(X2) -> proper#(dbl(X)) -> proper#(X) proper#(first(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(first(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> proper#(X1) proper#(first(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> proper#(X2) proper#(first(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(first(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(first(X1,X2)) -> proper#(X2) -> proper#(sqr(X)) -> sqr#(proper(X)) proper#(first(X1,X2)) -> proper#(X2) -> proper#(sqr(X)) -> proper#(X) proper#(first(X1,X2)) -> proper#(X2) -> proper#(recip(X)) -> recip#(proper(X)) proper#(first(X1,X2)) -> proper#(X2) -> proper#(recip(X)) -> proper#(X) proper#(first(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(first(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(first(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(first(X1,X2)) -> proper#(X2) -> proper#(terms(X)) -> terms#(proper(X)) proper#(first(X1,X2)) -> proper#(X2) -> proper#(terms(X)) -> proper#(X) proper#(first(X1,X2)) -> proper#(X1) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(first(X1,X2)) -> proper#(X1) -> proper#(first(X1,X2)) -> proper#(X1) proper#(first(X1,X2)) -> proper#(X1) -> proper#(first(X1,X2)) -> proper#(X2) proper#(first(X1,X2)) -> proper#(X1) -> proper#(dbl(X)) -> dbl#(proper(X)) proper#(first(X1,X2)) -> proper#(X1) -> proper#(dbl(X)) -> proper#(X) proper#(first(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(first(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> proper#(X1) proper#(first(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> proper#(X2) proper#(first(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(first(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(first(X1,X2)) -> proper#(X1) -> proper#(sqr(X)) -> sqr#(proper(X)) proper#(first(X1,X2)) -> proper#(X1) -> proper#(sqr(X)) -> proper#(X) proper#(first(X1,X2)) -> proper#(X1) -> proper#(recip(X)) -> recip#(proper(X)) proper#(first(X1,X2)) -> proper#(X1) -> proper#(recip(X)) -> proper#(X) proper#(first(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(first(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(first(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(first(X1,X2)) -> proper#(X1) -> proper#(terms(X)) -> terms#(proper(X)) proper#(first(X1,X2)) -> proper#(X1) -> proper#(terms(X)) -> proper#(X) proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) -> first#(ok(X1),ok(X2)) -> first#(X1,X2) proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) -> first#(X1,mark(X2)) -> first#(X1,X2) proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) -> first#(mark(X1),X2) -> first#(X1,X2) proper#(add(X1,X2)) -> proper#(X2) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(add(X1,X2)) -> proper#(X2) -> proper#(first(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X2) -> proper#(first(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X2) -> proper#(dbl(X)) -> dbl#(proper(X)) proper#(add(X1,X2)) -> proper#(X2) -> proper#(dbl(X)) -> proper#(X) proper#(add(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(add(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(add(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(add(X1,X2)) -> proper#(X2) -> proper#(sqr(X)) -> sqr#(proper(X)) proper#(add(X1,X2)) -> proper#(X2) -> proper#(sqr(X)) -> proper#(X) proper#(add(X1,X2)) -> proper#(X2) -> proper#(recip(X)) -> recip#(proper(X)) proper#(add(X1,X2)) -> proper#(X2) -> proper#(recip(X)) -> proper#(X) proper#(add(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(add(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X2) -> proper#(terms(X)) -> terms#(proper(X)) proper#(add(X1,X2)) -> proper#(X2) -> proper#(terms(X)) -> proper#(X) proper#(add(X1,X2)) -> proper#(X1) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(add(X1,X2)) -> proper#(X1) -> proper#(first(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X1) -> proper#(first(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X1) -> proper#(dbl(X)) -> dbl#(proper(X)) proper#(add(X1,X2)) -> proper#(X1) -> proper#(dbl(X)) -> proper#(X) proper#(add(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(add(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(add(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(add(X1,X2)) -> proper#(X1) -> proper#(sqr(X)) -> sqr#(proper(X)) proper#(add(X1,X2)) -> proper#(X1) -> proper#(sqr(X)) -> proper#(X) proper#(add(X1,X2)) -> proper#(X1) -> proper#(recip(X)) -> recip#(proper(X)) proper#(add(X1,X2)) -> proper#(X1) -> proper#(recip(X)) -> proper#(X) proper#(add(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(add(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(add(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X1) -> proper#(terms(X)) -> terms#(proper(X)) proper#(add(X1,X2)) -> proper#(X1) -> proper#(terms(X)) -> proper#(X) proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) -> add#(ok(X1),ok(X2)) -> add#(X1,X2) proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) -> add#(X1,mark(X2)) -> add#(X1,X2) proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) -> add#(mark(X1),X2) -> add#(X1,X2) proper#(dbl(X)) -> proper#(X) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(dbl(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X1) proper#(dbl(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X2) proper#(dbl(X)) -> proper#(X) -> proper#(dbl(X)) -> dbl#(proper(X)) proper#(dbl(X)) -> proper#(X) -> proper#(dbl(X)) -> proper#(X) proper#(dbl(X)) -> proper#(X) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(dbl(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X1) proper#(dbl(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X2) proper#(dbl(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(dbl(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(dbl(X)) -> proper#(X) -> proper#(sqr(X)) -> sqr#(proper(X)) proper#(dbl(X)) -> proper#(X) -> proper