YES Problem: a__incr(nil()) -> nil() a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__adx(nil()) -> nil() a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) a__nats() -> a__adx(a__zeros()) a__zeros() -> cons(0(),zeros()) a__head(cons(X,L)) -> mark(X) a__tail(cons(X,L)) -> mark(L) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__zeros() -> zeros() a__head(X) -> head(X) a__tail(X) -> tail(X) Proof: DP Processor: DPs: a__incr#(cons(X,L)) -> mark#(X) a__adx#(cons(X,L)) -> mark#(X) a__adx#(cons(X,L)) -> a__incr#(cons(mark(X),adx(L))) a__nats#() -> a__zeros#() a__nats#() -> a__adx#(a__zeros()) a__head#(cons(X,L)) -> mark#(X) a__tail#(cons(X,L)) -> mark#(L) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) mark#(adx(X)) -> mark#(X) mark#(adx(X)) -> a__adx#(mark(X)) mark#(nats()) -> a__nats#() mark#(zeros()) -> a__zeros#() mark#(head(X)) -> mark#(X) mark#(head(X)) -> a__head#(mark(X)) mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> a__tail#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__incr(nil()) -> nil() a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__adx(nil()) -> nil() a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) a__nats() -> a__adx(a__zeros()) a__zeros() -> cons(0(),zeros()) a__head(cons(X,L)) -> mark(X) a__tail(cons(X,L)) -> mark(L) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__zeros() -> zeros() a__head(X) -> head(X) a__tail(X) -> tail(X) TDG Processor: DPs: a__incr#(cons(X,L)) -> mark#(X) a__adx#(cons(X,L)) -> mark#(X) a__adx#(cons(X,L)) -> a__incr#(cons(mark(X),adx(L))) a__nats#() -> a__zeros#() a__nats#() -> a__adx#(a__zeros()) a__head#(cons(X,L)) -> mark#(X) a__tail#(cons(X,L)) -> mark#(L) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) mark#(adx(X)) -> mark#(X) mark#(adx(X)) -> a__adx#(mark(X)) mark#(nats()) -> a__nats#() mark#(zeros()) -> a__zeros#() mark#(head(X)) -> mark#(X) mark#(head(X)) -> a__head#(mark(X)) mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> a__tail#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__incr(nil()) -> nil() a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__adx(nil()) -> nil() a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) a__nats() -> a__adx(a__zeros()) a__zeros() -> cons(0(),zeros()) a__head(cons(X,L)) -> mark(X) a__tail(cons(X,L)) -> mark(L) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__zeros() -> zeros() a__head(X) -> head(X) a__tail(X) -> tail(X) graph: a__tail#(cons(X,L)) -> mark#(L) -> mark#(s(X)) -> mark#(X) a__tail#(cons(X,L)) -> mark#(L) -> mark#(cons(X1,X2)) -> mark#(X1) a__tail#(cons(X,L)) -> mark#(L) -> mark#(tail(X)) -> a__tail#(mark(X)) a__tail#(cons(X,L)) -> mark#(L) -> mark#(tail(X)) -> mark#(X) a__tail#(cons(X,L)) -> mark#(L) -> mark#(head(X)) -> a__head#(mark(X)) a__tail#(cons(X,L)) -> mark#(L) -> mark#(head(X)) -> mark#(X) a__tail#(cons(X,L)) -> mark#(L) -> mark#(zeros()) -> a__zeros#() a__tail#(cons(X,L)) -> mark#(L) -> mark#(nats()) -> a__nats#() a__tail#(cons(X,L)) -> mark#(L) -> mark#(adx(X)) -> a__adx#(mark(X)) a__tail#(cons(X,L)) -> mark#(L) -> mark#(adx(X)) -> mark#(X) a__tail#(cons(X,L)) -> mark#(L) -> mark#(incr(X)) -> a__incr#(mark(X)) a__tail#(cons(X,L)) -> mark#(L) -> mark#(incr(X)) -> mark#(X) a__head#(cons(X,L)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__head#(cons(X,L)