/export/starexec/sandbox/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(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#(terms(N)) -> mark#(cons(recip(sqr(N)),terms(s(N)))) active#(sqr(0())) -> mark#(0()) 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#(sqr(s(X))) -> mark#(s(add(sqr(X),dbl(X)))) active#(dbl(0())) -> mark#(0()) active#(dbl(s(X))) -> dbl#(X) active#(dbl(s(X))) -> s#(dbl(X)) active#(dbl(s(X))) -> s#(s(dbl(X))) active#(dbl(s(X))) -> mark#(s(s(dbl(X)))) active#(add(0(),X)) -> mark#(X) active#(add(s(X),Y)) -> add#(X,Y) active#(add(s(X),Y)) -> s#(add(X,Y)) active#(add(s(X),Y)) -> mark#(s(add(X,Y))) active#(first(0(),X)) -> mark#(nil()) active#(first(s(X),cons(Y,Z))) -> first#(X,Z) active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z))) active#(half(0())) -> mark#(0()) active#(half(s(0()))) -> mark#(0()) active#(half(s(s(X)))) -> half#(X) active#(half(s(s(X)))) -> s#(half(X)) active#(half(s(s(X)))) -> mark#(s(half(X))) active#(half(dbl(X))) -> mark#(X) mark#(terms(X)) -> mark#(X) mark#(terms(X)) -> terms#(mark(X)) mark#(terms(X)) -> active#(terms(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(recip(X)) -> mark#(X) mark#(recip(X)) -> recip#(mark(X)) mark#(recip(X)) -> active#(recip(mark(X))) mark#(sqr(X)) -> mark#(X) mark#(sqr(X)) -> sqr#(mark(X)) mark#(sqr(X)) -> active#(sqr(mark(X))) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(0()) -> active#(0()) mark#(add(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> add#(mark(X1),mark(X2)) mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2))) mark#(dbl(X)) -> mark#(X) mark#(dbl(X)) -> dbl#(mark(X)) mark#(dbl(X)) -> active#(dbl(mark(X))) mark#(first(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> first#(mark(X1),mark(X2)) mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2))) mark#(nil()) -> active#(nil()) mark#(half(X)) -> mark#(X) mark#(half(X)) -> half#(mark(X)) mark#(half(X)) -> active#(half(mark(X))) terms#(mark(X)) -> terms#(X) terms#(active(X)) -> terms#(X) cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) recip#(mark(X)) -> recip#(X) recip#(active(X)) -> recip#(X) sqr#(mark(X)) -> sqr#(X) sqr#(active(X)) -> sqr#(X) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) add#(mark(X1),X2) -> add#(X1,X2) add#(X1,mark(X2)) -> add#(X1,X2) add#(active(X1),X2) -> add#(X1,X2) add#(X1,active(X2)) -> add#(X1,X2) dbl#(mark(X)) -> dbl#(X) dbl#(active(X)) -> dbl#(X) first#(mark(X1),X2) -> first#(X1,X2) first#(X1,mark(X2)) -> first#(X1,X2) first#(active(X1),X2) -> first#(X1,X2) first#(X1,active(X2)) -> first#(X1,X2) half#(mark(X)) -> half#(X) half#(active(X)) -> half#(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(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#(terms(N)) -> mark#(cons(recip(sqr(N)),terms(s(N)))) active#(sqr(0())) -> mark#(0()) 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#(sqr(s(X))) -> mark#(s(add(sqr(X),dbl(X)))) active#(dbl(0())) -> mark#(0()) active#(dbl(s(X))) -> dbl#(X) active#(dbl(s(X))) -> s#(dbl(X)) active#(dbl(s(X))) -> s#(s(dbl(X))) active#(dbl(s(X))) -> mark#(s(s(dbl(X)))) active#(add(0(),X)) -> mark#(X) active#(add(s(X),Y)) -> add#(X,Y) active#(add(s(X),Y)) -> s#(add(X,Y)) active#(add(s(X),Y)) -> mark#(s(add(X,Y))) active#(first(0(),X)) -> mark#(nil()) active#(first(s(X),cons(Y,Z))) -> first#(X,Z) active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z))) active#(half(0())) -> mark#(0()) active#(half(s(0()))) -> mark#(0()) active#(half(s(s(X)))) -> half#(X) active#(half(s(s(X)))) -> s#(half(X)) active#(half(s(s(X)))) -> mark#(s(half(X))) active#(half(dbl(X))) -> mark#(X) mark#(terms(X)) -> mark#(X) mark#(terms(X)) -> terms#(mark(X)) mark#(terms(X)) -> active#(terms(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(recip(X)) -> mark#(X) mark#(recip(X)) -> recip#(mark(X)) mark#(recip(X)) -> active#(recip(mark(X))) mark#(sqr(X)) -> mark#(X) mark#(sqr(X)) -> sqr#(mark(X)) mark#(sqr(X)) -> active#(sqr(mark(X))) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(0()) -> active#(0()) mark#(add(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> add#(mark(X1),mark