/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: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(0()) -> 0() mark(nil()) -> nil() a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1,X2) -> add(X1,X2) a__dbl(X) -> dbl(X) a__first(X1,X2) -> first(X1,X2) Proof: DP Processor: DPs: a__terms#(N) -> mark#(N) a__terms#(N) -> a__sqr#(mark(N)) a__add#(0(),X) -> mark#(X) a__first#(s(X),cons(Y,Z)) -> mark#(Y) mark#(terms(X)) -> mark#(X) mark#(terms(X)) -> a__terms#(mark(X)) mark#(sqr(X)) -> mark#(X) mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(add(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(dbl(X)) -> mark#(X) mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(first(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) TRS: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(0()) -> 0() mark(nil()) -> nil() a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1,X2) -> add(X1,X2) a__dbl(X) -> dbl(X) a__first(X1,X2) -> first(X1,X2) TDG Processor: DPs: a__terms#(N) -> mark#(N) a__terms#(N) -> a__sqr#(mark(N)) a__add#(0(),X) -> mark#(X) a__first#(s(X),cons(Y,Z)) -> mark#(Y) mark#(terms(X)) -> mark#(X) mark#(terms(X)) -> a__terms#(mark(X)) mark#(sqr(X)) -> mark#(X) mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(add(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(dbl(X)) -> mark#(X) mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(first(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) TRS: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(0()) -> 0() mark(nil()) -> nil() a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1,X2) -> add(X1,X2) a__dbl(X) -> dbl(X) a__first(X1,X2) -> first(X1,X2) graph: a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(recip(X)) -> mark#(X) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(cons(X1,X2)) -> mark#(X1) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(first(X1,X2)) -> mark#(X1) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(first(X1,X2)) -> mark#(X2) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(dbl(X)) -> a__dbl#(mark(X)) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(dbl(X)) -> mark#(X) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(add(X1,X2)) -> mark#(X1) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(add(X1,X2)) -> mark#(X2) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(sqr(X)) -> a__sqr#(mark(X)) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(sqr(X)) -> mark#(X) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(terms(X)) -> a__terms#(mark(X)) a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(terms(X)) -> mark#(X) a__add#(0(),X) -> mark#(X) -> mark#(recip(X)) -> mark#(X) a__add#(0(),X) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__add#(0(),X) -> mark#(X) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) a__add#(0(),X) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1) a__add#(0(),X) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2) a__add#(0(),X) -> mark#(X) -> mark#(dbl(X)) -> a__dbl#(mark(X)) a__add#(0(),X) -> mark#(X) -> mark#(dbl(X)) -> mark#(X) a__add#(0(),X) -> mark#(X) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) a__add#(0(),X) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) a__add#(0(),X) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X2) a__add#(0(),X) -> mark#(X) -> mark#(sqr(X)) -> a__sqr#(mark(X)) a__add#(0(),X) -> mark#(X) -> mark#(sqr(X)) -> mark#(X) a__add#(0(),X) -> mark#(X) -> mark#(terms(X)) -> a__terms#(mark(X)) a__add#(0(),X) -> mark#(X) -> mark#(terms(X)) -> mark#(X) mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) -> a__first#(s(X),cons(Y,Z)) -> mark#(Y) mark#(first(X1,X2)) -> mark#(X2) -> mark#(recip(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X2) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(first(X1,X2)) -> mark#(X2) -> mark#(first(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X2) -> mark#(first(