#(sqr(X)) -> proper#(X) proper#(dbl(X)) -> proper#(X) -> proper#(recip(X)) -> recip#(proper(X)) proper#(dbl(X)) -> proper#(X) -> proper#(recip(X)) -> proper#(X) proper#(dbl(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(dbl(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(dbl(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(dbl(X)) -> proper#(X) -> proper#(terms(X)) -> terms#(proper(X)) proper#(dbl(X)) -> proper#(X) -> proper#(terms(X)) -> proper#(X) proper#(dbl(X)) -> dbl#(proper(X)) -> dbl#(ok(X)) -> dbl#(X) proper#(dbl(X)) -> dbl#(proper(X)) -> dbl#(mark(X)) -> dbl#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(first(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(first(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(dbl(X)) -> dbl#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(dbl(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(add(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(sqr(X)) -> sqr#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(sqr(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(recip(X)) -> recip#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(recip(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(terms(X)) -> terms#(proper(X)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(terms(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(first(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(first(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(dbl(X)) -> dbl#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(dbl(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(add(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> s#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(sqr(X)) -> sqr#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(sqr(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(recip(X)) -> recip#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(recip(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(terms(X)) -> terms#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(terms(X)) -> proper#(X) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(mark(X1),X2) -> cons#(X1,X2) proper#(s(X)) -> proper#(X) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(dbl(X)) -> dbl#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(dbl(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(sqr(X)) -> sqr#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(sqr(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(recip(X)) -> recip#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(recip(X)) -> proper#(X) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(s(X)) -> proper#(X) -> proper#(terms(X)) -> terms#(proper(X)) proper#(s(X)) -> proper#(X) -> proper#(terms(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) -> s#(ok(X)) -> s#(X) proper#(recip(X)) -> proper#(X) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(recip(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X1) proper#(recip(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X2) proper#(recip(X)) -> proper#(X) -> proper#(dbl(X)) -> dbl#(proper(X)) proper#(recip(X)) -> proper#(X) -> proper#(dbl(X)) -> proper#(X) proper#(recip(X)) -> proper#(X) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(recip(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X1) proper#(recip(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X2) proper#(recip(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(recip(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(recip(X)) -> proper#(X) -> proper#(sqr(X)) -> sqr#(proper(X)) proper#(recip(X)) -> proper#(X) -> proper#(sqr(X)) -> proper#(X) proper#(recip(X)) -> proper#(X) -> proper#(recip(X)) -> recip#(proper(X)) proper#(recip(X)) -> proper#(X) -> proper#(recip(X)) -> proper#(X) proper#(recip(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(recip(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(recip(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(recip(X)) -> proper#(X) -> proper#(terms(X)) -> terms#(proper(X)) proper#(recip(X)) -> proper#(X) -> proper#(terms(X)) -> proper#(X) proper#(recip(X)) -> recip#(proper(X)) -> recip#(ok(X)) -> recip#(X) proper#(recip(X)) -> recip#(proper(X)) -> recip#(mark(X)) -> recip#(X) proper#(sqr(X)) -> proper#(X) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(sqr(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X1) proper#(sqr(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X2) proper#(sqr(X)) -> proper#(X) -> proper#(dbl(X)) -> dbl#(proper(X)) proper#(sqr(X)) -> proper#(X) -> proper#(dbl(X)) -> proper#(X) proper#(sqr(X)) -> proper#(X) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(sqr(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X1) proper#(sqr(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X2) proper#(sqr(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(sqr(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(sqr(X)) -> proper#(X) -> proper#(sqr(X)) -> sqr#(proper(X)) proper#(sqr(X)) -> proper#(X) -> proper#(sqr(X)) -> proper#(X) proper#(sqr(X)) -> proper#(X) -> proper#(recip(X)) -> recip#(proper(X)) proper#(sqr(X)) -> proper#(X) -> proper#(recip(X)) -> proper#(X) proper#(sqr(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(sqr(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(sqr(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(sqr(X)) -> proper#(X) -> proper#(terms(X)) -> terms#(proper(X)) proper#(sqr(X)) -> proper#(X) -> proper#(terms(X)) -> proper#(X) proper#(sqr(X)) -> sqr#(proper(X)) -> sqr#(ok(X)) -> sqr#(X) proper#(sqr(X)) -> sqr#(proper(X)) -> sqr#(mark(X)) -> sqr#(X) proper#(terms(X)) -> proper#(X) -> proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) proper#(terms(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X1) proper#(terms(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X2) proper#(terms(X)) -> proper#(X) -> proper#(dbl(X)) -> dbl#(proper(X)) proper#(terms(X)) -> proper#(X) -> proper#(dbl(X)) -> proper#(X) proper#(terms(X)) -> proper#(X) -> proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) proper#(terms(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X1) proper#(terms(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X2) proper#(terms(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X)) proper#(terms(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X) proper#(terms(X)) -> proper#(X) -> proper#(sqr(X)) -> sqr#(proper(X)) proper#(terms(X)) -> proper#(X) -> proper#(sqr(X)) -> proper#(X) proper#(terms(X)) -> proper#(X) -> proper#(recip(X)) -> recip#(proper(X)) proper#(terms(X)) -> proper#(X) -> proper#(recip(X)) -> proper#(X) proper#(terms(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(terms(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(terms(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(terms(X)) -> proper#(X) -> proper#(terms(X)) -> terms#(proper(X)) proper#(terms(X)) -> proper#(X) -> proper#(terms(X)) -> proper#(X) proper#(terms(X)) -> terms#(proper(X)) -> terms#(ok(X)) -> terms#(X) proper#(terms(X)) -> terms#(proper(X)) -> terms#(mark(X)) -> terms#(X) first#(ok(X1),ok(X2)) -> first#(X1,X2) -> first#(ok(X1),ok(X2)) -> first#(X1,X2) first#(ok(X1),ok(X2)) -> first#(X1,X2) -> first#(X1,mark(X2)) -> first#(X1,X2) first#(ok(X1),ok(X2)) -> first#(X1,X2) -> first#(mark(X1),X2) -> first#(X1,X2) first#(mark(X1),X2) -> first#(X1,X2) -> first#(ok(X1),ok(X2)) -> first#(X1,X2) first#(mark(X1),X2) -> first#(X1,X2) -> first#(X1,mark(X2)) -> first#(X1,X2) first#(mark(X1),X2) -> first#(X1,X2) -> first#(mark(X1),X2) -> first#(X1,X2) first#(X1,mark(X2)) -> first#(X1,X2) -> first#(ok(X1),ok(X2)) -> first#(X1,X2) first#(X1,mark(X2)) -> first#(X1,X2) -> first#(X1,mark(X2)) -> first#(X1,X2) first#(X1,mark(X2)) -> first#(X1,X2) -> first#(mark(X1),X2) -> first#(X1,X2) add#(ok(X1),ok(X2)) -> add#(X1,X2) -> add#(ok(X1),ok(X2)) -> add#(X1,X2) add#(ok(X1),ok(X2)) -> add#(X1,X2) -> add#(X1,mark(X2)) -> add#(X1,X2) add#(ok(X1),ok(X2)) -> add#(X1,X2) -> add#(mark(X1),X2) -> add#(X1,X2) add#(mark(X1),X2) -> add#(X1,X2) -> add#(ok(X1),ok(X2)) -> add#(X1,X2) add#(mark(X1),X2) -> add#(X1,X2) -> add#(X1,mark(X2)) -> add#(X1,X2) add#(mark(X1),X2) -> add#(X1,X2) -> add#(mark(X1),X2) -> add#(X1,X2) add#(X1,mark(X2)) -> add#(X1,X2) -> add#(ok(X1),ok(X2)) -> add#(X1,X2) add#(X1,mark(X2)) -> add#(X1,X2) -> add#(X1,mark(X2)) -> add#(X1,X2) add#(X1,mark(X2)) -> add#(X1,X2) -> add#(mark(X1),X2) -> add#(X1,X2) dbl#(ok(X)) -> dbl#(X) -> dbl#(ok(X)) -> dbl#(X) dbl#(ok(X)) -> dbl#(X) -> dbl#(mark(X)) -> dbl#(X) dbl#(mark(X)) -> dbl#(X) -> dbl#(ok(X)) -> dbl#(X) dbl#(mark(X)) -> dbl#(X) -> dbl#(mark(X)) -> dbl#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) recip#(ok(X)) -> recip#(X) -> recip#(ok(X)) -> recip#(X) recip#(ok(X)) -> recip#(X) -> recip#(mark(X)) -> recip#(X) recip#(mark(X)) -> recip#(X) -> recip#(ok(X)) -> recip#(X) recip#(mark(X)) -> recip#(X) -> recip#(mark(X)) -> recip#(X) sqr#(ok(X)) -> sqr#(X) -> sqr#(ok(X)) -> sqr#(X) sqr#(ok(X)) -> sqr#(X) -> sqr#(mark(X)) -> sqr#(X) sqr#(mark(X)) -> sqr#(X) -> sqr#(ok(X)) -> sqr#(X) sqr#(mark(X)) -> sqr#(X) -> sqr#(mark(X)) -> sqr#(X) terms#(ok(X)) -> terms#(X) -> terms#(ok(X)) -> terms#(X) terms#(ok(X)) -> terms#(X) -> terms#(mark(X)) -> terms#(X) terms#(mark(X)) -> terms#(X) -> terms#(ok(X)) -> terms#(X) terms#(mark(X)) -> terms#(X) -> terms#(mark(X)) -> terms#(X) s#(ok(X)) -> s#(X) -> s#(ok(X)) -> s#(X) active#(first(s(X),cons(Y,Z))) -> first#(X,Z) -> first#(ok(X1),ok(X2)) -> first#(X1,X2) active#(first(s(X),cons(Y,Z))) -> first#(X,Z) -> first#(X1,mark(X2)) -> first#(X1,X2) active#(first(s(X),cons(Y,Z))) -> first#(X,Z) -> first#(mark(X1),X2) -> first#(X1,X2) active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(first(X1,X2)) -> first#(active(X1),X2) -> first#(ok(X1),ok(X2)) -> first#(X1,X2) active#(first(X1,X2)) -> first#(active(X1),X2) -> first#(X1,mark(X2)) -> first#(X1,X2) active#(first(X1,X2)) -> first#(active(X1),X2) -> first#(mark(X1),X2) -> first#(X1,X2) active#(first(X1,X2)) -> first#(X1,active(X2)) -> first#(ok(X1),ok(X2)) -> first#(X1,X2) active#(first(X1,X2)) -> first#(X1,active(X2)) -> first#(X1,mark(X2)) -> first#(X1,X2) active#(first(X1,X2)) -> first#(X1,active(X2)) -> first#(mark(X1),X2) -> first#(X1,X2) active#(first(X1,X2)) -> active#(X2) -> active#(first(X1,X2)) -> first#(X1,active(X2)) active#(first(X1,X2)) -> active#(X2) -> active#(first(X1,X2)) -> active#(X2) active#(first(X1,X2)) -> active#(X2) -> active#(first(X1,X2)) -> first#(active(X1),X2) active#(first(X1,X2)) -> active#(X2) -> active#(first(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> active#(X2) -> active#(dbl(X)) -> dbl#(active(X)) active#(first(X1,X2)) -> active#(X2) -> active#(dbl(X)) -> active#(X) active#(first(X1,X2)) -> active#(X2) -> active#(add(X1,X2)) -> add#(X1,active(X2)) active#(first(X1,X2)) -> active#(X2) -> active#(add(X1,X2)) -> active#(X2) active#(first(X1,X2)) -> active#(X2) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(first(X1,X2)) -> active#(X2) -> active#(add(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> active#(X2) -> active#(sqr(X)) -> sqr#(active(X)) active#(first(X1,X2)) -> active#(X2) -> active#(sqr(X)) -> active#(X) active#(first(X1,X2)) -> active#(X2) -> active#(recip(X)) -> recip#(active(X)) active#(first(X1,X2)) -> active#(X2) -> active#(recip(X)) -> active#(X) active#(first(X1,X2)) -> active#(X2) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(first(X1,X2)) -> active#(X2) -> active#(cons(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> active#(X2) -> active#(terms(X)) -> terms#(active(X)) active#(first(X1,X2)) -> active#(X2) -> active#(terms(X)) -> active#(X) active#(first(X1,X2)) -> active#(X2) -> active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(first(X1,X2)) -> active#(X2) -> active#(first(s(X),cons