(X2)) mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2))) mark#(dbl(X)) -> mark#(X) mark#(dbl(X)) -> dbl#(mark(X)) mark#(dbl(X)) -> active#(dbl(mark(X))) mark#(first(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> first#(mark(X1),mark(X2)) mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2))) mark#(nil()) -> active#(nil()) mark#(half(X)) -> mark#(X) mark#(half(X)) -> half#(mark(X)) mark#(half(X)) -> active#(half(mark(X))) terms#(mark(X)) -> terms#(X) terms#(active(X)) -> terms#(X) cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) recip#(mark(X)) -> recip#(X) recip#(active(X)) -> recip#(X) sqr#(mark(X)) -> sqr#(X) sqr#(active(X)) -> sqr#(X) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) add#(mark(X1),X2) -> add#(X1,X2) add#(X1,mark(X2)) -> add#(X1,X2) add#(active(X1),X2) -> add#(X1,X2) add#(X1,active(X2)) -> add#(X1,X2) dbl#(mark(X)) -> dbl#(X) dbl#(active(X)) -> dbl#(X) first#(mark(X1),X2) -> first#(X1,X2) first#(X1,mark(X2)) -> first#(X1,X2) first#(active(X1),X2) -> first#(X1,X2) first#(X1,active(X2)) -> first#(X1,X2) half#(mark(X)) -> half#(X) half#(active(X)) -> half#(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) graph: ... SCC Processor: #sccs: 10 #rules: 59 #arcs: 1227/7396 DPs: mark#(half(X)) -> mark#(X) mark#(terms(X)) -> mark#(X) mark#(terms(X)) -> active#(terms(mark(X))) active#(terms(N)) -> mark#(cons(recip(sqr(N)),terms(s(N)))) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(sqr(0())) -> mark#(0()) mark#(recip(X)) -> mark#(X) mark#(recip(X)) -> active#(recip(mark(X))) active#(sqr(s(X))) -> mark#(s(add(sqr(X),dbl(X)))) mark#(sqr(X)) -> mark#(X) mark#(sqr(X)) -> active#(sqr(mark(X))) active#(dbl(0())) -> mark#(0()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(dbl(s(X))) -> mark#(s(s(dbl(X)))) mark#(0()) -> active#(0()) active#(add(0(),X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2))) active#(add(s(X),Y)) -> mark#(s(add(X,Y))) mark#(dbl(X)) -> mark#(X) mark#(dbl(X)) -> active#(dbl(mark(X))) active#(first(0(),X)) -> mark#(nil()) mark#(first(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2))) active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z))) mark#(nil()) -> active#(nil()) active#(half(0())) -> mark#(0()) mark#(half(X)) -> active#(half(mark(X))) active#(half(s(0()))) -> mark#(0()) active#(half(s(s(X)))) -> mark#(s(half(X))) active#(half(dbl(X))) -> mark#(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) EDG Processor: DPs: mark#(half(X)) -> mark#(X) mark#(terms(X)) -> mark#(X) mark#(terms(X)) -> active#(terms(mark(X))) active#(terms(N)) -> mark#(cons(recip(sqr(N)),terms(s(N)))) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(sqr(0())) -> mark#(0()) mark#(recip(X)) -> mark#(X) mark#(recip(X)) -> active#(recip(mark(X))) active#(sqr(s(X))) -> mark#(s(add(sqr(X),dbl(X)))) mark#(sqr(X)) -> mark#(X) mark#(sqr(X)) -> active#(sqr(mark(X))) active#(dbl(0())) -> mark#(0()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(dbl(s(X))) -> mark#(s(s(dbl(X)))) mark#(0()) -> active#(0()) active#(add(0(),X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2))) active#(add(s(X),Y)) -> mark#(s(add(X,Y))) mark#(dbl(X)) -> mark#(X) mark#(dbl(X)) -> active#(dbl(mark(X))) active#(first(0(),X)) -> mark#(nil()) mark#(first(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2))) active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z))) mark#(nil()) -> active#(nil()) active#(half(0())) -> mark#(0()) mark#(half(X)) -> active#(half(mark(X))) active#(half(s(0()))) -> mark#(0()) active#(half(s(s(X)))) -> mark#(s(half(X))) active#(half(dbl(X))) -> mark#(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) graph: mark#(half(X)) -> mark#(X) -> mark#(terms(X)) -> mark#(X) mark#(half(X)) -> mark#(X) -> mark#(terms(X)) -> active#(terms(mark(X))) mark#(half(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(half(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(half(X)) -> mark#(X) -> mark#(recip(X)) -> mark#(X) mark#(half(X)) -> mark#(X) -> mark#(recip(X)) -> active#(recip(mark(X))) mark#(half(X)) -> mark#(X) -> mark#(sqr(X)) -> mark#(X) mark#(half(X)) -> mark#(X) -> mark#(sqr(X)) -> active#(sqr(mark(X))) mark#(half(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(half(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(half(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(half(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X2) mark#(half(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) mark#(half(X)) -> mark#(X) -> mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2))) mark#(half(X)) -> mark#(X) -> mark#(dbl(X)) -> mark#(X) mark#(half(X)) -> mark#(X) -> mark#(dbl(X)) -> active#(dbl(mark(X))) mark#(half(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2) mark#(half(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1) mark#(half(X)) -> mark#(X) -> mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2))) mark#(half(X)) -> mark#(X) -> mark#(nil()) -> active#(nil()) mark#(half(X)) -> mark#(X) -> mark#(half(X)) -> mark#(X) mark#(half(X)) -> mark#(X) -> mark#(half(X)) -> active#(half(mark(X))) mark#(half(X)) -> active#(half(mark(X))) -> active#(half(0())) -> mark#(0()) mark#(half(X)) -> active#(half(mark(X))) -> active#(half(s(0()))) -> mark#(0()) mark#(half(X)) -> active#(half(mark(X))) -> active#(half(s(s(X)))) -> mark#(s(half(X))) mark#(half(X)) -> active#(half(mark(X))) -> active#(half(dbl(X))) -> mark#(X) mark#(first(X1,X2)) -> mark#(X2) -> mark#(terms(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X2) -> mark#(terms(X)) -> active#(terms(mark(X))) mark#(first(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(first(X1,X2)) -> mark#(X2) -> mark#(recip(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X2) -> mark#(recip(X)) -> active#(recip(mark(X))) mark#(first(X1,X2)) -> mark#(X2) -> mark#(sqr(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X2) -> mark#(sqr(X)) -> active#(sqr(mark(X))) mark#(first(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> active#(s(mark(X))) mark#(first(X1,X2)) -> mark#(X2) -> mark#(0()) -> active#(0()) mark#(first(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2))) mark#(first(X1,X2)) -> mark#(X2) -> mark#(dbl(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X2) -> mark#(dbl(X)) -> active#(dbl(mark(X))) mark#(first(X1,X2)) -> mark#(X2) -> mark#(first(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X2) -> mark#(first(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X2) -> mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2))) mark#(first(X1,X2)) -> mark#(X2) -> mark#(nil()) -> active#(nil()) mark#(first(X1,X2)) -> mark#(X2) -> mark#(half(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X2) -> mark#(half(X)) -> active#(half(mark(X))) mark#(first(X1,X2)) -> mark#(X1) -> mark#(terms(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X1) -> mark#(terms(X)) -> active#(terms(mark(X))) mark#(first(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(first(X1,X2)) -> mark#(X1) -> mark#(recip(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X1) -> mark#(recip(X)) -> active#(recip(mark(X))) mark#(first(X1,X2)) -> mark#(X1) -> mark#(sqr(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X1) -> mark#(sqr(X)) -> active#(sqr(mark(X))) mark#(first(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> active#(s(mark(X))) mark#(first(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(first(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2))) mark#(first(X1,X2)) -> mark#(X1) -> mark#(dbl(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X1) -> mark#(dbl(X)) -> active#(dbl(mark(X))) mark#(first(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2))) mark#(first(X1,X2)) -> mark#(X1) -> mark#(nil()) -> active#(nil()) mark#(first(X1,X2)) -> mark#(X1) -> mark#(half(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X1) -> mark#(half(X)) -> active#(half(mark(X))) mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2))) -> active#(first(0(),X)) -> mark#(nil()) mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2))) -> active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z))) mark#(add(X1,X2)) -> mark#(X2) -> mark#(terms(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