/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: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) half(0()) -> 0() half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) half(dbl(X)) -> X terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Proof: DP Processor: DPs: terms#(N) -> sqr#(N) sqr#(s(X)) -> dbl#(X) sqr#(s(X)) -> sqr#(X) sqr#(s(X)) -> add#(sqr(X),dbl(X)) sqr#(s(X)) -> s#(add(sqr(X),dbl(X))) dbl#(s(X)) -> dbl#(X) dbl#(s(X)) -> s#(dbl(X)) dbl#(s(X)) -> s#(s(dbl(X))) add#(s(X),Y) -> add#(X,Y) add#(s(X),Y) -> s#(add(X,Y)) first#(s(X),cons(Y,Z)) -> activate#(Z) half#(s(s(X))) -> half#(X) half#(s(s(X))) -> s#(half(X)) activate#(n__terms(X)) -> activate#(X) activate#(n__terms(X)) -> terms#(activate(X)) activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> s#(activate(X)) activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) half(0()) -> 0() half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) half(dbl(X)) -> X terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X TDG Processor: DPs: terms#(N) -> sqr#(N) sqr#(s(X)) -> dbl#(X) sqr#(s(X)) -> sqr#(X) sqr#(s(X)) -> add#(sqr(X),dbl(X)) sqr#(s(X)) -> s#(add(sqr(X),dbl(X))) dbl#(s(X)) -> dbl#(X) dbl#(s(X)) -> s#(dbl(X)) dbl#(s(X)) -> s#(s(dbl(X))) add#(s(X),Y) -> add#(X,Y) add#(s(X),Y) -> s#(add(X,Y)) first#(s(X),cons(Y,Z)) -> activate#(Z) half#(s(s(X))) -> half#(X) half#(s(s(X))) -> s#(half(X)) activate#(n__terms(X)) -> activate#(X) activate#(n__terms(X)) -> terms#(activate(X)) activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> s#(activate(X)) activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) half(0()) -> 0() half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) half(dbl(X)) -> X terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X graph: half#(s(s(X))) -> half#(X) -> half#(s(s(X))) -> s#(half(X)) half#(s(s(X))) -> half#(X) -> half#(s(s(X))) -> half#(X) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__s(X)) -> activate#(X) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__terms(X)) -> terms#(activate(X)) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__terms(X)) -> activate#(X) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> activate#(X) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__terms(X)) -> terms#(activate(X)) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__terms(X)) -> activate#(X) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) -> first#(s(X),cons(Y,Z)) -> activate#(Z) activate#(n__terms(X)) -> activate#(X) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) activate#(n__terms(X)) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__terms(X)) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__terms(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__terms(X)) -> activate#(X) -> activate#(n__s(X)) -> activate#(X) activate#(n__terms(X)) -> activate#(X) -> activate#(n__terms(X)) -> terms#(activate(X)) activate#(n__terms(X)) -> activate#(X) -> activate#(n__terms(X)) -> activate#(X) activate#(n__terms(X)) -> terms#(activate(X)) -> terms#(N) -> sqr#(N) activate#(n__s(X)) -> activate#(X) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) activate#(n__s(X)) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__s(X)) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__s(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__s(X)) -> activate#(X) -> activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> activate#(X) -> activate#(n__terms(X)) -> terms#(activate(X)) activate#(n__s(X)) -> activate#(X) -> activate#(n__terms(X)) -> activate#(X) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__first(X1,X2)) -> activate#(X1) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__first(X1,X2)) -> activate#(X2) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__s(X)) -> s#(activate(X)) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__s(X)) -> activate#(X) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__terms(X)) -> terms#(activate(X)) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__terms(X)) -> activate#(X) add#(s(X),Y) -> add#(X,Y) -> add#(s(X),Y) -> s#(add(X,Y)) add#(s(X),Y) -> add#(X,Y) -> add#(s(X),Y) -> add#(X,Y) dbl#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> s#(s(dbl(X))) dbl#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> s#(dbl(X)) dbl#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> dbl#(X) sqr#(s(X)) -> add#(sqr(X),dbl(X)) -> add#(s(X),Y) -> s#(add(X,Y)) sqr#(s(X)) -> add#(sqr(X),dbl(X)) -> add#(s(X),Y) -> add#(X,Y) sqr#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> s#(s(dbl(X))) sqr#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> s#(dbl(X)) sqr#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> dbl#(X) sqr#(s(X)) -> sqr#(X) -> sqr#(s(X)) -> s#(add(sqr(X),dbl(X))) sqr#(s(X)) -> sqr#(X) -> sqr#(s(X)) -> add#(sqr(X),dbl(X)) sqr#(s(X)) -> sqr#(X) -> sqr#(s(X)) -> sqr#(X) sqr#(s(X)) -> sqr#(X) -> sqr#(s(X)) -> dbl#(X) terms#(N) -> sqr#(N) -> sqr#(s(X)) -> s#(add(sqr(X),dbl(X))) terms#(N) -> sqr#(N) -> sqr#(s(X)) -> add#(sqr(X),dbl(X)) terms#(N) -> sqr#(N) -> sqr#(s(X)) -> sqr#(X) terms#(N) -> sqr#(N) -> sqr#(s(X)) -> dbl#(X) SCC