(Y,Z))) -> first#(X,Z) active#(first(X1,X2)) -> active#(X2) -> active#(add(s(X),Y)) -> s#(add(X,Y)) active#(first(X1,X2)) -> active#(X2) -> active#(add(s(X),Y)) -> add#(X,Y) active#(first(X1,X2)) -> active#(X2) -> active#(dbl(s(X))) -> s#(s(dbl(X))) active#(first(X1,X2)) -> active#(X2) -> active#(dbl(s(X))) -> s#(dbl(X)) active#(first(X1,X2)) -> active#(X2) -> active#(dbl(s(X))) -> dbl#(X) active#(first(X1,X2)) -> active#(X2) -> active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X))) active#(first(X1,X2)) -> active#(X2) -> active#(sqr(s(X))) -> add#(sqr(X),dbl(X)) active#(first(X1,X2)) -> active#(X2) -> active#(sqr(s(X))) -> sqr#(X) active#(first(X1,X2)) -> active#(X2) -> active#(sqr(s(X))) -> dbl#(X) active#(first(X1,X2)) -> active#(X2) -> active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N))) active#(first(X1,X2)) -> active#(X2) -> active#(terms(N)) -> recip#(sqr(N)) active#(first(X1,X2)) -> active#(X2) -> active#(terms(N)) -> sqr#(N) active#(first(X1,X2)) -> active#(X2) -> active#(terms(N)) -> terms#(s(N)) active#(first(X1,X2)) -> active#(X2) -> active#(terms(N)) -> s#(N) active#(first(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> first#(X1,active(X2)) active#(first(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> active#(X2) active#(first(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> first#(active(X1),X2) active#(first(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> active#(X1) -> active#(dbl(X)) -> dbl#(active(X)) active#(first(X1,X2)) -> active#(X1) -> active#(dbl(X)) -> active#(X) active#(first(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> add#(X1,active(X2)) active#(first(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> active#(X2) active#(first(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(first(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> active#(X1) -> active#(sqr(X)) -> sqr#(active(X)) active#(first(X1,X2)) -> active#(X1) -> active#(sqr(X)) -> active#(X) active#(first(X1,X2)) -> active#(X1) -> active#(recip(X)) -> recip#(active(X)) active#(first(X1,X2)) -> active#(X1) -> active#(recip(X)) -> active#(X) active#(first(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(first(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> active#(X1) -> active#(terms(X)) -> terms#(active(X)) active#(first(X1,X2)) -> active#(X1) -> active#(terms(X)) -> active#(X) active#(first(X1,X2)) -> active#(X1) -> active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(first(X1,X2)) -> active#(X1) -> active#(first(s(X),cons(Y,Z))) -> first#(X,Z) active#(first(X1,X2)) -> active#(X1) -> active#(add(s(X),Y)) -> s#(add(X,Y)) active#(first(X1,X2)) -> active#(X1) -> active#(add(s(X),Y)) -> add#(X,Y) active#(first(X1,X2)) -> active#(X1) -> active#(dbl(s(X))) -> s#(s(dbl(X))) active#(first(X1,X2)) -> active#(X1) -> active#(dbl(s(X))) -> s#(dbl(X)) active#(first(X1,X2)) -> active#(X1) -> active#(dbl(s(X))) -> dbl#(X) active#(first(X1,X2)) -> active#(X1) -> active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X))) active#(first(X1,X2)) -> active#(X1) -> active#(sqr(s(X))) -> add#(sqr(X),dbl(X)) active#(first(X1,X2)) -> active#(X1) -> active#(sqr(s(X))) -> sqr#(X) active#(first(X1,X2)) -> active#(X1) -> active#(sqr(s(X))) -> dbl#(X) active#(first(X1,X2)) -> active#(X1) -> active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N))) active#(first(X1,X2)) -> active#(X1) -> active#(terms(N)) -> recip#(sqr(N)) active#(first(X1,X2)) -> active#(X1) -> active#(terms(N)) -> sqr#(N) active#(first(X1,X2)) -> active#(X1) -> active#(terms(N)) -> terms#(s(N)) active#(first(X1,X2)) -> active#(X1) -> active#(terms(N)) -> s#(N) active#(add(s(X),Y)) -> add#(X,Y) -> add#(ok(X1),ok(X2)) -> add#(X1,X2) active#(add(s(X),Y)) -> add#(X,Y) -> add#(X1,mark(X2)) -> add#(X1,X2) active#(add(s(X),Y)) -> add#(X,Y) -> add#(mark(X1),X2) -> add#(X1,X2) active#(add(s(X),Y)) -> s#(add(X,Y)) -> s#(ok(X)) -> s#(X) active#(add(X1,X2)) -> add#(active(X1),X2) -> add#(ok(X1),ok(X2)) -> add#(X1,X2) active#(add(X1,X2)) -> add#(active(X1),X2) -> add#(X1,mark(X2)) -> add#(X1,X2) active#(add(X1,X2)) -> add#(active(X1),X2) -> add#(mark(X1),X2) -> add#(X1,X2) active#(add(X1,X2)) -> add#(X1,active(X2)) -> add#(ok(X1),ok(X2)) -> add#(X1,X2) active#(add(X1,X2)) -> add#(X1,active(X2)) -> add#(X1,mark(X2)) -> add#(X1,X2) active#(add(X1,X2)) -> add#(X1,active(X2)) -> add#(mark(X1),X2) -> add#(X1,X2) active#(add(X1,X2)) -> active#(X2) -> active#(first(X1,X2)) -> first#(X1,active(X2)) active#(add(X1,X2)) -> active#(X2) -> active#(first(X1,X2)) -> active#(X2) active#(add(X1,X2)) -> active#(X2) -> active#(first(X1,X2)) -> first#(active(X1),X2) active#(add(X1,X2)) -> active#(X2) -> active#(first(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> active#(X2) -> active#(dbl(X)) -> dbl#(active(X)) active#(add(X1,X2)) -> active#(X2) -> active#(dbl(X)) -> active#(X) active#(add(X1,X2)) -> active#(X2) -> active#(add(X1,X2)) -> add#(X1,active(X2)) active#(add(X1,X2)) -> active#(X2) -> active#(add(X1,X2)) -> active#(X2) active#(add(X1,X2)) -> active#(X2) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(add(X1,X2)) -> active#(X2) -> active#(add(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> active#(X2) -> active#(sqr(X)) -> sqr#(active(X)) active#(add(X1,X2)) -> active#(X2) -> active#(sqr(X)) -> active#(X) active#(add(X1,X2)) -> active#(X2) -> active#(recip(X)) -> recip#(active(X)) active#(add(X1,X2)) -> active#(X2) -> active#(recip(X)) -> active#(X) active#(add(X1,X2)) -> active#(X2) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(add(X1,X2)) -> active#(X2) -> active#(cons(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> active#(X2) -> active#(terms(X)) -> terms#(active(X)) active#(add(X1,X2)) -> active#(X2) -> active#(terms(X)) -> active#(X) active#(add(X1,X2)) -> active#(X2) -> active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(add(X1,X2)) -> active#(X2) -> active