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X2) -> mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(first(X1,X2)) -> mark#(X2) -> mark#(dbl(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(first(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X2) -> mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(first(X1,X2)) -> mark#(X2) -> mark#(sqr(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X2) -> mark#(terms(X)) -> a__terms#(mark(X)) mark#(first(X1,X2)) -> mark#(X2) -> mark#(terms(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X1) -> mark#(recip(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(first(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X1) -> mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(first(X1,X2)) -> mark#(X1) -> mark#(dbl(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(first(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X1) -> mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(first(X1,X2)) -> mark#(X1) -> mark#(sqr(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X1) -> mark#(terms(X)) -> a__terms#(mark(X)) mark#(first(X1,X2)) -> mark#(X1) -> mark#(terms(X)) -> mark#(X) mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) -> a__add#(0(),X) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) -> mark#(recip(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X2) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(add(X1,X2)) -> mark#(X2) -> mark#(first(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X2) -> mark#(first(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X2) -> mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(add(X1,X2)) -> mark#(X2) -> mark#(dbl(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(add(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X2) -> mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(add(X1,X2)) -> mark#(X2) -> mark#(sqr(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) -> mark#(terms(X)) -> a__terms#(mark(X)) mark#(add(X1,X2)) -> mark#(X2) -> mark#(terms(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) -> mark#(recip(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(add(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X1) -> mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(add(X1,X2)) -> mark#(X1) -> mark#(dbl(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(add(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X1) -> mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(add(X1,X2)) -> mark#(X1) -> mark#(sqr(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) -> mark#(terms(X)) -> a__terms#(mark(X)) mark#(add(X1,X2)) -> mark#(X1) -> mark#(terms(X)) -> mark#(X) mark#(dbl(X)) -> mark#(X) -> mark#(recip(X)) -> mark#(X) mark#(dbl(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(dbl(X)) -> mark#(X) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(dbl(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1) mark#(dbl(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2) mark#(dbl(X)) -> mark#(X) -> mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(dbl(X)) -> mark#(X) -> mark#(dbl(X)) -> mark#(X) mark#(dbl(X)) -> mark#(X) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(dbl(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) mark#(dbl(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X2) mark#(dbl(X)) -> mark#(X) -> mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(dbl(X)) -> mark#(X) -> mark#(sqr(X)) -> mark#(X) mark#(dbl(X)) -> mark#(X) -> mark#(terms(X)) -> a__terms#(mark(X)) mark#(dbl(X)) -> mark#(X) -> mark#(terms(X)) -> mark#(X) mark#(sqr(X)) -> mark#(X) -> mark#(recip(X)) -> mark#(X) mark#(sqr(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(sqr(X)) -> mark#(X) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(sqr(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1) mark#(sqr(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2) mark#(sqr(X)) -> mark#(X) -> mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(sqr(X)) -> mark#(X) -> mark#(dbl(X)) -> mark#(X) mark#(sqr(X)) -> mark#(X) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(sqr(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) mark#(sqr(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X2) mark#(sqr(X)) -> mark#(X) -> mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(sqr(X)) -> mark#(X) -> mark#(sqr(X)) -> mark#(X) mark#(sqr(X)) -> mark#(X) -> mark#(terms(X)) -> a__terms#(mark(X)) mark#(sqr(X)) -> mark#(X) -> mark#(terms(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(recip(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(dbl(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(sqr(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(terms(X)) -> a__terms#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(terms(X)) -> mark#(X) mark#(terms(X)) -> mark#(X) -> mark#(recip(X)) -> mark#(X) mark#(terms(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(terms(X)) -> mark#(X) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(terms(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1) mark#(terms(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2) mark#(terms(X)) -> mark#(X) -> mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(terms(X)) -> mark#(X) -> mark#(dbl(X)) -> mark#(X) mark#(terms(X)) -> mark#(X) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(terms(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) mark#(terms(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X2) mark#(terms(X)) -> mark#(X) -> mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(terms(X)) -> mark#(X) -> mark#(sqr(X)) -> mark#(X) mark#(terms(X)) -> mark#(X) -> mark#(terms(X)) -> a__terms#(mark(X)) mark#(terms(X)) -> mark#(X) -> mark#(terms(X)) -> mark#(X) mark#(terms(X)) -> a__terms#(mark(X)) -> a__terms#(N) -> a__sqr#(mark(N)) mark#(terms(X)) -> a__terms#(mark(X)) -> a__terms#(N) -> mark#(N) mark#(recip(X)) -> mark#(X) -> mark#(recip(X)) -> mark#(X) mark#(recip(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(recip(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2) mark#(recip(X)) -> mark#(X) -> mark#(dbl(X)) -> a__dbl#(mark(X)) mark#(recip(X)) -> mark#(X) -> mark#(dbl(X)) -> mark#(X) mark#(recip(X)) -> mark#(X) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(recip(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X2) mark#(recip(X)) -> mark#(X) -> mark#(sqr(X)) -> a__sqr#(mark(X)) mark#(recip(X)) -> mark#(X) -> mark#(sqr(X)) -> mark#(X) mark#(recip(X)) -> mark#(X) -> mark#(terms(X)) -> a__terms#(mark(X)) mark#(recip(X)) -> mark#(X) -> mark#(terms(X)) -> mark#(X) a__terms#(N) -> mark#(N) -> mark#(recip(X)) -> mark#(X) a__terms#(N) -> mark#(N) -> mark#(cons(X1,X2)) -> mark#(X1) a__terms#(N) -> mark#(N) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) a__terms#(N) -> mark#(N) -> mark#(first(X1,X2)) -> mark#(X1) a__terms#(N) -> mark#(N) -> mark#(first(X1,X2)) -> mark#(X2) a__terms#(N) -> mark#(N) -> mark#(dbl(X)) -> a__dbl#(mark(X)) a__terms#(N) -> mark#(N) -> mark#(dbl(X)) -> mark#(X) a__terms#(N) -> mark#(N) -> mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) a__terms#(N) -> mark#(N) -> mark#(add(X1,X2)) -> mark#(X1) a__terms#(N) -> mark#(N) -> mark#(add(X1,X2)) -> mark#(X2) a__terms#(N) -> mark#(N) -> mark#(sqr(X)) -> a__sqr#(mark(X)) a__terms#(N) -> mark#(N) -> mark#(sqr(X)) -> mark#(X) a__terms#(N) -> mark#(N) -> mark#(terms(X)) -> a__terms#(mark(X)) a__terms#(N) -> mark#(N) -> mark#(terms(X)) -> mark#(X) SCC Processor: #sccs: 1 #rules: 15 #arcs: 172/324 DPs: a__first#(s(X),cons(Y,Z)) -> mark#(Y) mark#(terms(X)) -> mark#(X) mark#(terms(X)) -> a__terms#(mark(X)) a__terms#(N) -> mark#(N) mark#(sqr(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) a__add#(0(),X) -> mark#(X) mark#(dbl(X)) -> mark#(X) mark#(first(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) TRS: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(0()) -> 