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: Matrix Interpretation Processor: dim=1 interpretation: [tail](x0) = 6x0, [head](x0) = x0 + 4, [nats] = 0, [a__tail](x0) = 6x0, [a__head](x0) = x0 + 4, [zeros] = 0, [0] = 0, [a__zeros] = 0, [a__nats] = 0, [adx](x0) = 2x0, [a__adx](x0) = 2x0, [incr](x0) = x0, [s](x0) = x0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [a__incr](x0) = x0, [nil] = 0 orientation: a__incr(nil()) = 0 >= 0 = nil() a__incr(cons(X,L)) = L + X >= L + X = cons(s(mark(X)),incr(L)) a__adx(nil()) = 0 >= 0 = nil() a__adx(cons(X,L)) = 2L + 2X >= 2L + X = a__incr(cons(mark(X),adx(L))) a__nats() = 0 >= 0 = a__adx(a__zeros()) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__head(cons(X,L)) = L + X + 4 >= X = mark(X) a__tail(cons(X,L)) = 6L + 6X >= L = mark(L) mark(incr(X)) = X >= X = a__incr(mark(X)) mark(adx(X)) = 2X >= 2X = a__adx(mark(X)) mark(nats()) = 0 >= 0 = a__nats() mark(zeros()) = 0 >= 0 = a__zeros() mark(head(X)) = X + 4 >= X + 4 = a__head(mark(X)) mark(tail(X)) = 6X >= 6X = a__tail(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(cons(X1,X2)) = X1 + X2 >= X1 + 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) = 2X >= 2X = adx(X) a__nats() = 0 >= 0 = nats() a__zeros() = 0 >= 0 = zeros() a__head(X) = X + 4 >= X + 4 = head(X) a__tail(X) = 6X >= 6X = tail(X) 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__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) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [tail](x0) = [0 0 0]x0 + [1] [1 0 0] [0], [1 0 0] [1] [head](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1] [nats] = [0] [0], [1 0 0] [0] [a__tail](x0) = [0 0 0]x0 + [1] [1 0 0] [0], [1 0 0] [1] [a__head](x0) = [0 1 0]x0 + [0] [1 1 0] [0], [0] [zeros] = [0] [0], [0] [0] = [0] [0], [0] [a__zeros] = [0] [0], [1] [a__nats] = [1] [1], [1 0 0] [adx](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [a__adx](x0) = [0 0 0]x0 + [1] [1 0 0] [0], [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [s](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 0] [0] [mark](x0) = [0 0 0]x0 + [1] [1 1 0] [0], [1 0 0] [1 1 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [1 0 0] [0 0 0] , [1 0 0] [0] [a__incr](x0) = [0 0 0]x0 + [1] [1 0 0] [0], [0] [nil] = [1] [0] orientation: [0] [0] a__incr(nil()) = [1] >= [1] = nil() [0] [0] [1 1 0] [1 0 0] [0] [1 0 0] [1 0 0] a__incr(cons(X,L)) = [0 0 0]L + [0 0 0]X + [1] >= [0 0 0]L + [0 0 0]X = cons(s(mark(X)),incr(L)) [1 1 0] [1 0 0] [0] [0 0 0] [1 0 0] [0] [0] a__adx(nil()) = [1] >= [1] = nil() [0] [0] [1 1 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] a__adx(cons(X,L)) = [0 0 0]L + [0 0 0]X + [1] >= [0 0 0]L + [0 0 0]X + [1] = a__incr(cons(mark(X),adx(L))) [1 1 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] [1] [0] a__nats() = [1] >= [1] = a__adx(a__zeros()) [1] [0] [0] [0] a__zeros() = [0] >= [0] = cons(0(),zeros()) [0] [0] [1 1 0] [1 0 0] [0] [1 0 0] [0] a__tail(cons(X,L)) = [0 0 0]L + [0 0 0]X + [1] >= [0 0 0]L + [1] = mark(L) [1 1 0] [1 0 0] [0] [1 1 0] [0] [1 0 0] [0] [1 0 0] [0] mark(incr(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = a__incr(mark(X)) [1 0 0] [0] [1 0 0] [0] [1 0 0] [0] [1 0 0] [0] mark(adx(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = a__adx(mark(X)) [1 0 0] [0] [1 0 0] [0] [1] [1] mark(nats()) = [1] >= [1] = a__nats() [1] [1] [0] [0] mark(zeros()) = [1] >= [0] = a__zeros() [0] [0] [1 0 0] [1] [1 0 