#(first(s(X),cons(Y,Z))) -> first#(X,Z) active#(add(X1,X2)) -> active#(X2) -> active#(add(s(X),Y)) -> s#(add(X,Y)) active#(add(X1,X2)) -> active#(X2) -> active#(add(s(X),Y)) -> add#(X,Y) active#(add(X1,X2)) -> active#(X2) -> active#(dbl(s(X))) -> s#(s(dbl(X))) active#(add(X1,X2)) -> active#(X2) -> active#(dbl(s(X))) -> s#(dbl(X)) active#(add(X1,X2)) -> active#(X2) -> active#(dbl(s(X))) -> dbl#(X) active#(add(X1,X2)) -> active#(X2) -> active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X))) active#(add(X1,X2)) -> active#(X2) -> active#(sqr(s(X))) -> add#(sqr(X),dbl(X)) active#(add(X1,X2)) -> active#(X2) -> active#(sqr(s(X))) -> sqr#(X) active#(add(X1,X2)) -> active#(X2) -> active#(sqr(s(X))) -> dbl#(X) active#(add(X1,X2)) -> active#(X2) -> active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N))) active#(add(X1,X2)) -> active#(X2) -> active#(terms(N)) -> recip#(sqr(N)) active#(add(X1,X2)) -> active#(X2) -> active#(terms(N)) -> sqr#(N) active#(add(X1,X2)) -> active#(X2) -> active#(terms(N)) -> terms#(s(N)) active#(add(X1,X2)) -> active#(X2) -> active#(terms(N)) -> s#(N) active#(add(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> first#(X1,active(X2)) active#(add(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> active#(X2) active#(add(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> first#(active(X1),X2) active#(add(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> active#(X1) -> active#(dbl(X)) -> dbl#(active(X)) active#(add(X1,X2)) -> active#(X1) -> active#(dbl(X)) -> active#(X) active#(add(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> add#(X1,active(X2)) active#(add(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> active#(X2) active#(add(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(add(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> active#(X1) -> active#(sqr(X)) -> sqr#(active(X)) active#(add(X1,X2)) -> active#(X1) -> active#(sqr(X)) -> active#(X) active#(add(X1,X2)) -> active#(X1) -> active#(recip(X)) -> recip#(active(X)) active#(add(X1,X2)) -> active#(X1) -> active#(recip(X)) -> active#(X) active#(add(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(add(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> active#(X1) -> active#(terms(X)) -> terms#(active(X)) active#(add(X1,X2)) -> active#(X1) -> active#(terms(X)) -> active#(X) active#(add(X1,X2)) -> active#(X1) -> active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(add(X1,X2)) -> active#(X1) -> active#(first(s(X),cons(Y,Z))) -> first#(X,Z) active#(add(X1,X2)) -> active#(X1) -> active#(add(s(X),Y)) -> s#(add(X,Y)) active#(add(X1,X2)) -> active#(X1) -> active#(add(s(X),Y)) -> add#(X,Y) active#(add(X1,X2)) -> active#(X1) -> active#(dbl(s(X))) -> s#(s(dbl(X))) active#(add(X1,X2)) -> active#(X1) -> active#(dbl(s(X))) -> s#(dbl(X)) active#(add(X1,X2)) -> active#(X1) -> active#(dbl(s(X))) -> dbl#(X) active#(add(X1,X2)) -> active#(X1) -> active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X))) active#(add(X1,X2)) -> active#(X1) -> active#(sqr(s(X))) -> add#(sqr(X),dbl(X)) active#(add(X1,X2)) -> active#(X1) -> active#(sqr(s(X))) -> sqr#(X) active#(add(X1,X2)) -> active#(X1) -> active#(sqr(s(X))) -> dbl#(X) active#(add(X1,X2)) -> active#(X1) -> active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N))) active#(add(X1,X2)) -> active#(X1) -> active#(terms(N)) -> recip#(sqr(N)) active#(add(X1,X2)) -> active#(X1) -> active#(terms(N)) -> sqr#(N) active#(add(X1,X2)) -> active#(X1) -> active#(terms(N)) -> terms#(s(N)) active#(add(X1,X2)) -> active#(X1) -> active#(terms(N)) -> s#(N) active#(dbl(s(X))) -> dbl#(X) -> dbl#(ok(X)) -> dbl#(X) active#(dbl(s(X))) -> dbl#(X) -> dbl#(mark(X)) -> dbl#(X) active#(dbl(s(X))) -> s#(dbl(X)) -> s#(ok(X)) -> s#(X) active#(dbl(s(X))) -> s#(s(dbl(X))) -> s#(ok(X)) -> s#(X) active#(dbl(X)) -> dbl#(active(X)) -> dbl#(ok(X)) -> dbl#(X) active#(dbl(X)) -> dbl#(active(X)) -> dbl#(mark(X)) -> dbl#(X) active#(dbl(X)) -> active#(X) -> active#(first(X1,X2)) -> first#(X1,active(X2)) active#(dbl(X)) -> active#(X) -> active#(first(X1,X2)) -> active#(X2) active#(dbl(X)) -> active#(X) -> active#(first(X1,X2)) -> first#(active(X1),X2) active#(dbl(X)) -> active#(X) -> active#(first(X1,X2)) -> active#(X1) active#(dbl(X)) -> active#(X) -> active#(dbl(X)) -> dbl#(active(X)) active#(dbl(X)) -> active#(X) -> active#(dbl(X)) -> active#(X) active#(dbl(X)) -> active#(X) -> active#(add(X1,X2)) -> add#(X1,active(X2)) active#(dbl(X)) -> active#(X) -> active#(add(X1,X2)) -> active#(X2) active#(dbl(X)) -> active#(X) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(dbl(X)) -> active#(X) -> active#(add(X1,X2)) -> active#(X1) active#(dbl(X)) -> active#(X) -> active#(sqr(X)) -> sqr#(active(X)) active#(dbl(X)) -> active#(X) -> active#(sqr(X)) -> active#(X) active#(dbl(X)) -> active#(X) -> active#(recip(X)) -> recip#(active(X)) active#(dbl(X)) -> active#(X) -> active#(recip(X)) -> active#(X) active#(dbl(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(dbl(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(dbl(X)) -> active#(X) -> active#(terms(X)) -> terms#(active(X)) active#(dbl(X)) -> active#(X) -> active#(terms(X)) -> active#(X) active#(dbl(X)) -> active#(X) -> active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(dbl(X)) -> active#(X) -> active#(first(s(X),cons(Y,Z))) -> first#(X,Z) active#(dbl(X)) -> active#(X) -> active#(add(s(X),Y)) -> s#(add(X,Y)) active#(dbl(X)) -> active#(X) -> active#(add(s(X),Y)) -> add#(X,Y) active#(dbl(X)) -> active#(X) -> active#(dbl(s(X))) -> s#(s(dbl(X))) active#(dbl(X)) -> active#(X) -> active#(dbl(s(X))) -> s#(dbl(X)) active#(dbl(X)) -> active#(X) -> active#(dbl(s(X))) -> dbl#(X) active#(dbl(X)) -> active#(X) -> active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X))) active#(dbl(X)) -> active#(X) -> active#(sqr(s(X))) -> add#(sqr(X),dbl(X)) active#(dbl(X)) -> active#(X) -> active#(sqr(s(X))) -> sqr#(X) active#(dbl