Processor: #sccs: 5 #rules: 10 #arcs: 57/400 DPs: activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__terms(X)) -> activate#(X) activate#(n__s(X)) -> activate#(X) activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) -> activate#(Z) TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) half(0()) -> 0() half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) half(dbl(X)) -> X terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Usable Rule Processor: DPs: activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__terms(X)) -> activate#(X) activate#(n__s(X)) -> activate#(X) activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) -> activate#(Z) TRS: activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) -> n__terms(X) s(X) -> n__s(X) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) first(X1,X2) -> n__first(X1,X2) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) Arctic Interpretation Processor: dimension: 1 usable rules: activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) -> n__terms(X) s(X) -> n__s(X) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) first(X1,X2) -> n__first(X1,X2) interpretation: [activate#](x0) = x0, [first#](x0, x1) = x0 + x1, [n__first](x0, x1) = 2x0 + x1, [activate](x0) = x0, [nil] = 0, [first](x0, x1) = 2x0 + x1, [add](x0, x1) = x1 + 0, [dbl](x0) = x0, [s](x0) = x0 + 0, [0] = 0, [cons](x0, x1) = 4x0 + x1 + 4, [n__terms](x0) = 4x0 + 4, [n__s](x0) = x0 + 0, [recip](x0) = 0, [sqr](x0) = 4x0 + 0, [terms](x0) = 4x0 + 4 orientation: activate#(n__first(X1,X2)) = 2X1 + X2 >= X2 = activate#(X2) activate#(n__terms(X)) = 4X + 4 >= X = activate#(X) activate#(n__s(X)) = X + 0 >= X = activate#(X) activate#(n__first(X1,X2)) = 2X1 + X2 >= X1 = activate#(X1) activate#(n__first(X1,X2)) = 2X1 + X2 >= X1 + X2 = first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) = X + 4Y + Z + 4 >= Z = activate#(Z) activate(n__terms(X)) = 4X + 4 >= 4X + 4 = terms(activate(X)) activate(n__s(X)) = X + 0 >= X + 0 = s(activate(X)) activate(n__first(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = first(activate(X1),activate(X2)) activate(X) = X >= X = X terms(N) = 4N + 4 >= 4N + 4 = cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) = 4X + 4 >= 4X + 4 = n__terms(X) s(X) = X + 0 >= X + 0 = n__s(X) first(0(),X) = X + 2 >= 0 = nil() first(s(X),cons(Y,Z)) = 2X + 4Y + Z + 4 >= 2X + 4Y + Z + 4 = cons(Y,n__first(X,activate(Z))) first(X1,X2) = 2X1 + X2 >= 2X1 + X2 = n__first(X1,X2) sqr(0()) = 4 >= 0 = 0() sqr(s(X)) = 4X + 4 >= X + 0 = s(add(sqr(X),dbl(X))) dbl(0()) = 0 >= 0 = 0() dbl(s(X)) = X + 0 >= X + 0 = s(s(dbl(X))) add(0(),X) = X + 0 >= X = X add(s(X),Y) = Y + 0 >= Y + 0 = s(add(X,Y)) problem: DPs: activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__s(X)) -> activate#(X) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) -> activate#(Z) TRS: activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) -> n__terms(X) s(X) -> n__s(X) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) first(X1,X2) -> n__first(X1,X2) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) Restore Modifier: DPs: activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__s(X)) -> activate#(X) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) -> activate#(Z) TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) half(0()) -> 0() half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) half(dbl(X)) -> X terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Usable Rule Processor: DPs: activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__s(X)) -> activate#(X) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) -> activate#(Z) TRS: activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) -> n__terms(X) s(X) -> n__s(X) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) first(X1,X2) -> n__first(X1,X2) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) Arctic Interpretation Processor: dimension: 1 usable rules: activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) -> n__terms(X) s(X) -> n__s(X) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) first(X1,X2) -> n__first(X1,X2) interpretation: [activate#](x0) = x0, [first#](x0, x1) = 1x1, [n__first](x0, x1) = x0 + 1x1, [activate](x0) = x0, [nil] = 2, [first](x0, x1) = x0 + 1x1, [add](x0, x1) = x0 + x1 + 0, [dbl](x0) = 4x0 + 7, [s](x0) = x0 + 0, [0] = 2, [cons](x0, x1) = x0 + x1 + 0, [n__terms](x0) = 4, [n__s](x0) = x0 + 0, [recip](x0) = 0, [sqr](x0) = 5x0 + 1, [terms](x0) = 4 orientation: activate#(n__first(X1,X2)) = X1 + 1X2 >= X2 = activate#(X2) activate#(n__s(X)) = X + 0 >= X = activate#(X) activate#(n__first(X1,X2)) = X1 + 1X2 >= 1X2 = first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) = 1Y + 1Z + 1 >= Z = activate#(Z) activate(n__terms(X)) = 4 >= 4 = terms(activate(X)) activate(n__s(X)) = X + 0 >= X + 0 = s(activate(X)) activate(n__first(X1,X2)) = X1 + 1X2 >= X1 + 1X2 = first(activate(X1),activate(X2)) activate(X) = X >= X = X terms(N) = 4 >= 4 = cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) = 4 >= 4 = n__terms(X) s(X) = X + 0 >= X + 0 = n__s(X) first(0(),X) = 1X + 2 >= 2 = nil() first(s(X),cons(Y,Z)) = X + 1Y + 1Z + 1 >= X + Y + 1Z + 0 = cons(Y,n__first(X,activate(Z))) first(X1,X2) = X1 + 1X2 >= X1 + 1X2 = n__first(X1,X2) sqr(0()) = 7 >= 2 = 0() sqr(s(X)) = 5X + 5 >= 5X + 7 = s(add(sqr(X),dbl(X))) dbl(0()) = 7 >= 2 = 0() dbl(s(X)) = 4X + 7 >= 4X + 7 = s(s(dbl(X))) add(0(),X) = X + 2 >= X = X add(s(X),Y) = X + Y + 0 >= X + Y + 0 = s(add(X,Y)) problem: DPs: activate#(n__s(X)) -> activate#(X) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) TRS: activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) -> n__terms(X) s(X) -> n__s(X) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) first(X1,X2) -> n__first(X1,X2) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) Restore Modifier: DPs: activate#(n__s(X)) -> activate#(X) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) half(0()) -> 0() half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) half(dbl(X)) -> X terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X SCC Processor: #sccs: 1 #rules: 1 #arcs: 26/4 DPs: activate#(n__s(X)) -> activate#(X) TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) half(0()) -> 0() half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) half(dbl(X)) -> X terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Size-Change Termination Processor: DPs: TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) half(0()) -> 0() half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) half(dbl(X)) -> X terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X The DP: activate#(n__s(X)) -> activate#(X) has the edges: 0 > 0 Qed DPs: sqr#(s(X)) -> sqr#(X) TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) half(0()) -> 0() half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) half(dbl(X)) -> X terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Subterm Criterion Processor: simple projection: pi(sqr#) = 0 problem: DPs: TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) half(0()) -> 0() half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) half(dbl(X)) -> X terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Qed DPs: add#(s(X),Y) -> add#(X,Y) TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) half(0()) -> 0() half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) half(dbl(X)) -> X terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Subterm Criterion Processor: simple projection: pi(add#) = 0 problem: DPs: TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) half(0()) -> 0() half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) half(dbl(X)) -> X terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Qed DPs: dbl#(s(X)) -> dbl#(X) TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) half(0()) -> 0() half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) half(dbl(X)) -> X terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Subterm Criterion Processor: simple projection: pi(dbl#) = 0 problem: DPs: TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) half(0()) -> 0() half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) half(dbl(X)) -> X terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Qed DPs: half#(s(s(X))) -> half#(X) TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) half(0()) -> 0() half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) half(dbl(X)) -> X terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Subterm Criterion Processor: simple projection: pi(half#) = 0 problem: DPs: TRS: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) half(0()) -> 0() half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) half(dbl(X)) -> X terms(X) -> n__terms(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__terms(X)) -> terms(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Qed