) -> mark#(terms(X)) -> active#(terms(mark(X))) mark#(add(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(add(X1,X2)) -> mark#(X2) -> mark#(recip(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) -> mark#(recip(X)) -> active#(recip(mark(X))) mark#(add(X1,X2)) -> mark#(X2) -> mark#(sqr(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) -> mark#(sqr(X)) -> active#(sqr(mark(X))) mark#(add(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> active#(s(mark(X))) mark#(add(X1,X2)) -> mark#(X2) -> mark#(0()) -> active#(0()) mark#(add(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2))) mark#(add(X1,X2)) -> mark#(X2) -> mark#(dbl(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) -> mark#(dbl(X)) -> active#(dbl(mark(X))) mark#(add(X1,X2)) -> mark#(X2) -> mark#(first(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X2) -> mark#(first(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X2) -> mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2))) mark#(add(X1,X2)) -> mark#(X2) -> mark#(nil()) -> active#(nil()) mark#(add(X1,X2)) -> mark#(X2) -> mark#(half(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) -> mark#(half(X)) -> active#(half(mark(X))) mark#(add(X1,X2)) -> mark#(X1) -> mark#(terms(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) -> mark#(terms(X)) -> active#(terms(mark(X))) mark#(add(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(add(X1,X2)) -> mark#(X1) -> mark#(recip(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) -> mark#(recip(X)) -> active#(recip(mark(X))) mark#(add(X1,X2)) -> mark#(X1) -> mark#(sqr(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) -> mark#(sqr(X)) -> active#(sqr(mark(X))) mark#(add(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> active#(s(mark(X))) mark#(add(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(add(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2))) mark#(add(X1,X2)) -> mark#(X1) -> mark#(dbl(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) -> mark#(dbl(X)) -> active#(dbl(mark(X))) mark#(add(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2))) mark#(add(X1,X2)) -> mark#(X1) -> mark#(nil()) -> active#(nil()) mark#(add(X1,X2)) -> mark#(X1) -> mark#(half(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) -> mark#(half(X)) -> active#(half(mark(X))) mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2))) -> active#(add(0(),X)) -> mark#(X) mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2))) -> active#(add(s(X),Y)) -> mark#(s(add(X,Y))) mark#(dbl(X)) -> mark#(X) -> mark#(terms(X)) -> mark#(X) mark#(dbl(X)) -> mark#(X) -> mark#(terms(X)) -> active#(terms(mark(X))) mark#(dbl(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(dbl(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(dbl(X)) -> mark#(X) -> mark#(recip(X)) -> mark#(X) mark#(dbl(X)) -> mark#(X) -> mark#(recip(X)) -> active#(recip(mark(X))) mark#(dbl(X)) -> mark#(X) -> mark#(sqr(X)) -> mark#(X) mark#(dbl(X)) -> mark#(X) -> mark#(sqr(X)) -> active#(sqr(mark(X))) mark#(dbl(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(dbl(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(dbl(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(dbl(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X2) mark#(dbl(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) mark#(dbl(X)) -> mark#(X) -> mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2))) mark#(dbl(X)) -> mark#(X) -> mark#(dbl(X)) -> mark#(X) mark#(dbl(X)) -> mark#(X) -> mark#(dbl(X)) -> active#(dbl(mark(X))) mark#(dbl(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2) mark#(dbl(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1) mark#(dbl(X)) -> mark#(X) -> mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2))) mark#(dbl(X)) -> mark#(X) -> mark#(nil()) -> active#(nil()) mark#(dbl(X)) -> mark#(X) -> mark#(half(X)) -> mark#(X) mark#(dbl(X)) -> mark#(X) -> mark#(half(X)) -> active#(half(mark(X))) mark#(dbl(X)) -> active#(dbl(mark(X))) -> active#(dbl(0())) -> mark#(0()) mark#(dbl(X)) -> active#(dbl(mark(X))) -> active#(dbl(s(X))) -> mark#(s(s(dbl(X)))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(terms(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(terms(X)) -> active#(terms(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(recip(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(recip(X)) -> active#(recip(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(sqr(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(sqr(X)) -> active#(sqr(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> active#(s(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(dbl(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(dbl(X)) -> active#(dbl(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(nil()) -> active#(nil()) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(half(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(half(X)) -> active#(half(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(terms(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(terms(X)) -> active#(terms(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(s(X)) -> mark#(X) -> mark#(recip(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(recip(X)) -> active#(recip(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(sqr(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(sqr(X)) -> active#(sqr(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(s(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2))) mark#(s(X)) -> mark#(X) -> mark#(dbl(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(dbl(X)) -> active#(dbl(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2))) mark#(s(X)) -> mark#(X) -> mark#(nil()) -> active#(nil()) mark#(s(X)) -> mark#(X) -> mark#(half(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(half(X)) -> active#(half(mark(X))) mark#(recip(X)) -> mark#(X) -> mark#(terms(X)) -> mark#(X) mark#(recip(X)) -> mark#(X) -> mark#(terms(X)) -> active#(terms(mark(X))) mark#(recip(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(recip(X)) -> mark#(X) -> mark#(recip(X)) -> mark#(X) mark#(recip(X)) -> mark#(X) -> mark#(recip(X)) -> active#(recip(mark(X))) mark#(recip(X)) -> mark#(X) -> mark#(sqr(X)) -> mark#(X) mark#(recip(X)) -> mark#(X) -> mark#(sqr(X)) -> active#(sqr(mark(X))) mark#(recip(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(recip(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(recip(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(recip(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X2) mark#(recip(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) -> mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2))) mark#(recip(X)) -> mark#(X) -> mark#(dbl(X)) -> mark#(X) mark#(recip(X)) -> mark#(X) -> mark#(dbl(X)) -> active#(dbl(mark(X))) mark#(recip(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2) mark#(recip(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) -> mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2))) mark#(recip(X)) -> mark#(X) -> mark#(nil()) -> active#(nil()) mark#(recip(X)) -> mark#(X) -> mark#(half(X)) -> mark#(X) mark#(recip(X)) -> mark#(X) -> mark#(half(X)) -> active#(half(mark(X))) mark#(sqr(X)) -> mark#(X) -> mark#(terms(X)) -> mark#(X) mark#(sqr(X)) -> mark#(X) -> mark#(terms(X)) -> active#(terms(mark(X))) mark#(sqr(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(sqr(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(sqr(X)) -> mark#(X) -> mark#(recip(X)) -> mark#(X) mark#(sqr(X)) -> mark#(X) -> mark#(recip(X)) -> active#(recip(mark(X))) mark#(sqr(X)) -> mark#(X) -> mark#(sqr(X)) -> mark#(X) mark#(sqr(X)) -> mark#(X) -> mark#(sqr(X)) -> active#(sqr(mark(X))) mark#(sqr(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(sqr(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(sqr(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(sqr(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X2) mark#(sqr(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) mark#(sqr(X)) -> mark#(X) -> mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2))) mark#(sqr(X)) -> mark#(X) -> mark#(dbl(X)) -> mark#(X) mark#(sqr(X)) -> mark#(X) -> mark#(dbl(X)) -> active#(dbl