(X)) -> active#(X) -> active#(sqr(s(X))) -> dbl#(X) active#(dbl(X)) -> active#(X) -> active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N))) active#(dbl(X)) -> active#(X) -> active#(terms(N)) -> recip#(sqr(N)) active#(dbl(X)) -> active#(X) -> active#(terms(N)) -> sqr#(N) active#(dbl(X)) -> active#(X) -> active#(terms(N)) -> terms#(s(N)) active#(dbl(X)) -> active#(X) -> active#(terms(N)) -> s#(N) active#(cons(X1,X2)) -> cons#(active(X1),X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(cons(X1,X2)) -> cons#(active(X1),X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(cons(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> first#(X1,active(X2)) active#(cons(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> active#(X2) active#(cons(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> first#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(first(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(dbl(X)) -> dbl#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(dbl(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> add#(X1,active(X2)) active#(cons(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> active#(X2) active#(cons(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(add(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(sqr(X)) -> sqr#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(sqr(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(recip(X)) -> recip#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(recip(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(terms(X)) -> terms#(active(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(terms(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(cons(X1,X2)) -> active#(X1) -> active#(first(s(X),cons(Y,Z))) -> first#(X,Z) active#(cons(X1,X2)) -> active#(X1) -> active#(add(s(X),Y)) -> s#(add(X,Y)) active#(cons(X1,X2)) -> active#(X1) -> active#(add(s(X),Y)) -> add#(X,Y) active#(cons(X1,X2)) -> active#(X1) -> active#(dbl(s(X))) -> s#(s(dbl(X))) active#(cons(X1,X2)) -> active#(X1) -> active#(dbl(s(X))) -> s#(dbl(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(dbl(s(X))) -> dbl#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X))) active#(cons(X1,X2)) -> active#(X1) -> active#(sqr(s(X))) -> add#(sqr(X),dbl(X)) active#(cons(X1,X2)) -> active#(X1) -> active#(sqr(s(X))) -> sqr#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(sqr(s(X))) -> dbl#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N))) active#(cons(X1,X2)) -> active#(X1) -> active#(terms(N)) -> recip#(sqr(N)) active#(cons(X1,X2)) -> active#(X1) -> active#(terms(N)) -> sqr#(N) active#(cons(X1,X2)) -> active#(X1) -> active#(terms(N)) -> terms#(s(N)) active#(cons(X1,X2)) -> active#(X1) -> active#(terms(N)) -> s#(N) active#(recip(X)) -> recip#(active(X)) -> recip#(ok(X)) -> recip#(X) active#(recip(X)) -> recip#(active(X)) -> recip#(mark(X)) -> recip#(X) active#(recip(X)) -> active#(X) -> active#(first(X1,X2)) -> first#(X1,active(X2)) active#(recip(X)) -> active#(X) -> active#(first(X1,X2)) -> active#(X2) active#(recip(X)) -> active#(X) -> active#(first(X1,X2)) -> first#(active(X1),X2) active#(recip(X)) -> active#(X) -> active#(first(X1,X2)) -> active#(X1) active#(recip(X)) -> active#(X) -> active#(dbl(X)) -> dbl#(active(X)) active#(recip(X)) -> active#(X) -> active#(dbl(X)) -> active#(X) active#(recip(X)) -> active#(X) -> active#(add(X1,X2)) -> add#(X1,active(X2)) active#(recip(X)) -> active#(X) -> active#(add(X1,X2)) -> active#(X2) active#(recip(X)) -> active#(X) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(recip(X)) -> active#(X) -> active#(add(X1,X2)) -> active#(X1) active#(recip(X)) -> active#(X) -> active#(sqr(X)) -> sqr#(active(X)) active#(recip(X)) -> active#(X) -> active#(sqr(X)) -> active#(X) active#(recip(X)) -> active#(X) -> active#(recip(X)) -> recip#(active(X)) active#(recip(X)) -> active#(X) -> active#(recip(X)) -> active#(X) active#(recip(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(recip(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(recip(X)) -> active#(X) -> active#(terms(X)) -> terms#(active(X)) active#(recip(X)) -> active#(X) -> active#(terms(X)) -> active#(X) active#(recip(X)) -> active#(X) -> active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(recip(X)) -> active#(X) -> active#(first(s(X),cons(Y,Z))) -> first#(X,Z) active#(recip(X)) -> active#(X) -> active#(add(s(X),Y)) -> s#(add(X,Y)) active#(recip(X)) -> active#(X) -> active#(add(s(X),Y)) -> add#(X,Y) active#(recip(X)) -> active#(X) -> active#(dbl(s(X))) -> s#(s(dbl(X))) active#(recip(X)) -> active#(X) -> active#(dbl(s(X))) -> s#(dbl(X)) active#(recip(X)) -> active#(X) -> active#(dbl(s(X))) -> dbl#(X) active#(recip(X)) -> active#(X) -> active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X))) active#(recip(X)) -> active#(X) -> active#(sqr(s(X))) -> add#(sqr(X),dbl(X)) active#(recip(X)) -> active#(X) -> active#(sqr(s(X))) -> sqr#(X) active#(recip(X)) -> active#(X) -> active#(sqr(s(X))) -> dbl#(X) active#(recip(X)) -> active#(X) -> active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N))) active#(recip(X)) -> active#(X) -> active#(terms(N)) -> recip#(sqr(N)) active#(recip(X)) -> active#(X) -> active#(terms(N)) -> sqr#(N) active#(recip(X)) -> active#(X) -> active#(terms(N)) -> terms#(s(N)) active#(recip(X)) -> active#(X) -> active#(terms(N)) -> s#(N) active#(sqr(s(X))) -> add#(sqr(X),dbl(X)) -> add#(ok(X1),ok(X2)) -> add#(X1,X2) active#(sqr(s(X))) -> add#(sqr(X),dbl(X)) -> add#(X1,mark(X2)) -> add#(X1,X2) active#(sqr(s(X))) -> add#(sqr(X),dbl(X)) -> add#(mark(X1),X2) -> add#(X1,X2) active#(sqr(s(X))) -> dbl#(X) -> dbl#(ok(X)) -> dbl#(X) active#(sqr(s(X))) -> dbl#(X) -> dbl#(mark(X)) -> dbl#(X) active#(sqr(s(X))) -> sqr#(X) -> sqr#(ok(X)) -> sqr#(X) active#(sqr(s(X))) -> sqr#(X) -> sqr#(mark(X)) -> sqr#(X) active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X))) -> s#(ok(X)) -> s#(X) active#(sqr(X)