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__head#(cons(X,L)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) a__head#(cons(X,L)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__head#(cons(X,L)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) a__head#(cons(X,L)) -> mark#(X) -> mark#(head(X)) -> mark#(X) a__head#(cons(X,L)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() a__head#(cons(X,L)) -> mark#(X) -> mark#(nats()) -> a__nats#() a__head#(cons(X,L)) -> mark#(X) -> mark#(adx(X)) -> a__adx#(mark(X)) a__head#(cons(X,L)) -> mark#(X) -> mark#(adx(X)) -> mark#(X) a__head#(cons(X,L)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) a__head#(cons(X,L)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) a__nats#() -> a__adx#(a__zeros()) -> a__adx#(cons(X,L)) -> a__incr#(cons(mark(X),adx(L))) a__nats#() -> a__adx#(a__zeros()) -> a__adx#(cons(X,L)) -> mark#(X) a__adx#(cons(X,L)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__adx#(cons(X,L)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__adx#(cons(X,L)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) a__adx#(cons(X,L)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__adx#(cons(X,L)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) a__adx#(cons(X,L)) -> mark#(X) -> mark#(head(X)) -> mark#(X) a__adx#(cons(X,L)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() a__adx#(cons(X,L)) -> mark#(X) -> mark#(nats()) -> a__nats#() a__adx#(cons(X,L)) -> mark#(X) -> mark#(adx(X)) -> a__adx#(mark(X)) a__adx#(cons(X,L)) -> mark#(X) -> mark#(adx(X)) -> mark#(X) a__adx#(cons(X,L)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) a__adx#(cons(X,L)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) a__adx#(cons(X,L)) -> a__incr#(cons(mark(X),adx(L))) -> a__incr#(cons(X,L)) -> mark#(X) mark#(tail(X)) -> a__tail#(mark(X)) -> a__tail#(cons(X,L)) -> mark#(L) mark#(tail(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(tail(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(tail(X)) -> mark#(X) -> mark#(nats()) -> a__nats#() mark#(tail(X)) -> mark#(X) -> mark#(adx(X)) -> a__adx#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(adx(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(head(X)) -> a__head#(mark(X)) -> a__head#(cons(X,L)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(head(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(head(X)) -> mark#(X) -> mark#(nats()) -> a__nats#() mark#(head(X)) -> mark#(X) -> mark#(adx(X)) -> a__adx#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(adx(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(nats()) -> a__nats#() -> a__nats#() -> a__adx#(a__zeros()) mark#(nats()) -> a__nats#() -> a__nats#() -> a__zeros#() mark#(adx(X)) -> a__adx#(mark(X)) -> a__adx#(cons(X,L)) -> a__incr#(cons(mark(X),adx(L))) mark#(adx(X)) -> a__adx#(mark(X)) -> a__adx#(cons(X,L)) -> mark#(X) mark#(adx(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(adx(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(adx(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(adx(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(adx(X)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) mark#(adx(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(adx(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(adx(X)) -> mark#(X) -> mark#(nats()) -> a__nats#() 