0] [1] mark(head(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = a__head(mark(X)) [1 0 0] [1] [1 0 0] [1] [1 0 0] [0] [1 0 0] [0] mark(tail(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = a__tail(mark(X)) [1 0 0] [1] [1 0 0] [0] [0] [0] mark(nil()) = [1] >= [1] = nil() [1] [0] [1 0 0] [1 1 0] [0] [1 0 0] [1 1 0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 = cons(mark(X1),X2) [1 0 0] [1 1 0] [0] [1 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] mark(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(mark(X)) [1 0 0] [1] [0 0 0] [0] [0] [0] mark(0()) = [1] >= [0] = 0() [0] [0] [1 0 0] [0] [1 0 0] a__incr(X) = [0 0 0]X + [1] >= [0 0 0]X = incr(X) [1 0 0] [0] [0 0 0] [1 0 0] [0] [1 0 0] a__adx(X) = [0 0 0]X + [1] >= [0 0 0]X = adx(X) [1 0 0] [0] [0 0 0] [1] [1] a__nats() = [1] >= [0] = nats() [1] [0] [0] [0] a__zeros() = [0] >= [0] = zeros() [0] [0] [1 0 0] [1] [1 0 0] [1] a__head(X) = [0 1 0]X + [0] >= [0 0 0]X + [0] = head(X) [1 1 0] [0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] a__tail(X) = [0 0 0]X + [1] >= [0 0 0]X + [1] = tail(X) [1 0 0] [0] [1 0 0] [0] 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__zeros() -> cons(0(),zeros()) 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) 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__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__zeros() -> cons(0(),zeros()) 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__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__zeros() -> cons(0(),zeros()) 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__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)) -> 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#(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: 13 #arcs: 113/256 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#(tail(X)) -> 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__zeros() -> cons(0(),zeros()) 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) = 5x0 + 0, [a__adx#](x0) = x0 + 0, [mark#](x0) = x0, [a__incr#](x0) = x0 + 0, [tail](x0) = 5x0 + 6, [head](x0) = x0, [nats] = 6, [a__tail](x0) = 5x0 + 6, [a__head](x0) = x0 + 0, [zeros] = 0, [0] = 0, [a__zeros] = 0, [a__nats] = 6, [adx](x0) = x0 + 0, [a__adx](x0) = x0 + 0, [incr](x0) = x0 + 0, [s](x0) = x0, [mark](x0) = x0 + 0, [cons](x0, x1) = x0 + x1 + 0, [a__incr](x0) = x0 + 0, [nil] = 2 orientation: a__tail#(cons(X,L)) = 5L + 5X + 5 >= L = mark#(L) mark#(incr(X)) = X + 0 >= X = mark#(X) mark#(incr(X)) = X + 0 >= X + 0 = a__incr#(mark(X)) a__incr#(cons(X,L)) = L + X + 0 >= X = mark#(X) mark#(adx(X)) = X + 0 >= X = mark#(X) mark#(adx(X)) = X + 0 >= X + 0 = a__adx#(mark(X)) a__adx#(cons(X,L)) = L + X + 0 >= X = mark#(X) mark#(head(X)) = X >= X = mark#(X) mark#(tail(X)) = 5X + 6 >= X = mark#(X) mark#(tail(X)) = 5X + 6 >= 5X + 5 = a__tail#(mark(X)) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(s(X)) = X >= X = mark#(X) a__adx#(cons(X,L)) = L + X + 0 >= L + X + 0 = a__incr#(cons(mark(X),adx(L))) a__incr(nil()) = 2 >= 2 = nil() a__incr(cons(X,L)) = L + X + 0 >= L + X + 0 = cons(s(mark(X)),incr(L)) a__adx(nil()) = 2 >= 2 = nil() a__adx(cons(X,L)) = L + X + 0 >= L + X + 0 = a__incr(cons(mark(X),adx(L))) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__tail(cons(X,L)) = 5L + 5X + 6 >= 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()) = 6 >= 6 = a__nats() mark(zeros()) = 0 >= 0 = a__zeros() mark(head(X)) = X + 0 >= X + 0 = a__head(mark(X)) mark(tail(X)) = 5X + 6 >= 5X + 6 = a__tail(mark(X)) mark(nil()) = 2 >= 2 = 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 + 0 >= X + 0 = incr(X) a__adx(X) = X + 0 >= X + 0 = adx(X) a__nats() = 6 >= 6 = nats() a__zeros() = 0 >= 0 = zeros() a__head(X) = X + 0 >= X = head(X) a__tail(X) = 5X + 6 >= 5X + 6 = tail(X) problem: DPs: 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#(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__zeros() -> cons(0(),zeros()) 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: 10 #arcs: 86/121 DPs: a__adx#(cons(X,L)) -> mark#(X) 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)) -> a__incr#(cons(mark(X),adx(L))) mark#(head(X)) -> 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__zeros() -> cons(0(),zeros()) 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__adx#](x0) = x0 + 0, [mark#](x0) = x0 + 0, [a__incr#](x0) = x0 + 0, [tail](x0) = x0 + 0, [head](x0) = 1x0 + 1, [nats] = 0, [a__tail](x0) = x0 + 0, [a__head](x0) = 1x0 + 1, [zeros] = 4, [0] = 0, [a__zeros] = 4, [a__nats] = 0, [adx](x0) = x0 + 0, [a__adx](x0) = x0 + 0, [incr](x0) = x0 + 0, [s](x0) = x0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1 + 0, [a__incr](x0) = x0 + 0, [nil] = 2 orientation: a__adx#(cons(X,L)) = L + X + 0 >= X + 0 = mark#(X) 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 >= L + X + 0 = a__incr#(cons(mark(X),adx(L))) mark#(head(X)) = 1X + 1 >= X + 0 = mark#(X) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) a__incr(nil()) = 2 >= 2 = nil() a__incr(cons(X,L)) = L + X + 0 >= L + X + 0 = cons(s(mark(X)),incr(L)) a__adx(nil()) = 2 >= 2 = nil() a__adx(cons(X,L)) = L + X + 0 >= L + X + 0 = a__incr(cons(mark(X),adx(L))) a__zeros() = 4 >= 4 = cons(0(),zeros()) a__tail(cons(X,L)) = L + X + 0 >= L = 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()) = 0 >= 0 = a__nats() mark(zeros()) = 4 >= 4 = a__zeros() mark(head(X)) = 1X + 1 >= 1X + 1 = a__head(mark(X)) mark(tail(X)) = X + 0 >= X + 0 = a__tail(mark(X)) mark(nil()) = 2 >= 2 = nil() mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(s(X)) = X >= X = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__incr(X) = X + 0 >= X + 0 = incr(X) a__adx(X) = X + 0 >= X + 0 = adx(X) a__nats() = 0 >= 0 = nats() a__zeros() = 4 >= 4 = zeros() a__head(X) = 1X + 1 >= 1X + 1 = head(X) a__tail(X) = X + 0 >= X + 0 = tail(X) problem: DPs: a__adx#(cons(X,L)) -> mark#(X) 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)) -> a__incr#(cons(mark(X),adx(L))) 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__zeros() -> cons(0(),zeros()) 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__adx#](x0) = 7x0 + 0, [mark#](x0) = 5x0, [a__incr#](x0) = 5x0 + 0, [tail](x0) = 2x0 + 2, [head](x0) = x0, [nats] = 0, [a__tail](x0) = 2x0 + 2, [a__head](x0) = x0 + 0, [zeros] = 0, [0] = 0, [a__zeros] = 0, [a__nats] = 0, [adx](x0) = 2x0 + 2, [a__adx](x0) = 2x0 + 2, [incr](x0) = x0 + 1, [s](x0) = x0, [mark](x0) = x0 + 0, [cons](x0, x1) = x0 + x1 + 0, [a__incr](x0) = x0 + 1, [nil] = 0 orientation: a__adx#(cons(X,L)) = 7L + 7X + 7 >= 5X = mark#(X) mark#(incr(X)) = 5X + 6 >= 5X = mark#(X) mark#(incr(X)) = 5X + 6 >= 5X + 5 = a__incr#(mark(X)) a__incr#(cons(X,L)) = 5L + 5X + 5 >= 5X = mark#(X) mark#(adx(X)) = 7X + 7 >= 5X = mark#(X) mark#(adx(X)) = 7X + 7 >= 7X + 7 = a__adx#(mark(X)) a__adx#(cons(X,L)) = 7L + 7X + 7 >= 7L + 5X + 7 = a__incr#(cons(mark(X),adx(L))) mark#(cons(X1,X2)) = 5X1 + 5X2 + 5 >= 5X1 = mark#(X1) mark#(s(X)) = 5X >= 5X = mark#(X) a__incr(nil()) = 1 >= 0 = nil() a__incr(cons(X,L)) = L + X + 1 >= L + X + 1 = cons(s(mark(X)),incr(L)) a__adx(nil()) = 2 >= 0 = nil() a__adx(cons(X,L)) = 2L + 2X + 2 >= 2L + X + 2 = a__incr(cons(mark(X),adx(L))) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__tail(cons(X,L)) = 2L + 2X + 2 >= L + 0 = mark(L) mark(incr(X)) = X + 1 >= X + 1 = a__incr(mark(X)) mark(adx(X)) = 2X + 2 >= 2X + 2 = a__adx(mark(X)) mark(nats()) = 0 >= 0 = a__nats() mark(zeros()) = 0 >= 0 = a__zeros() mark(head(X)) = X + 0 >= X + 0 = a__head(mark(X)) mark(tail(X)) = 2X + 2 >= 2X + 2 = 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()) = 0 >= 0 = 0() a__incr(X) = X + 1 >= X + 1 = incr(X) a__adx(X) = 2X + 2 >= 2X + 2 = adx(X) a__nats() = 0 >= 0 = nats() a__zeros() = 0 >= 0 = zeros() a__head(X) = X + 0 >= X = head(X) a__tail(X) = 2X + 2 >= 2X + 2 = tail(X) problem: DPs: mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,L)) -> mark#(X) mark#(adx(X)) -> a__adx#(mark(X)) a__adx#(cons(X,L)) -> a__incr#(cons(mark(X),adx(L))) 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__zeros() -> cons(0(),zeros()) 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__adx#](x0) = x0, [mark#](x0) = x0 + 0, [a__incr#](x0) = x0 + 0, [tail](x0) = 1x0, [head](x0) = 0, [nats] = 4, [a__tail](x0) = 1x0, [a__head](x0) = 0, [zeros] = 4, [0] = 0, [a__zeros] = 4, [a__nats] = 4, [adx](x0) = x0 + 1, [a__adx](x0) = x0 + 1, [incr](x0) = x0 + 0, [s](x0) = x0 + 0, [mark](x0) = x0, [cons](x0, x1) = 4x0 + x1 + 4, [a__incr](x0) = x0 + 0, [nil] = 0 orientation: 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 + 4X + 4 >= X + 0 = mark#(X) mark#(adx(X)) = X + 1 >= X = a__adx#(mark(X)) a__adx#(cons(X,L)) = L + 4X + 4 >= L + 4X + 4 = a__incr#(cons(mark(X),adx(L))) mark#(cons(X1,X2)) = 4X1 + X2 + 4 >= X1 + 0 = mark#(X1) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) a__incr(nil()) = 0 >= 0 = nil() a__incr(cons(X,L)) = L + 4X + 4 >= L + 4X + 4 = cons(s(mark(X)),incr(L)) a__adx(nil()) = 1 >= 0 = nil() a__adx(cons(X,L)) = L + 4X + 4 >= L + 4X + 4 = a__incr(cons(mark(X),adx(L))) a__zeros() = 4 >= 4 = cons(0(),zeros()) a__tail(cons(X,L)) = 1L + 5X + 5 >= L = mark(L) mark(incr(X)) = X + 0 >= X + 0 = a__incr(mark(X)) mark(adx(X)) = X + 1 >= X + 1 = a__adx(mark(X)) mark(nats()) = 4 >= 4 = a__nats() mark(zeros()) = 4 >= 4 = a__zeros() mark(head(X)) = 0 >= 0 = a__head(mark(X)) mark(tail(X)) = 1X >= 1X = a__tail(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(cons(X1,X2)) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = cons(mark(X1),X2) mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__incr(X) = X + 0 >= X + 0 = incr(X) a__adx(X) = X + 1 >= X + 1 = adx(X) a__nats() = 4 >= 4 = nats() a__zeros() = 4 >= 4 = zeros() a__head(X) = 0 >= 0 = head(X) a__tail(X) = 1X >= 1X = tail(X) problem: DPs: mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) mark#(adx(X)) -> a__adx#(mark(X)) a__adx#(cons(X,L)) -> a__incr#(cons(mark(X),adx(L))) 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__zeros() -> cons(0(),zeros()) 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: 2 #arcs: 53/25 DPs: 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__zeros() -> cons(0(),zeros()) 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__zeros() -> cons(0(),zeros()) 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