0() mark(nil()) -> nil() a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1,X2) -> add(X1,X2) a__dbl(X) -> dbl(X) a__first(X1,X2) -> first(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(0()) -> 0() mark(nil()) -> nil() a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1,X2) -> add(X1,X2) a__dbl(X) -> dbl(X) a__first(X1,X2) -> first(X1,X2) interpretation: [a__first#](x0, x1) = x0 + 7x1 + 0, [a__add#](x0, x1) = x0 + x1 + 0, [mark#](x0) = x0, [a__terms#](x0) = x0 + 0, [first](x0, x1) = 7x0 + 7x1 + 7, [nil] = 0, [a__first](x0, x1) = 7x0 + 7x1 + 7, [a__add](x0, x1) = x0 + x1 + 0, [a__dbl](x0) = x0 + 0, [add](x0, x1) = x0 + x1 + 0, [dbl](x0) = x0 + 0, [sqr](x0) = x0 + 0, [0] = 0, [cons](x0, x1) = x0 + x1, [terms](x0) = x0 + 0, [s](x0) = x0 + 0, [recip](x0) = x0 + 0, [a__sqr](x0) = x0 + 0, [mark](x0) = x0 + 0, [a__terms](x0) = x0 + 0 orientation: a__first#(s(X),cons(Y,Z)) = X + 7Y + 7Z + 0 >= Y = mark#(Y) mark#(terms(X)) = X + 0 >= X = mark#(X) mark#(terms(X)) = X + 0 >= X + 0 = a__terms#(mark(X)) a__terms#(N) = N + 0 >= N = mark#(N) mark#(sqr(X)) = X + 0 >= X = mark#(X) mark#(add(X1,X2)) = X1 + X2 + 0 >= X2 = mark#(X2) mark#(add(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(add(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__add#(mark(X1),mark(X2)) a__add#(0(),X) = X + 0 >= X = mark#(X) mark#(dbl(X)) = X + 0 >= X = mark#(X) mark#(first(X1,X2)) = 7X1 + 7X2 + 7 >= X2 = mark#(X2) mark#(first(X1,X2)) = 7X1 + 7X2 + 7 >= X1 = mark#(X1) mark#(first(X1,X2)) = 7X1 + 7X2 + 7 >= X1 + 7X2 + 7 = a__first#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(recip(X)) = X + 0 >= X = mark#(X) a__terms(N) = N + 0 >= N + 0 = cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) = 0 >= 0 = 0() a__sqr(s(X)) = X + 0 >= X + 0 = s(add(sqr(X),dbl(X))) a__dbl(0()) = 0 >= 0 = 0() a__dbl(s(X)) = X + 0 >= X + 0 = s(s(dbl(X))) a__add(0(),X) = X + 0 >= X + 0 = mark(X) a__add(s(X),Y) = X + Y + 0 >= X + Y + 0 = s(add(X,Y)) a__first(0(),X) = 7X + 7 >= 0 = nil() a__first(s(X),cons(Y,Z)) = 7X + 7Y + 7Z + 7 >= 7X + Y + 7Z + 7 = cons(mark(Y),first(X,Z)) mark(terms(X)) = X + 0 >= X + 0 = a__terms(mark(X)) mark(sqr(X)) = X + 0 >= X + 0 = a__sqr(mark(X)) mark(add(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__add(mark(X1),mark(X2)) mark(dbl(X)) = X + 0 >= X + 0 = a__dbl(mark(X)) mark(first(X1,X2)) = 7X1 + 7X2 + 7 >= 7X1 + 7X2 + 7 = a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(recip(X)) = X + 0 >= X + 0 = recip(mark(X)) mark(s(X)) = X + 0 >= X + 0 = s(X) mark(0()) = 0 >= 0 = 0() mark(nil()) = 0 >= 0 = nil() a__terms(X) = X + 0 >= X + 0 = terms(X) a__sqr(X) = X + 0 >= X + 0 = sqr(X) a__add(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = add(X1,X2) a__dbl(X) = X + 0 >= X + 0 = dbl(X) a__first(X1,X2) = 7X1 + 7X2 + 7 >= 7X1 + 7X2 + 7 = first(X1,X2) problem: DPs: mark#(terms(X)) -> mark#(X) mark#(terms(X)) -> a__terms#(mark(X)) a__terms#(N) -> mark#(N) mark#(sqr(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) a__add#(0(),X) -> mark#(X) mark#(dbl(X)) -> mark#(X) mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) TRS: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(0()) -> 0() mark(nil()) -> nil() a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1,X2) -> add(X1,X2) a__dbl(X) -> dbl(X) a__first(X1,X2) -> first(X1,X2) Restore Modifier: DPs: mark#(terms(X)) -> mark#(X) mark#(terms(X)) -> a__terms#(mark(X)) a__terms#(N) -> mark#(N) mark#(sqr(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) a__add#(0(),X) -> mark#(X) mark#(dbl(X)) -> mark#(X) mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) TRS: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(0()) -> 0() mark(nil()) -> nil() a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1,X2) -> add(X1,X2) a__dbl(X) -> dbl(X) a__first(X1,X2) -> first(X1,X2) SCC Processor: #sccs: 1 #rules: 11 #arcs: 147/144 DPs: a__add#(0(),X) -> mark#(X) mark#(terms(X)) -> mark#(X) mark#(terms(X)) -> a__terms#(mark(X)) a__terms#(N) -> mark#(N) mark#(sqr(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2)) mark#(dbl(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) TRS: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(0()) -> 0() mark(nil()) -> nil() a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1,X2) -> add(X1,X2) a__dbl(X) -> dbl(X) a__first(X1,X2) -> first(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(0()) -> 0() mark(nil()) -> nil() a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1,X2) -> add(X1,X2) a__dbl(X) -> dbl(X) a__first(X1,X2) -> first(X1,X2) interpretation: [a__add#](x0, x1) = 3x1 + 0, [mark#](x0) = x0 + 0, [a__terms#](x0) = x0 + 0, [first](x0, x1) = 1x1 + 3, [nil] = 3, [a__first](x0, x1) = 1x1 + 3, [a__add](x0, x1) = x0 + 4x1 + 4, [a__dbl](x0) = 1x0 + 0, [add](x0, x1) = x0 + 4x1 + 4, [dbl](x0) = 1x0 + 0, [sqr](x0) = x0, [0] = 0, [cons](x0, x1) = x0 + x1, [terms](x0) = x0 + 0, [s](x0) = 0, [recip](x0) = x0, [a__sqr](x0) = x0, [mark](x0) = x0, [a__terms](x0) = x0 + 0 orientation: a__add#(0(),X) = 3X + 0 >= X + 0 = mark#(X) mark#(terms(X)) = X + 0 >= X + 0 = mark#(X) mark#(terms(X)) = X + 0 >= X + 0 = a__terms#(mark(X)) a__terms#(N) = N + 0 >= N + 0 = mark#(N) mark#(sqr(X)) = X + 0 >= X + 0 = mark#(X) mark#(add(X1,X2)) = X1 + 4X2 + 4 >= X2 + 0 = mark#(X2) mark#(add(X1,X2)) = X1 + 4X2 + 4 >= X1 + 0 = mark#(X1) mark#(add(X1,X2)) = X1 + 4X2 + 4 >= 3X2 + 0 = a__add#(mark(X1),mark(X2)) mark#(dbl(X)) = 1X + 0 >= X + 0 = mark#(X) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(recip(X)) = X + 0 >= X + 0 = mark#(X) a__terms(N) = N + 0 >= N + 0 = cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) = 0 >= 0 = 0() a__sqr(s(X)) = 0 >= 0 = s(add(sqr(X),dbl(X))) a__dbl(0()) = 1 >= 0 = 0() a__dbl(s(X)) = 1 >= 0 = s(s(dbl(X))) a__add(0(),X) = 4X + 4 >= X = mark(X) a__add(s(X),Y) = 4Y + 4 >= 0 = s(add(X,Y)) a__first(0(),X) = 1X + 3 >= 3 = nil() a__first(s(X),cons(Y,Z)) = 1Y + 1Z + 3 >= Y + 1Z + 3 = cons(mark(Y),first(X,Z)) mark(terms(X)) = X + 0 >= X + 0 = a__terms(mark(X)) mark(sqr(X)) = X >= X = a__sqr(mark(X)) mark(add(X1,X2)) = X1 + 4X2 + 4 >= X1 + 4X2 + 4 = a__add(mark(X1),mark(X2)) mark(dbl(X)) = 1X + 0 >= 1X + 0 = a__dbl(mark(X)) mark(first(X1,X2)) = 1X2 + 3 >= 1X2 + 3 = a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(recip(X)) = X >= X = recip(mark(X)) mark(s(X)) = 0 >= 0 = s(X) mark(0()) = 0 >= 0 = 0() mark(nil()) = 3 >= 3 = nil() a__terms(X) = X + 0 >= X + 0 = terms(X) a__sqr(X) = X >= X = sqr(X) a__add(X1,X2) = X1 + 4X2 + 4 >= X1 + 4X2 + 4 = add(X1,X2) a__dbl(X) = 1X + 0 >= 1X + 0 = dbl(X) a__first(X1,X2) = 1X2 + 3 >= 1X2 + 3 = first(X1,X2) problem: DPs: a__add#(0(),X) -> mark#(X) mark#(terms(X)) -> mark#(X) mark#(terms(X)) -> a__terms#(mark(X)) a__terms#(N) -> mark#(N) mark#(sqr(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) mark#(dbl(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) TRS: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(0()) -> 0() mark(nil()) -> nil() a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1,X2) -> add(X1,X2) a__dbl(X) -> dbl(X) a__first(X1,X2) -> first(X1,X2) Restore Modifier: DPs: a__add#(0(),X) -> mark#(X) mark#(terms(X)) -> mark#(X) mark#(terms(X)) -> a__terms#(mark(X)) a__terms#(N) -> mark#(N) mark#(sqr(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) mark#(dbl(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) TRS: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(0()) -> 0() mark(nil()) -> nil() a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1,X2) -> add(X1,X2) a__dbl(X) -> dbl(X) a__first(X1,X2) -> first(X1,X2) SCC Processor: #sccs: 1 #rules: 8 #arcs: 83/81 DPs: mark#(terms(X)) -> mark#(X) mark#(terms(X)) -> a__terms#(mark(X)) a__terms#(N) -> mark#(N) mark#(sqr(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) mark#(dbl(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) TRS: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(0()) -> 