) -> sqr#(active(X)) -> sqr#(ok(X)) -> sqr#(X) active#(sqr(X)) -> sqr#(active(X)) -> sqr#(mark(X)) -> sqr#(X) active#(sqr(X)) -> active#(X) -> active#(first(X1,X2)) -> first#(X1,active(X2)) active#(sqr(X)) -> active#(X) -> active#(first(X1,X2)) -> active#(X2) active#(sqr(X)) -> active#(X) -> active#(first(X1,X2)) -> first#(active(X1),X2) active#(sqr(X)) -> active#(X) -> active#(first(X1,X2)) -> active#(X1) active#(sqr(X)) -> active#(X) -> active#(dbl(X)) -> dbl#(active(X)) active#(sqr(X)) -> active#(X) -> active#(dbl(X)) -> active#(X) active#(sqr(X)) -> active#(X) -> active#(add(X1,X2)) -> add#(X1,active(X2)) active#(sqr(X)) -> active#(X) -> active#(add(X1,X2)) -> active#(X2) active#(sqr(X)) -> active#(X) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(sqr(X)) -> active#(X) -> active#(add(X1,X2)) -> active#(X1) active#(sqr(X)) -> active#(X) -> active#(sqr(X)) -> sqr#(active(X)) active#(sqr(X)) -> active#(X) -> active#(sqr(X)) -> active#(X) active#(sqr(X)) -> active#(X) -> active#(recip(X)) -> recip#(active(X)) active#(sqr(X)) -> active#(X) -> active#(recip(X)) -> active#(X) active#(sqr(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(sqr(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(sqr(X)) -> active#(X) -> active#(terms(X)) -> terms#(active(X)) active#(sqr(X)) -> active#(X) -> active#(terms(X)) -> active#(X) active#(sqr(X)) -> active#(X) -> active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(sqr(X)) -> active#(X) -> active#(first(s(X),cons(Y,Z))) -> first#(X,Z) active#(sqr(X)) -> active#(X) -> active#(add(s(X),Y)) -> s#(add(X,Y)) active#(sqr(X)) -> active#(X) -> active#(add(s(X),Y)) -> add#(X,Y) active#(sqr(X)) -> active#(X) -> active#(dbl(s(X))) -> s#(s(dbl(X))) active#(sqr(X)) -> active#(X) -> active#(dbl(s(X))) -> s#(dbl(X)) active#(sqr(X)) -> active#(X) -> active#(dbl(s(X))) -> dbl#(X) active#(sqr(X)) -> active#(X) -> active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X))) active#(sqr(X)) -> active#(X) -> active#(sqr(s(X))) -> add#(sqr(X),dbl(X)) active#(sqr(X)) -> active#(X) -> active#(sqr(s(X))) -> sqr#(X) active#(sqr(X)) -> active#(X) -> active#(sqr(s(X))) -> dbl#(X) active#(sqr(X)) -> active#(X) -> active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N))) active#(sqr(X)) -> active#(X) -> active#(terms(N)) -> recip#(sqr(N)) active#(sqr(X)) -> active#(X) -> active#(terms(N)) -> sqr#(N) active#(sqr(X)) -> active#(X) -> active#(terms(N)) -> terms#(s(N)) active#(sqr(X)) -> active#(X) -> active#(terms(N)) -> s#(N) active#(terms(X)) -> terms#(active(X)) -> terms#(ok(X)) -> terms#(X) active#(terms(X)) -> terms#(active(X)) -> terms#(mark(X)) -> terms#(X) active#(terms(X)) -> active#(X) -> active#(first(X1,X2)) -> first#(X1,active(X2)) active#(terms(X)) -> active#(X) -> active#(first(X1,X2)) -> active#(X2) active#(terms(X)) -> active#(X) -> active#(first(X1,X2)) -> first#(active(X1),X2) active#(terms(X)) -> active#(X) -> active#(first(X1,X2)) -> active#(X1) active#(terms(X)) -> active#(X) -> active#(dbl(X)) -> dbl#(active(X)) active#(terms(X)) -> active#(X) -> active#(dbl(X)) -> active#(X) active#(terms(X)) -> active#(X) -> active#(add(X1,X2)) -> add#(X1,active(X2)) active#(terms(X)) -> active#(X) -> active#(add(X1,X2)) -> active#(X2) active#(terms(X)) -> active#(X) -> active#(add(X1,X2)) -> add#(active(X1),X2) active#(terms(X)) -> active#(X) -> active#(add(X1,X2)) -> active#(X1) active#(terms(X)) -> active#(X) -> active#(sqr(X)) -> sqr#(active(X)) active#(terms(X)) -> active#(X) -> active#(sqr(X)) -> active#(X) active#(terms(X)) -> active#(X) -> active#(recip(X)) -> recip#(active(X)) active#(terms(X)) -> active#(X) -> active#(recip(X)) -> active#(X) active#(terms(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(terms(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(terms(X)) -> active#(X) -> active#(terms(X)) -> terms#(active(X)) active#(terms(X)) -> active#(X) -> active#(terms(X)) -> active#(X) active#(terms(X)) -> active#(X) -> active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(terms(X)) -> active#(X) -> active#(first(s(X),cons(Y,Z))) -> first#(X,Z) active#(terms(X)) -> active#(X) -> active#(add(s(X),Y)) -> s#(add(X,Y)) active#(terms(X)) -> active#(X) -> active#(add(s(X),Y)) -> add#(X,Y) active#(terms(X)) -> active#(X) -> active#(dbl(s(X))) -> s#(s(dbl(X))) active#(terms(X)) -> active#(X) -> active#(dbl(s(X))) -> s#(dbl(X)) active#(terms(X)) -> active#(X) -> active#(dbl(s(X))) -> dbl#(X) active#(terms(X)) -> active#(X) -> active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X))) active#(terms(X)) -> active#(X) -> active#(sqr(s(X))) -> add#(sqr(X),dbl(X)) active#(terms(X)) -> active#(X) -> active#(sqr(s(X))) -> sqr#(X) active#(terms(X)) -> active#(X) -> active#(sqr(s(X))) -> dbl#(X) active#(terms(X)) -> active#(X) -> active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N))) active#(terms(X)) -> active#(X) -> active#(terms(N)) -> recip#(sqr(N)) active#(terms(X)) -> active#(X) -> active#(terms(N)) -> sqr#(N) active#(terms(X)) -> active#(X) -> active#(terms(N)) -> terms#(s(N)) active#(terms(X)) -> active#(X) -> active#(terms(N)) -> s#(N) active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N))) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N))) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(terms(N)) -> recip#(sqr(N)) -> recip#(ok(X)) -> recip#(X) active#(terms(N)) -> recip#(sqr(N)) -> recip#(mark(X)) -> recip#(X) active#(terms(N)) -> sqr#(N) -> sqr#(ok(X)) -> sqr#(X) active#(terms(N)) -> sqr#(N) -> sqr#(mark(X)) -> sqr#(X) active#(terms(N)) -> terms#(s(N)) -> terms#(ok(X)) -> terms#(X) active#(terms(N)) -> terms#(s(N)) -> terms#(mark(X)) -> terms#(X) active#(terms(N)) -> s#(N) -> s#(ok(X)) -> s#(X) SCC Processor: #sccs: 11 #rules: 39 #arcs: 684/5476 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Usable Rule Processor: DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) s(ok(X)) -> ok(s(X)) terms(mark(X)) -> mark(terms(X)) terms(ok(X)) -> ok(terms(X)) sqr(mark(X)) -> mark(sqr(X)) sqr(ok(X)) -> ok(sqr(X)) recip(mark(X)) -> mark(recip(X)) recip(ok(X)) -> ok(recip(X)) 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)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) LPO Processor: argument filtering: pi(terms) = [0] pi(active) = 0 pi(sqr) = [0] pi(recip) = 0 pi(s) = [] pi(cons) = 0 pi(mark) = [0] pi(0) = [] pi(dbl) = [0] pi(add) = [0,1] pi(first) = [0,1] pi(nil) = [] pi(proper) = 0 pi(ok) = 0 pi(top#) = 0 usable rules: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) s(ok(X)) -> ok(s(X)) terms(mark(X)) -> mark(terms(X)) terms(ok(X)) -> ok(terms(X)) sqr(mark(X)) -> mark(sqr(X)) sqr(ok(X)) -> ok(sqr(X)) recip(mark(X)) -> mark(recip(X)) recip(ok(X)) -> ok(recip(X)) 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)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) precedence: terms > sqr > first ~ add ~ dbl > top# ~ ok ~ proper ~ nil ~ 0 ~ mark ~ cons ~ s ~ recip ~ active problem: DPs: top#(ok(X)) -> top#(active(X)) TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) s(ok(X)) -> ok(s(X)) terms(mark(X)) -> mark(terms(X)) terms(ok(X)) -> ok(terms(X)) sqr(mark(X)) -> mark(sqr(X)) sqr(ok(X)) -> ok(sqr(X)) recip(mark(X)) -> mark(recip(X)) recip(ok(X)) -> ok(recip(X)) 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)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) Restore Modifier: DPs: top#(ok(X)) -> top#(active(X)) TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Usable Rule Processor: DPs: top#(ok(X)) -> top#(active(X)) TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) s(ok(X)) -> ok(s(X)) terms(mark(X)) -> mark(terms(X)) terms(ok(X)) -> ok(terms(X)) sqr(mark(X)) -> mark(sqr(X)) sqr(ok(X)) -> ok(sqr(X)) recip(mark(X)) -> mark(recip(X)) recip(ok(X)) -> ok(recip(X)) 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)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) KBO Processor: argument filtering: pi(terms) = [0] pi(active) = 0 pi(sqr) = 0 pi(recip) = 0 pi(s) = [] pi(cons) = 0 pi(mark) = [] pi(0) = [] pi(dbl) = 0 pi(add) = [1] pi(first) = [0] pi(nil) = [] pi(ok) = [0] pi(top#) = 0 usable rules: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) terms(ok(X)) -> ok(terms(X)) sqr(mark(X)) -> mark(sqr(X)) sqr(ok(X)) -> ok(sqr(X)) recip(mark(X)) -> mark(recip(X)) recip(ok(X)) -> ok(recip(X)) 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)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) weight function: w0 = 1 w(top#) = w(ok) = w(nil) = w(first) = w(add) = w(0) = w(mark) = w( s) = w(recip) = w(terms) = 1 w(dbl) = w(cons) = w(sqr) = w(active) = 0 precedence: 0 > first ~ add ~ s ~ terms > top# ~ ok ~ nil ~ dbl ~ mark ~ cons ~ recip ~ sqr ~ active problem: DPs: TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) s(ok(X)) -> ok(s(X)) terms(mark(X)) -> mark(terms(X)) terms(ok(X)) -> ok(terms(X)) sqr(mark(X)) -> mark(sqr(X)) sqr(ok(X)) -> ok(sqr(X)) recip(mark(X)) -> mark(recip(X)) recip(ok(X)) -> ok(recip(X)) 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)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) Qed DPs: active#(terms(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) active#(recip(X)) -> active#(X) active#(sqr(X)) -> active#(X) active#(add(X1,X2)) -> active#(X1) active#(add(X1,X2)) -> active#(X2) active#(dbl(X)) -> active#(X) active#(first(X1,X2)) -> active#(X1) active#(first(X1,X2)) -> active#(X2) TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(active#) = 0 problem: DPs: TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: proper#(terms(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(recip(X)) -> proper#(X) proper#(sqr(X)) -> proper#(X) proper#(s(X)) -> proper#(X) proper#(add(X1,X2)) -> proper#(X2) proper#(add(X1,X2)) -> proper#(X1) proper#(dbl(X)) -> proper#(X) proper#(first(X1,X2)) -> proper#(X2) proper#(first(X1,X2)) -> proper#(X1) TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(proper#) = 0 problem: DPs: TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: first#(mark(X1),X2) -> first#(X1,X2) first#(X1,mark(X2)) -> first#(X1,X2) first#(ok(X1),ok(X2)) -> first#(X1,X2) TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(first#) = 0 problem: DPs: first#(X1,mark(X2)) -> first#(X1,X2) TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(first#) = 1 problem: DPs: TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: dbl#(mark(X)) -> dbl#(X) dbl#(ok(X)) -> dbl#(X) TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(dbl#) = 0 problem: DPs: TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: add#(mark(X1),X2) -> add#(X1,X2) add#(X1,mark(X2)) -> add#(X1,X2) add#(ok(X1),ok(X2)) -> add#(X1,X2) TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(add#) = 0 problem: DPs: add#(X1,mark(X2)) -> add#(X1,X2) TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(add#) = 1 problem: DPs: TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: s#(ok(X)) -> s#(X) TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(s#) = 0 problem: DPs: TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: sqr#(mark(X)) -> sqr#(X) sqr#(ok(X)) -> sqr#(X) TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(sqr#) = 0 problem: DPs: TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: recip#(mark(X)) -> recip#(X) recip#(ok(X)) -> recip#(X) TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(recip#) = 0 problem: DPs: TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: cons#(mark(X1),X2) -> cons#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(cons#) = 0 problem: DPs: TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: terms#(mark(X)) -> terms#(X) terms#(ok(X)) -> terms#(X) TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(terms#) = 0 problem: DPs: TRS: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1,X2)) add(X1,mark(X2)) -> mark(add(X1,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed