YES Problem: a__nats() -> a__adx(a__zeros()) a__zeros() -> cons(0(),zeros()) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__hd(cons(X,Y)) -> mark(X) a__tl(cons(X,Y)) -> mark(Y) mark(nats()) -> a__nats() mark(adx(X)) -> a__adx(mark(X)) mark(zeros()) -> a__zeros() mark(incr(X)) -> a__incr(mark(X)) mark(hd(X)) -> a__hd(mark(X)) mark(tl(X)) -> a__tl(mark(X)) mark(cons(X1,X2)) -> cons(X1,X2) mark(0()) -> 0() mark(s(X)) -> s(X) a__nats() -> nats() a__adx(X) -> adx(X) a__zeros() -> zeros() a__incr(X) -> incr(X) a__hd(X) -> hd(X) a__tl(X) -> tl(X) Proof: DP Processor: DPs: a__nats#() -> a__zeros#() a__nats#() -> a__adx#(a__zeros()) a__adx#(cons(X,Y)) -> a__incr#(cons(X,adx(Y))) a__hd#(cons(X,Y)) -> mark#(X) a__tl#(cons(X,Y)) -> mark#(Y) mark#(nats()) -> a__nats#() mark#(adx(X)) -> mark#(X) mark#(adx(X)) -> a__adx#(mark(X)) mark#(zeros()) -> a__zeros#() mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) mark#(hd(X)) -> mark#(X) mark#(hd(X)) -> a__hd#(mark(X)) mark#(tl(X)) -> mark#(X) mark#(tl(X)) -> a__tl#(mark(X)) TRS: a__nats() -> a__adx(a__zeros()) a__zeros() -> cons(0(),zeros()) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__hd(cons(X,Y)) -> mark(X) a__tl(cons(X,Y)) -> mark(Y) mark(nats()) -> a__nats() mark(adx(X)) -> a__adx(mark(X)) mark(zeros()) -> a__zeros() mark(incr(X)) -> a__incr(mark(X)) mark(hd(X)) -> a__hd(mark(X)) mark(tl(X)) -> a__tl(mark(X)) mark(cons(X1,X2)) -> cons(X1,X2) mark(0()) -> 0() mark(s(X)) -> s(X) a__nats() -> nats() a__adx(X) -> adx(X) a__zeros() -> zeros() a__incr(X) -> incr(X) a__hd(X) -> hd(X) a__tl(X) -> tl(X) TDG Processor: DPs: a__nats#() -> a__zeros#() a__nats#() -> a__adx#(a__zeros()) a__adx#(cons(X,Y)) -> a__incr#(cons(X,adx(Y))) a__hd#(cons(X,Y)) -> mark#(X) a__tl#(cons(X,Y)) -> mark#(Y) mark#(nats()) -> a__nats#() mark#(adx(X)) -> mark#(X) mark#(adx(X)) -> a__adx#(mark(X)) mark#(zeros()) -> a__zeros#() mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) mark#(hd(X)) -> mark#(X) mark#(hd(X)) -> a__hd#(mark(X)) mark#(tl(X)) -> mark#(X) mark#(tl(X)) -> a__tl#(mark(X)) TRS: a__nats() -> a__adx(a__zeros()) a__zeros() -> cons(0(),zeros()) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__hd(cons(X,Y)) -> mark(X) a__tl(cons(X,Y)) -> mark(Y) mark(nats()) -> a__nats() mark(adx(X)) -> a__adx(mark(X)) mark(zeros()) -> a__zeros() mark(incr(X)) -> a__incr(mark(X)) mark(hd(X)) -> a__hd(mark(X)) mark(tl(X)) -> a__tl(mark(X)) mark(cons(X1,X2)) -> cons(X1,X2) mark(0()) -> 0() mark(s(X)) -> s(X) a__nats() -> nats() a__adx(X) -> adx(X) a__zeros() -> zeros() a__incr(X) -> incr(X) a__hd(X) -> hd(X) a__tl(X) -> tl(X) graph: a__tl#(cons(X,Y)) -> mark#(Y) -> mark#(tl(X)) -> a__tl#(mark(X)) a__tl#(cons(X,Y)) -> mark#(Y) -> mark#(tl(X)) -> mark#(X) a__tl#(cons(X,Y)) -> mark#(Y) -> mark#(hd(X)) -> a__hd#(mark(X)) a__tl#(cons(X,Y)) -> mark#(Y) -> mark#(hd(X)) -> mark#(X) a__tl#(cons(X,Y)) -> mark#(Y) -> mark#(incr(X)) -> a__incr#(mark(X)) a__tl#(cons(X,Y)) -> mark#(Y) -> mark#(incr(X)) -> mark#(X) a__tl#(cons(X,Y)) -> mark#(Y) -> mark#(zeros()) -> a__zeros#() a__tl#(cons(X,Y)) -> mark#(Y) -> mark#(adx(X)) -> a__adx#(mark(X)) a__tl#(cons(X,Y)) -> mark#(Y) -> mark#(adx(X)) -> mark#(X) a__tl#(cons(X,Y)) -> mark#(Y) -> mark#(nats()) -> a__nats#() mark#(tl(X)) -> a__tl#(mark(X)) -> a__tl#(cons(X,Y)) -> mark#(Y) mark#(tl(X)) -> mark#(X) -> mark#(tl(X)) -> a__tl#(mark(X)) mark#(tl(X)) -> mark#(X) -> mark#(tl(X)) -> mark#(X) mark#(tl(X)) -> mark#(X) -> mark#(hd(X)) -> a__hd#(mark(X)) mark#(tl(X)) -> mark#(X) -> mark#(hd(X)) -> mark#(X) mark#(tl(X)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(tl(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(tl(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(tl(X)) -> mark#(X) -> mark#(adx(X)) -> a__adx#(mark(X)) mark#(tl(X)) -> mark#(X) -> mark#(adx(X)) -> mark#(X) mark#(tl(X)) -> mark#(X) -> mark#(nats()) -> a__nats#() mark#(hd(X)) -> mark#(X) -> mark#(tl(X)) -> a__tl#(mark(X)) mark#(hd(X)) -> mark#(X) -> mark#(tl(X)) -> mark#(X) mark#(hd(X)) -> mark#(X) -> mark#(hd(X)) -> a__hd#(mark(X)) mark#(hd(X)) -> mark#(X) -> mark#(hd(X)) -> mark#(X) mark#(hd(X)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(hd(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(hd(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(hd(X)) -> mark#(X) -> mark#(adx(X)) -> a__adx#(mark(X)) mark#(hd(X)) -> mark#(X) -> mark#(adx(X)) -> mark#(X) mark#(hd(X)) -> mark#(X) -> mark#(nats()) -> a__nats#() mark#(hd(X)) -> a__hd#(mark(X)) -> a__hd#(cons(X,Y)) -> mark#(X) mark#(nats()) -> a__nats#() -> a__nats#() -> a__adx#(a__zeros()) mark#(nats()) -> a__nats#() -> a__nats#() -> a__zeros#() mark#(adx(X)) -> mark#(X) -> mark#(tl(X)) -> a__tl#(mark(X)) mark#(adx(X)) -> mark#(X) -> mark#(tl(X)) -> mark#(X) mark#(adx(X)) -> mark#(X) -> mark#(hd(X)) -> a__hd#(mark(X)) mark#(adx(X)) -> mark#(X) -> mark#(hd(X)) -> mark#(X) mark#(adx(X)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(adx(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(adx(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(adx(X)) -> mark#(X) -> mark#(adx(X)) -> a__adx#(mark(X)) mark#(adx(X)) -> mark#(X) -> mark#(adx(X)) -> mark#(X) mark#(adx(X)) -> mark#(X) -> mark#(nats()) -> a__nats#() mark#(adx(X)) -> a__adx#(mark(X)) -> a__adx#(cons(X,Y)) -> a__incr#(cons(X,adx(Y))) mark#(incr(X)) -> mark#(X) -> mark#(tl(X)) -> a__tl#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(tl(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(hd(X)) -> a__hd#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(hd(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(incr(X)) -> mark#(X) -> mark#(adx(X)) -> a__adx#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(adx(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(nats()) -> a__nats#() a__hd#(cons(X,Y)) -> mark#(X) -> mark#(tl(X)) -> a__tl#(mark(X)) a__hd#(cons(X,Y)) -> mark#(X) -> mark#(tl(X)) -> mark#(X) a__hd#(cons(X,Y)) -> mark#(X) -> mark#(hd(X)) -> a__hd#(mark(X)) a__hd#(cons(X,Y)) -> mark#(X) -> mark#(hd(X)) -> mark#(X) a__hd#(cons(X,Y)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) a__hd#(cons(X,Y)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) a__hd#(cons(X,Y)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() a__hd#(cons(X,Y)) -> mark#(X) -> mark#(adx(X)) -> a__adx#(mark(X)) a__hd#(cons(X,Y)) -> mark#(X) -> mark#(adx(X)) -> mark#(X) a__hd#(cons(X,Y)) -> mark#(X) -> mark#(nats()) -> a__nats#() a__nats#() -> a__adx#(a__zeros()) -> a__adx#(cons(X,Y)) -> a__incr#(cons(X,adx(Y))) CDG