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#(incr(X)) -> a__incr#(mark(X)) mark#(adx(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(incr(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(incr(X)) -> mark#(X) -> mark#(nats()) -> a__nats#() 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#(incr(X)) -> a__incr#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) -> a__incr#(cons(X,L)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(s(X)) -> mark#(X) -> mark#(nats()) -> a__nats#() mark#(s(X)) -> mark#(X) -> mark#(adx(X)) -> a__adx#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(adx(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> a__head#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(cons(X1,X2)) -> mark#(X1) -> mark#(nats()) -> a__nats#() mark#(cons(X1,X2)) -> mark#(X1) -> mark#(adx(X)) -> a__adx#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(adx(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> mark#(X) a__incr#(cons(X,L)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__incr#(cons(X,L)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__incr#(cons(X,L)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) a__incr#(cons(X,L)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__incr#(cons(X,L)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) a__incr#(cons(X,L)) -> mark#(X) -> mark#(head(X)) -> mark#(X) a__incr#(cons(X,L)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() a__incr#(cons(X,L)) -> mark#(X) -> mark#(nats()) -> a__nats#() a__incr#(cons(X,L)) -> mark#(X) -> mark#(adx(X)) -> a__adx#(mark(X)) a__incr#(cons(X,L)) -> mark#(X) -> mark#(adx(X)) -> mark#(X) a__incr#(cons(X,L)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,L)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) SCC Processor: #sccs: 1 #rules: 17 #arcs: 130/361 DPs: a__tail#(cons(X,L)) -> mark#(L) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,L)) -> mark#(X) mark#(adx(X)) -> mark#(X) mark#(adx(X)) -> a__adx#(mark(X)) a__adx#(cons(X,L)) -> mark#(X) mark#(nats()) -> a__nats#() a__nats#() -> a__adx#(a__zeros()) a__adx#(cons(X,L)) -> a__incr#(cons(mark(X),adx(L))) mark#(head(X)) -> mark#(X) mark#(head(X)) -> a__head#(mark(X)) a__head#(cons(X,L)) -> mark#(X) mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> a__tail#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__incr(nil()) -> nil() a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__adx(nil()) -> nil() a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) a__nats() -> a__adx(a__zeros()) a__zeros() -> cons(0(),zeros()) a__head(cons(X,L)) -> mark(X) a__tail(cons(X,L)) -> mark(L) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__zeros() -> zeros() a__head(X) -> head(X) a__tail(X) -> tail(X) Arctic Interpretation Processor: dimension: 1 interpretation: [a__tail#](x0) = x0 + 1, [a__head#](x0) = x0, [a__nats#] = 2, [a__adx#](x0) = x0, [mark#](x0) = x0, [a__incr#](x0) = x0, [tail](x0) = 4x0 + 1, [head](x0) = x0 + 0, [nats] = 2, [a__tail](x0) = 4x0 + 1, [a__head](x0) = x0 + 0, [zeros] = 0, [0] = 0, [a__zeros] = 0, [a__nats] = 2, [adx](x0) = x0, [a__adx](x0) = x0, [incr](x0) = x0, [s](x0) = x0 + 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1 + 0, [a__incr](x0) = x0, [nil] = 1 orientation: a__tail#(cons(X,L)) = L + X + 1 >= L = mark#(L) mark#(incr(X)) = X >= X = mark#(X) mark#(incr(X)) = X >= X = a__incr#(mark(X)) a__incr#(cons(X,L)) = L + X + 0 >= X = mark#(X) mark#(adx(X)) = X >= X = mark#(X) mark#(adx(X)) = X >= X = a__adx#(mark(X)) a__adx#(cons(X,L)) = L + X + 0 >= X = mark#(X) mark#(nats()) = 2 >= 2 = a__nats#() a__nats#() = 2 >= 0 = a__adx#(a__zeros()) a__adx#(cons(X,L)) = L + X + 0 >= L + X + 0 = a__incr#(cons(mark(X),adx(L))) mark#(head(X)) = X + 0 >= X = mark#(X) mark#(head(X)) = X + 0 >= X = a__head#(mark(X)) a__head#(cons(X,L)) = L + X + 0 >= X = mark#(X) mark#(tail(X)) = 4X + 1 >= X = mark#(X) mark#(tail(X)) = 4X + 1 >= X + 1 = a__tail#(mark(X)) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(s(X)) = X + 0 >= X = mark#(X) a__incr(nil()) = 1 >= 1 = nil() a__incr(cons(X,L)) = L + X + 0 >= L + X + 0 = cons(s(mark(X)),incr(L)) a__adx(nil()) = 1 >= 1 = nil() a__adx(cons(X,L)) = L + X + 0 >= L + X + 0 = a__incr(cons(mark(X),adx(L))) a__nats() = 2 >= 0 = a__adx(a__zeros()) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__head(cons(X,L)) = L + X + 0 >= X = mark(X) a__tail(cons(X,L)) = 4L + 4X + 4 >= L = mark(L) mark(incr(X)) = X >= X = a__incr(mark(X)) mark(adx(X)) = X >= X = a__adx(mark(X)) mark(nats()) = 2 >= 2 = a__nats() mark(zeros()) = 0 >= 0 = a__zeros() mark(head(X)) = X + 0 >= X + 0 = a__head(mark(X)) mark(tail(X)) = 4X + 1 >= 4X + 1 = a__tail(mark(X)) mark(nil()) = 1 >= 1 = nil() mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__incr(X) = X >= X = incr(X) a__adx(X) = X >= X = adx(X) a__nats() = 2 >= 2 = nats() a__zeros() = 0 >= 0 = zeros() a__head(X) = X + 0 >= X + 0 = head(X) a__tail(X) = 4X + 1 >= 4X + 1 = tail(X) problem: DPs: a__tail#(cons(X,L)) -> mark#(L) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,L)) -> mark#(X) mark#(adx(X)) -> mark#(X) mark#(adx(X)) -> a__adx#(mark(X)) a__adx#(cons(X,L)) -> mark#(X) mark#(nats()) -> a__nats#() a__adx#(cons(X,L)) -> a__incr#(cons(mark(X),adx(L))) mark#(head(X)) -> mark#(X) mark#(head(X)) -> a__head#(mark(X)) a__head#(cons(X,L)) -> mark#(X) mark#(tail(X)) -> a__tail#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__incr(nil()) -> nil() a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__adx(nil()) -> nil() a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) a__nats() -> a__adx(a__zeros()) a__zeros() -> cons(0(),zeros()) a__head(cons(X,L)) -> mark(X) a__tail(cons(X,L)) -> mark(L) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__zeros() -> zeros() a__head(X) -> head(X) a__tail(X) -> tail(X) SCC Processor: #sccs: 1 #rules: 14 #arcs: 119/225 DPs: a__tail#(cons(X,L)) -> mark#(L) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,L)) -> mark#(X) mark#(adx(X)) -> mark#(X) mark#(adx(X)) -> a__adx#(mark(X)) a__adx#(cons(X,L)) -> mark#(X) mark#(head(X)) -> mark#(X) mark#(head(X)) -> a__head#(mark(X)) a__head#(cons(X,L)) -> mark#(X) mark#(tail(X)) -> a__tail#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) a__adx#(cons(X,L)) -> a__incr#(cons(mark(X),adx(L))) TRS: a__incr(nil()) -> nil() a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__adx(nil()) -> nil() a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) a__nats() -> a__adx(a__zeros()) a__zeros() -> cons(0(),zeros()) a__head(cons(X,L)) -> mark(X) a__tail(cons(X,L)) -> mark(L) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__zeros() -> zeros() a__head(X) -> head(X) a__tail(X) -> tail(X) Arctic