(mark(X))) mark#(sqr(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2) mark#(sqr(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1) mark#(sqr(X)) -> mark#(X) -> mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2))) mark#(sqr(X)) -> mark#(X) -> mark#(nil()) -> active#(nil()) mark#(sqr(X)) -> mark#(X) -> mark#(half(X)) -> mark#(X) mark#(sqr(X)) -> mark#(X) -> mark#(half(X)) -> active#(half(mark(X))) mark#(sqr(X)) -> active#(sqr(mark(X))) -> active#(sqr(0())) -> mark#(0()) mark#(sqr(X)) -> active#(sqr(mark(X))) -> active#(sqr(s(X))) -> mark#(s(add(sqr(X),dbl(X)))) mark#(terms(X)) -> mark#(X) -> mark#(terms(X)) -> mark#(X) mark#(terms(X)) -> mark#(X) -> mark#(terms(X)) -> active#(terms(mark(X))) mark#(terms(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(terms(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(terms(X)) -> mark#(X) -> mark#(recip(X)) -> mark#(X) mark#(terms(X)) -> mark#(X) -> mark#(recip(X)) -> active#(recip(mark(X))) mark#(terms(X)) -> mark#(X) -> mark#(sqr(X)) -> mark#(X) mark#(terms(X)) -> mark#(X) -> mark#(sqr(X)) -> active#(sqr(mark(X))) mark#(terms(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(terms(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(terms(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(terms(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X2) mark#(terms(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) mark#(terms(X)) -> mark#(X) -> mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2))) mark#(terms(X)) -> mark#(X) -> mark#(dbl(X)) -> mark#(X) mark#(terms(X)) -> mark#(X) -> mark#(dbl(X)) -> active#(dbl(mark(X))) mark#(terms(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2) mark#(terms(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1) mark#(terms(X)) -> mark#(X) -> mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2))) mark#(terms(X)) -> mark#(X) -> mark#(nil()) -> active#(nil()) mark#(terms(X)) -> mark#(X) -> mark#(half(X)) -> mark#(X) mark#(terms(X)) -> mark#(X) -> mark#(half(X)) -> active#(half(mark(X))) mark#(terms(X)) -> active#(terms(mark(X))) -> active#(terms(N)) -> mark#(cons(recip(sqr(N)),terms(s(N)))) active#(half(dbl(X))) -> mark#(X) -> mark#(terms(X)) -> mark#(X) active#(half(dbl(X))) -> mark#(X) -> mark#(terms(X)) -> active#(terms(mark(X))) active#(half(dbl(X))) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) active#(half(dbl(X))) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(half(dbl(X))) -> mark#(X) -> mark#(recip(X)) -> mark#(X) active#(half(dbl(X))) -> mark#(X) -> mark#(recip(X)) -> active#(recip(mark(X))) active#(half(dbl(X))) -> mark#(X) -> mark#(sqr(X)) -> mark#(X) active#(half(dbl(X))) -> mark#(X) -> mark#(sqr(X)) -> active#(sqr(mark(X))) active#(half(dbl(X))) -> mark#(X) -> mark#(s(X)) -> mark#(X) active#(half(dbl(X))) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) active#(half(dbl(X))) -> mark#(X) -> mark#(0()) -> active#(0()) active#(half(dbl(X))) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X2) active#(half(dbl(X))) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) active#(half(dbl(X))) -> mark#(X) -> mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2))) active#(half(dbl(X))) -> mark#(X) -> mark#(dbl(X)) -> mark#(X) active#(half(dbl(X))) -> mark#(X) -> mark#(dbl(X)) -> active#(dbl(mark(X))) active#(half(dbl(X))) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2) active#(half(dbl(X))) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1) active#(half(dbl(X))) -> mark#(X) -> mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2))) active#(half(dbl(X))) -> mark#(X) -> mark#(nil()) -> active#(nil()) active#(half(dbl(X))) -> mark#(X) -> mark#(half(X)) -> mark#(X) active#(half(dbl(X))) -> mark#(X) -> mark#(half(X)) -> active#(half(mark(X))) active#(half(0())) -> mark#(0()) -> mark#(0()) -> active#(0()) active#(half(s(0()))) -> mark#(0()) -> mark#(0()) -> active#(0()) active#(half(s(s(X)))) -> mark#(s(half(X))) -> mark#(s(X)) -> mark#(X) active#(half(s(s(X)))) -> mark#(s(half(X))) -> mark#(s(X)) -> active#(s(mark(X))) active#(first(0(),X)) -> mark#(nil()) -> mark#