Processor: DPs: a__nats#() -> a__zeros#() a__nats#() -> a__adx#(a__zeros()) a__adx#(cons(X,Y)) -> a__incr#(cons(X,adx(Y))) a__hd#(cons(X,Y)) -> mark#(X) a__tl#(cons(X,Y)) -> mark#(Y) mark#(nats()) -> a__nats#() mark#(adx(X)) -> mark#(X) mark#(adx(X)) -> a__adx#(mark(X)) mark#(zeros()) -> a__zeros#() mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) mark#(hd(X)) -> mark#(X) mark#(hd(X)) -> a__hd#(mark(X)) mark#(tl(X)) -> mark#(X) mark#(tl(X)) -> a__tl#(mark(X)) TRS: a__nats() -> a__adx(a__zeros()) a__zeros() -> cons(0(),zeros()) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__hd(cons(X,Y)) -> mark(X) a__tl(cons(X,Y)) -> mark(Y) mark(nats()) -> a__nats() mark(adx(X)) -> a__adx(mark(X)) mark(zeros()) -> a__zeros() mark(incr(X)) -> a__incr(mark(X)) mark(hd(X)) -> a__hd(mark(X)) mark(tl(X)) -> a__tl(mark(X)) mark(cons(X1,X2)) -> cons(X1,X2) mark(0()) -> 0() mark(s(X)) -> s(X) a__nats() -> nats() a__adx(X) -> adx(X) a__zeros() -> zeros() a__incr(X) -> incr(X) a__hd(X) -> hd(X) a__tl(X) -> tl(X) graph: a__tl#(cons(X,Y)) -> mark#(Y) -> mark#(adx(X)) -> mark#(X) a__tl#(cons(X,Y)) -> mark#(Y) -> mark#(adx(X)) -> a__adx#(mark(X)) a__tl#(cons(X,Y)) -> mark#(Y) -> mark#(zeros()) -> a__zeros#() a__tl#(cons(X,Y)) -> mark#(Y) -> mark#(incr(X)) -> mark#(X) a__tl#(cons(X,Y)) -> mark#(Y) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(tl(X)) -> a__tl#(mark(X)) -> a__tl#(cons(X,Y)) -> mark#(Y) mark#(tl(X)) -> mark#(X) -> mark#(adx(X)) -> mark#(X) mark#(tl(X)) -> mark#(X) -> mark#(adx(X)) -> a__adx#(mark(X)) mark#(tl(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(tl(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(tl(X)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(hd(X)) -> mark#(X) -> mark#(adx(X)) -> mark#(X) mark#(hd(X)) -> mark#(X) -> mark#(adx(X)) -> a__adx#(mark(X)) mark#(hd(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(hd(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(hd(X)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(hd(X)) -> a__hd#(mark(X)) -> a__hd#(cons(X,Y)) -> mark#(X) mark#(nats()) -> a__nats#() -> a__nats#() -> a__zeros#() mark#(nats()) -> a__nats#() -> a__nats#() -> a__adx#(a__zeros()) mark#(adx(X)) -> mark#(X) -> mark#(adx(X)) -> mark#(X) mark#(adx(X)) -> mark#(X) -> mark#(adx(X)) -> a__adx#(mark(X)) mark#(adx(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(adx(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(adx(X)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(adx(X)) -> a__adx#(mark(X)) -> a__adx#(cons(X,Y)) -> a__incr#(cons(X,adx(Y))) mark#(incr(X)) -> mark#(X) -> mark#(adx(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(adx(X)) -> a__adx#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(incr(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) a__hd#(cons(X,Y)) -> mark#(X) -> mark#(adx(X)) -> mark#(X) a__hd#(cons(X,Y)) -> mark#(X) -> mark#(adx(X)) -> a__adx#(mark(X)) a__hd#(cons(X,Y)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() a__hd#(cons(X,Y)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) a__hd#(cons(X,Y)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) a__nats#() -> a__adx#(a__zeros()) -> a__adx#(cons(X,Y)) -> a__incr#(cons(X,adx(Y))) SCC