Interpretation Processor: dimension: 1 interpretation: [a__tail#](x0) = x0 + 0, [a__head#](x0) = x0 + 6, [a__adx#](x0) = x0 + 0, [mark#](x0) = x0 + 0, [a__incr#](x0) = x0 + 0, [tail](x0) = x0 + 1, [head](x0) = 6x0 + 6, [nats] = 5, [a__tail](x0) = x0 + 1, [a__head](x0) = 6x0 + 6, [zeros] = 3, [0] = 1, [a__zeros] = 3, [a__nats] = 5, [adx](x0) = x0, [a__adx](x0) = x0 + 0, [incr](x0) = x0, [s](x0) = x0, [mark](x0) = x0 + 0, [cons](x0, x1) = x0 + x1 + 0, [a__incr](x0) = x0 + 0, [nil] = 0 orientation: a__tail#(cons(X,L)) = L + X + 0 >= L + 0 = mark#(L) mark#(incr(X)) = X + 0 >= X + 0 = mark#(X) mark#(incr(X)) = X + 0 >= X + 0 = a__incr#(mark(X)) a__incr#(cons(X,L)) = L + X + 0 >= X + 0 = mark#(X) mark#(adx(X)) = X + 0 >= X + 0 = mark#(X) mark#(adx(X)) = X + 0 >= X + 0 = a__adx#(mark(X)) a__adx#(cons(X,L)) = L + X + 0 >= X + 0 = mark#(X) mark#(head(X)) = 6X + 6 >= X + 0 = mark#(X) mark#(head(X)) = 6X + 6 >= X + 6 = a__head#(mark(X)) a__head#(cons(X,L)) = L + X + 6 >= X + 0 = mark#(X) mark#(tail(X)) = X + 1 >= X + 0 = a__tail#(mark(X)) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) a__adx#(cons(X,L)) = L + X + 0 >= L + X + 0 = a__incr#(cons(mark(X),adx(L))) a__incr(nil()) = 0 >= 0 = nil() a__incr(cons(X,L)) = L + X + 0 >= L + X + 0 = cons(s(mark(X)),incr(L)) a__adx(nil()) = 0 >= 0 = nil() a__adx(cons(X,L)) = L + X + 0 >= L + X + 0 = a__incr(cons(mark(X),adx(L))) a__nats() = 5 >= 3 = a__adx(a__zeros()) a__zeros() = 3 >= 3 = cons(0(),zeros()) a__head(cons(X,L)) = 6L + 6X + 6 >= X + 0 = mark(X) a__tail(cons(X,L)) = L + X + 1 >= L + 0 = mark(L) mark(incr(X)) = X + 0 >= X + 0 = a__incr(mark(X)) mark(adx(X)) = X + 0 >= X + 0 = a__adx(mark(X)) mark(nats()) = 5 >= 5 = a__nats() mark(zeros()) = 3 >= 3 = a__zeros() mark(head(X)) = 6X + 6 >= 6X + 6 = a__head(mark(X)) mark(tail(X)) = X + 1 >= X + 1 = a__tail(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(0()) = 1 >= 1 = 0() a__incr(X) = X + 0 >= X = incr(X) a__adx(X) = X + 0 >= X = adx(X) a__nats() = 5 >= 5 = nats() a__zeros() = 3 >= 3 = zeros() a__head(X) = 6X + 6 >= 6X + 6 = head(X) a__tail(X) = X + 1 >= X + 1 = tail(X) problem: DPs: a__tail#(cons(X,L)) -> mark#(L) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,L)) -> mark#(X) mark#(adx(X)) -> mark#(X) mark#(adx(X)) -> a__adx#(mark(X)) a__adx#(cons(X,L)) -> mark#(X) mark#(head(X)) -> a__head#(mark(X)) a__head#(cons(X,L)) -> mark#(X) mark#(tail(X)) -> a__tail#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) a__adx#(cons(X,L)) -> a__incr#(cons(mark(X),adx(L))) TRS: a__incr(nil()) -> nil() a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__adx(nil()) -> nil() a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) a__nats() -> a__adx(a__zeros()) a__zeros() -> cons(0(),zeros()) a__head(cons(X,L)) -> mark(X) a__tail(cons(X,L)) -> mark(L) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__zeros() -> zeros() a__head(X) -> head(X) a__tail(X) -> tail(X) Arctic Interpretation Processor: dimension: 1 interpretation: [a__tail#](x0) = 7x0 + 3, [a__head#](x0) = x0 + 2, [a__adx#](x0) = 2x0, [mark#](x0) = 2x0, [a__incr#](x0) = x0, [tail](x0) = 5x0 + 1, [head](x0) = 2x0 + 0, [nats] = 5, [a__tail](x0) = 5x0 + 1, [a__head](x0) = 2x0 + 0, [zeros] = 4, [0] = 0, [a__zeros] = 4, [a__nats] = 5, [adx](x0) = x0, [a__adx](x0) = x0, [incr](x0) = x0, [s](x0) = x0, [mark](x0) = x0, [cons](x0, x1) = 