(nil()) -> active#(nil()) active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z))) -> mark#(cons(X1,X2)) -> mark#(X1) active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z))) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(add(0(),X)) -> mark#(X) -> mark#(terms(X)) -> mark#(X) active#(add(0(),X)) -> mark#(X) -> mark#(terms(X)) -> active#(terms(mark(X))) active#(add(0(),X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) active#(add(0(),X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(add(0(),X)) -> mark#(X) -> mark#(recip(X)) -> mark#(X) active#(add(0(),X)) -> mark#(X) -> mark#(recip(X)) -> active#(recip(mark(X))) active#(add(0(),X)) -> mark#(X) -> mark#(sqr(X)) -> mark#(X) active#(add(0(),X)) -> mark#(X) -> mark#(sqr(X)) -> active#(sqr(mark(X))) active#(add(0(),X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) active#(add(0(),X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) active#(add(0(),X)) -> mark#(X) -> mark#(0()) -> active#(0()) active#(add(0(),X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X2) active#(add(0(),X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) active#(add(0(),X)) -> mark#(X) -> mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2))) active#(add(0(),X)) -> mark#(X) -> mark#(dbl(X)) -> mark#(X) active#(add(0(),X)) -> mark#(X) -> mark#(dbl(X)) -> active#(dbl(mark(X))) active#(add(0(),X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2) active#(add(0(),X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1) active#(add(0(),X)) -> mark#(X) -> mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2))) active#(add(0(),X)) -> mark#(X) -> mark#(nil()) -> active#(nil()) active#(add(0(),X)) -> mark#(X) -> mark#(half(X)) -> mark#(X) active#(add(0(),X)) -> mark#(X) -> mark#(half(X)) -> active#(half(mark(X))) active#(add(s(X),Y)) -> mark#(s(add(X,Y))) -> mark#(s(X)) -> mark#(X) active#(add(s(X),Y)) -> mark#(s(add(X,Y))) -> mark#(s(X)) -> active#(s(mark(X))) active#(dbl(0())) -> mark#(0()) -> mark#(0()) -> active#(0()) active#(dbl(s(X))) -> mark#(s(s(dbl(X)))) -> mark#(s(X)) -> mark#(X) active#(dbl(s(X))) -> mark#(s(s(dbl(X)))) -> mark#(s(X)) -> active#(s(mark(X))) active#(sqr(0())) -> mark#(0()) -> mark#(0()) -> active#(0()) active#(sqr(s(X))) -> mark#(s(add(sqr(X),dbl(X)))) -> mark#(s(X)) -> mark#(X) active#(sqr(s(X))) -> mark#(s(add(sqr(X),dbl(X)))) -> mark#(s(X)) -> active#(s(mark(X))) active#(terms(N)) -> mark#(cons(recip(sqr(N)),terms(s(N)))) -> mark#(cons(X1,X2)) -> mark#(X1) active#(terms(N)) -> mark#(cons(recip(sqr(N)),terms(s(N)))) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) SCC Processor: #sccs: 1 #rules: 25 #arcs: 316/1225 DPs: mark#(half(X)) -> mark#(X) mark#(half(X)) -> active#(half(mark(X))) active#(half(dbl(X))) -> mark#(X) mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2))) active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z))) mark#(cons(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X2) mark#(dbl(X)) -> active#(dbl(mark(X))) active#(dbl(s(X))) -> mark#(s(s(dbl(X)))) mark#(s(X)) -> mark#(X) mark#(dbl(X)) -> mark#(X) mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2))) active#(add(s(X),Y)) -> mark#(s(add(X,Y))) active#(add(0(),X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X2) mark#(sqr(X)) -> active#(sqr(mark(X))) active#(sqr(s(X))) -> mark#(s(add(sqr(X),dbl(X)))) mark#(sqr(X)) -> mark#(X) mark#(recip(X)) -> mark#(X) mark#(terms(X)) -> active#(terms(mark(X))) active#(terms(N)) -> mark#(cons(recip(sqr(N)),terms(s(N)))) mark#(terms(X)) -> mark#(X) active#(half(s(s(X)))) -> mark#(s(half(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) LPO Processor: argument filtering: pi(terms) = [0] pi(active) = 0 pi(sqr) = [0] pi(recip) = 0 pi(s) = [0] pi(cons) = 0 pi(mark) = 0 pi(0) = [] pi(dbl) = [0] pi(add) = [0,1] pi(first) = [0,1] pi(nil) = [] pi(half) = [0] pi(active#) = 0 pi(mark#) = 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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) precedence: terms > sqr > first > half ~ add ~ dbl > mark# ~ active# ~ nil ~ 0 ~ mark ~ cons ~ s ~ recip ~ active