Processor: #sccs: 1 #rules: 2 #arcs: 36/225 DPs: mark#(incr(X)) -> mark#(X) mark#(adx(X)) -> mark#(X) TRS: a__nats() -> a__adx(a__zeros()) a__zeros() -> cons(0(),zeros()) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__hd(cons(X,Y)) -> mark(X) a__tl(cons(X,Y)) -> mark(Y) mark(nats()) -> a__nats() mark(adx(X)) -> a__adx(mark(X)) mark(zeros()) -> a__zeros() mark(incr(X)) -> a__incr(mark(X)) mark(hd(X)) -> a__hd(mark(X)) mark(tl(X)) -> a__tl(mark(X)) mark(cons(X1,X2)) -> cons(X1,X2) mark(0()) -> 0() mark(s(X)) -> s(X) a__nats() -> nats() a__adx(X) -> adx(X) a__zeros() -> zeros() a__incr(X) -> incr(X) a__hd(X) -> hd(X) a__tl(X) -> tl(X) Matrix Interpretation Processor: dim=1 interpretation: [mark#](x0) = 2x0 + 2, [tl](x0) = 2x0 + 1, [hd](x0) = 2x0 + 2, [nats] = 1/2, [a__tl](x0) = 2x0 + 3/2, [mark](x0) = 2x0, [a__hd](x0) = 2x0 + 7/2, [adx](x0) = x0 + 1/2, [incr](x0) = x0 + 1/2, [s](x0) = x0, [a__incr](x0) = x0 + 1/2, [cons](x0, x1) = 2x0 + x1, [zeros] = 0, [0] = 0, [a__adx](x0) = x0 + 1, [a__zeros] = 0, [a__nats] = 1 orientation: mark#(incr(X)) = 2X + 3 >= 2X + 2 = mark#(X) mark#(adx(X)) = 2X + 3 >= 2X + 2 = mark#(X) a__nats() = 1 >= 1 = a__adx(a__zeros()) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__incr(cons(X,Y)) = 2X + Y + 1/2 >= 2X + Y + 1/2 = cons(s(X),incr(Y)) a__adx(cons(X,Y)) = 2X + Y + 1 >= 2X + Y + 1 = a__incr(cons(X,adx(Y))) a__hd(cons(X,Y)) = 4X + 2Y + 7/2 >= 2X = mark(X) a__tl(cons(X,Y)) = 4X + 2Y + 3/2 >= 2Y = mark(Y) mark(nats()) = 1 >= 1 = a__nats() mark(adx(X)) = 2X + 1 >= 2X + 1 = a__adx(mark(X)) mark(zeros()) = 0 >= 0 = a__zeros() mark(incr(X)) = 2X + 1 >= 2X + 1/2 = a__incr(mark(X)) mark(hd(X)) = 4X + 4 >= 4X + 7/2 = a__hd(mark(X)) mark(tl(X)) = 4X + 2 >= 4X + 3/2 = a__tl(mark(X)) mark(cons(X1,X2)) = 4X1 + 2X2 >= 2X1 + X2 = cons(X1,X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = 2X >= X = s(X) a__nats() = 1 >= 1/2 = nats() a__adx(X) = X + 1 >= X + 1/2 = adx(X) a__zeros() = 0 >= 0 = zeros() a__incr(X) = X + 1/2 >= X + 1/2 = incr(X) a__hd(X) = 2X + 7/2 >= 2X + 2 = hd(X) a__tl(X) = 2X + 3/2 >= 2X + 1 = tl(X) problem: DPs: TRS: a__nats() -> a__adx(a__zeros()) a__zeros() -> cons(0(),zeros()) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__hd(cons(X,Y)) -> mark(X) a__tl(cons(X,Y)) -> mark(Y) mark(nats()) -> a__nats() mark(adx(X)) -> a__adx(mark(X)) mark(zeros()) -> a__zeros() mark(incr(X)) -> a__incr(mark(X)) mark(hd(X)) -> a__hd(mark(X)) mark(tl(X)) -> a__tl(mark(X)) mark(cons(X1,X2)) -> cons(X1,X2) mark(0()) -> 0() mark(s(X)) -> s(X) a__nats() -> nats() a__adx(X) -> adx(X) a__zeros() -> zeros() a__incr(X) -> incr(X) a__hd(X) -> hd(X) a__tl(X) -> tl(X) Qed