4x0 + x1, [a__incr](x0) = x0, [nil] = 1 orientation: a__tail#(cons(X,L)) = 7L + 11X + 3 >= 2L = mark#(L) mark#(incr(X)) = 2X >= 2X = mark#(X) mark#(incr(X)) = 2X >= X = a__incr#(mark(X)) a__incr#(cons(X,L)) = L + 4X >= 2X = mark#(X) mark#(adx(X)) = 2X >= 2X = mark#(X) mark#(adx(X)) = 2X >= 2X = a__adx#(mark(X)) a__adx#(cons(X,L)) = 2L + 6X >= 2X = mark#(X) mark#(head(X)) = 4X + 2 >= X + 2 = a__head#(mark(X)) a__head#(cons(X,L)) = L + 4X + 2 >= 2X = mark#(X) mark#(tail(X)) = 7X + 3 >= 7X + 3 = a__tail#(mark(X)) mark#(cons(X1,X2)) = 6X1 + 2X2 >= 2X1 = mark#(X1) mark#(s(X)) = 2X >= 2X = mark#(X) a__adx#(cons(X,L)) = 2L + 6X >= L + 4X = a__incr#(cons(mark(X),adx(L))) a__incr(nil()) = 1 >= 1 = nil() a__incr(cons(X,L)) = L + 4X >= L + 4X = cons(s(mark(X)),incr(L)) a__adx(nil()) = 1 >= 1 = nil() a__adx(cons(X,L)) = L + 4X >= L + 4X = a__incr(cons(mark(X),adx(L))) a__nats() = 5 >= 4 = a__adx(a__zeros()) a__zeros() = 4 >= 4 = cons(0(),zeros()) a__head(cons(X,L)) = 2L + 6X + 0 >= X = mark(X) a__tail(cons(X,L)) = 5L + 9X + 1 >= L = mark(L) mark(incr(X)) = X >= X = a__incr(mark(X)) mark(adx(X)) = X >= X = a__adx(mark(X)) mark(nats()) = 5 >= 5 = a__nats() mark(zeros()) = 4 >= 4 = a__zeros() mark(head(X)) = 2X + 0 >= 2X + 0 = a__head(mark(X)) mark(tail(X)) = 5X + 1 >= 5X + 1 = a__tail(mark(X)) mark(nil()) = 1 >= 1 = nil() mark(cons(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = cons(mark(X1),X2) mark(s(X)) = X >= X = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__incr(X) = X >= X = incr(X) a__adx(X) = X >= X = adx(X) a__nats() = 5 >= 5 = nats() a__zeros() = 4 >= 4 = zeros() a__head(X) = 2X + 0 >= 2X + 0 = head(X) a__tail(X) = 5X + 1 >= 5X + 1 = tail(X) problem: DPs: mark#(incr(X)) -> mark#(X) mark#(adx(X)) -> mark#(X) mark#(adx(X)) -> a__adx#(mark(X)) mark#(head(X)) -> a__head#(mark(X)) mark#(tail(X)) -> a__tail#(mark(X)) mark#(s(X)) -> mark#(X) TRS: a__incr(nil()) -> nil() a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__adx(nil()) -> nil() a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) a__nats() -> a__adx(a__zeros()) a__zeros() -> cons(0(),zeros()) a__head(cons(X,L)) -> mark(X) a__tail(cons(X,L)) -> mark(L) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__zeros() -> zeros() a__head(X) -> head(X) a__tail(X) -> tail(X) SCC Processor: #sccs: 1 #rules: 3 #arcs: 87/36 DPs: mark#(adx(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) mark#(s(X)) -> mark#(X) TRS: a__incr(nil()) -> nil() a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__adx(nil()) -> nil() a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) a__nats() -> a__adx(a__zeros()) a__zeros() -> cons(0(),zeros()) a__head(cons(X,L)) -> mark(X) a__tail(cons(X,L)) -> mark(L) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__zeros() -> zeros() a__head(X) -> head(X) a__tail(X) -> tail(X) Subterm Criterion Processor: simple projection: pi(mark#) = 0 problem: DPs: TRS: a__incr(nil()) -> nil() a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__adx(nil()) -> nil() a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) a__nats() -> a__adx(a__zeros()) a__zeros() -> cons(0(),zeros()) a__head(cons(X,L)) -> mark(X) a__tail(cons(X,L)) -> mark(L) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__zeros() -> zeros() a__head(X) -> head(X) a__tail(X) -> tail(X) Qed