problem: DPs: mark#(half(X)) -> active#(half(mark(X))) mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2))) mark#(cons(X1,X2)) -> mark#(X1) mark#(dbl(X)) -> active#(dbl(mark(X))) mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2))) mark#(sqr(X)) -> active#(sqr(mark(X))) mark#(recip(X)) -> mark#(X) mark#(terms(X)) -> active#(terms(mark(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) Restore Modifier: DPs: mark#(half(X)) -> active#(half(mark(X))) mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2))) mark#(cons(X1,X2)) -> mark#(X1) mark#(dbl(X)) -> active#(dbl(mark(X))) mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2))) mark#(sqr(X)) -> active#(sqr(mark(X))) mark#(recip(X)) -> mark#(X) mark#(terms(X)) -> active#(terms(mark(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) SCC Processor: #sccs: 1 #rules: 2 #arcs: 235/64 DPs: mark#(cons(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) Size-Change Termination Processor: 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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) The DP: mark#(cons(X1,X2)) -> mark#(X1) has the edges: 0 > 0 The DP: mark#(recip(X)) -> mark#(X) has the edges: 0 > 0 Qed DPs: cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) Subterm Criterion Processor: simple projection: pi(cons#) = 0 problem: DPs: cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(X1,active(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) Subterm Criterion Processor: simple projection: pi(cons#) = 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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) Qed DPs: recip#(mark(X)) -> recip#(X) recip#(active(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) Qed DPs: sqr#(mark(X)) -> sqr#(X) sqr#(active(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) Qed DPs: s#(mark(X)) -> s#(X) s#(active(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) Qed DPs: terms#(mark(X)) -> terms#(X) terms#(active(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) Qed DPs: dbl#(mark(X)) -> dbl#(X) dbl#(active(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) Qed DPs: add#(mark(X1),X2) -> add#(X1,X2) add#(X1,mark(X2)) -> add#(X1,X2) add#(active(X1),X2) -> add#(X1,X2) add#(X1,active(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) Subterm Criterion Processor: simple projection: pi(add#) = 0 problem: DPs: add#(X1,mark(X2)) -> add#(X1,X2) add#(X1,active(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) Qed DPs: first#(mark(X1),X2) -> first#(X1,X2) first#(X1,mark(X2)) -> first#(X1,X2) first#(active(X1),X2) -> first#(X1,X2) first#(X1,active(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) Subterm Criterion Processor: simple projection: pi(first#) = 0 problem: DPs: first#(X1,mark(X2)) -> first#(X1,X2) first#(X1,active(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) Qed DPs: half#(mark(X)) -> half#(X) half#(active(X)) -> half#(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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) Subterm Criterion Processor: simple projection: pi(half#) = 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(half(0())) -> mark(0()) active(half(s(0()))) -> mark(0()) active(half(s(s(X)))) -> mark(s(half(X))) active(half(dbl(X))) -> mark(X) mark(terms(X)) -> active(terms(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(recip(X)) -> active(recip(mark(X))) mark(sqr(X)) -> active(sqr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(0()) -> active(0()) mark(add(X1,X2)) -> active(add(mark(X1),mark(X2))) mark(dbl(X)) -> active(dbl(mark(X))) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(half(X)) -> active(half(mark(X))) terms(mark(X)) -> terms(X) terms(active(X)) -> terms(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) recip(mark(X)) -> recip(X) recip(active(X)) -> recip(X) sqr(mark(X)) -> sqr(X) sqr(active(X)) -> sqr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) dbl(mark(X)) -> dbl(X) dbl(active(X)) -> dbl(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) half(mark(X)) -> half(X) half(active(X)) -> half(X) Qed