0() mark(nil()) -> nil() a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1,X2) -> add(X1,X2) a__dbl(X) -> dbl(X) a__first(X1,X2) -> first(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(0()) -> 0() mark(nil()) -> nil() a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1,X2) -> add(X1,X2) a__dbl(X) -> dbl(X) a__first(X1,X2) -> first(X1,X2) interpretation: [mark#](x0) = x0 + 0, [a__terms#](x0) = 2x0 + 1, [first](x0, x1) = 1x0 + 4x1, [nil] = 3, [a__first](x0, x1) = 1x0 + 4x1, [a__add](x0, x1) = x0 + x1 + 0, [a__dbl](x0) = x0, [add](x0, x1) = x0 + x1 + 0, [dbl](x0) = x0, [sqr](x0) = x0, [0] = 2, [cons](x0, x1) = x0, [terms](x0) = 3x0 + 1, [s](x0) = 0, [recip](x0) = x0 + 0, [a__sqr](x0) = x0, [mark](x0) = x0, [a__terms](x0) = 3x0 + 1 orientation: mark#(terms(X)) = 3X + 1 >= X + 0 = mark#(X) mark#(terms(X)) = 3X + 1 >= 2X + 1 = a__terms#(mark(X)) a__terms#(N) = 2N + 1 >= N + 0 = mark#(N) mark#(sqr(X)) = X + 0 >= X + 0 = mark#(X) mark#(add(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(dbl(X)) = X + 0 >= X + 0 = mark#(X) mark#(cons(X1,X2)) = X1 + 0 >= X1 + 0 = mark#(X1) mark#(recip(X)) = X + 0 >= X + 0 = mark#(X) a__terms(N) = 3N + 1 >= N + 0 = cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) = 2 >= 2 = 0() a__sqr(s(X)) = 0 >= 0 = s(add(sqr(X),dbl(X))) a__dbl(0()) = 2 >= 2 = 0() a__dbl(s(X)) = 0 >= 0 = s(s(dbl(X))) a__add(0(),X) = X + 2 >= X = mark(X) a__add(s(X),Y) = Y + 0 >= 0 = s(add(X,Y)) a__first(0(),X) = 4X + 3 >= 3 = nil() a__first(s(X),cons(Y,Z)) = 4Y + 1 >= Y = cons(mark(Y),first(X,Z)) mark(terms(X)) = 3X + 1 >= 3X + 1 = a__terms(mark(X)) mark(sqr(X)) = X >= X = a__sqr(mark(X)) mark(add(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__add(mark(X1),mark(X2)) mark(dbl(X)) = X >= X = a__dbl(mark(X)) mark(first(X1,X2)) = 1X1 + 4X2 >= 1X1 + 4X2 = a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 >= X1 = cons(mark(X1),X2) mark(recip(X)) = X + 0 >= X + 0 = recip(mark(X)) mark(s(X)) = 0 >= 0 = s(X) mark(0()) = 2 >= 2 = 0() mark(nil()) = 3 >= 3 = nil() a__terms(X) = 3X + 1 >= 3X + 1 = terms(X) a__sqr(X) = X >= X = sqr(X) a__add(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = add(X1,X2) a__dbl(X) = X >= X = dbl(X) a__first(X1,X2) = 1X1 + 4X2 >= 1X1 + 4X2 = first(X1,X2) problem: DPs: mark#(terms(X)) -> a__terms#(mark(X)) mark#(sqr(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) mark#(dbl(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) TRS: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(0()) -> 0() mark(nil()) -> nil() a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1,X2) -> add(X1,X2) a__dbl(X) -> dbl(X) a__first(X1,X2) -> first(X1,X2) Restore Modifier: DPs: mark#(terms(X)) -> a__terms#(mark(X)) mark#(sqr(X)) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) mark#(dbl(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) TRS: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(0()) -> 0() mark(nil()) -> nil() a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1,X2) -> add(X1,X2) a__dbl(X) -> dbl(X) a__first(X1,X2) -> first(X1,X2) SCC Processor: #sccs: 1 #rules: 5 #arcs: 50/36 DPs: mark#(add(X1,X2)) -> mark#(X1) mark#(sqr(X)) -> mark#(X) mark#(dbl(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(recip(X)) -> mark#(X) TRS: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(0()) -> 0() mark(nil()) -> nil() a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1,X2) -> add(X1,X2) a__dbl(X) -> dbl(X) a__first(X1,X2) -> first(X1,X2) Size-Change Termination Processor: DPs: TRS: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(0()) -> 0() mark(nil()) -> nil() a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1,X2) -> add(X1,X2) a__dbl(X) -> dbl(X) a__first(X1,X2) -> first(X1,X2) The DP: mark#(add(X1,X2)) -> mark#(X1) has the edges: 0 > 0 The DP: mark#(sqr(X)) -> mark#(X) has the edges: 0 > 0 The DP: mark#(dbl(X)) -> mark#(X) has the edges: 0 > 0 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