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))) SCC Processor: #sccs: 1 #rules: 8 #arcs: 66/225 DPs: a__tl#(cons(X,Y)) -> mark#(Y) mark#(adx(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) mark#(hd(X)) -> mark#(X) mark#(hd(X)) -> a__hd#(mark(X)) a__hd#(cons(X,Y)) -> 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) Arctic Interpretation Processor: dimension: 1 interpretation: [a__tl#](x0) = x0 + 0, [mark#](x0) = x0 + 0, [a__hd#](x0) = 4x0 + 0, [tl](x0) = x0, [hd](x0) = 4x0 + 4, [nats] = 3, [a__tl](x0) = x0 + 0, [mark](x0) = x0 + 0, [a__hd](x0) = 4x0 + 4, [adx](x0) = x0, [incr](x0) = x0, [s](x0) = x0, [a__incr](x0) = x0 + 0, [cons](x0, x1) = 1x0 + x1, [zeros] = 2, [0] = 0, [a__adx](x0) = x0 + 0, [a__zeros] = 2, [a__nats] = 3 orientation: a__tl#(cons(X,Y)) = 1X + Y + 0 >= Y + 0 = mark#(Y) mark#(adx(X)) = X + 0 >= X + 0 = mark#(X) mark#(incr(X)) = X + 0 >= X + 0 = mark#(X) mark#(hd(X)) = 4X + 4 >= X + 0 = mark#(X) mark#(hd(X)) = 4X + 4 >= 4X + 4 = a__hd#(mark(X)) a__hd#(cons(X,Y)) = 5X + 4Y + 0 >= X + 0 = mark#(X) mark#(tl(X)) = X + 0 >= X + 0 = mark#(X) mark#(tl(X)) = X + 0 >= X + 0 = a__tl#(mark(X)) a__nats() = 3 >= 2 = a__adx(a__zeros()) a__zeros() = 2 >= 2 = cons(0(),zeros()) a__incr(cons(X,Y)) = 1X + Y + 0 >= 1X + Y = cons(s(X),incr(Y)) a__adx(cons(X,Y)) = 1X + Y + 0 >= 1X + Y + 0 = a__incr(cons(X,adx(Y))) a__hd(cons(X,Y)) = 5X + 4Y + 4 >= X + 0 = mark(X) a__tl(cons(X,Y)) = 1X + Y + 0 >= Y + 0 = mark(Y) mark(nats()) = 3 >= 3 = a__nats() mark(adx(X)) = X + 0 >= X + 0 = a__adx(mark(X)) mark(zeros()) = 2 >= 2 = a__zeros() mark(incr(X)) = X + 0 >= X + 0 = a__incr(mark(X)) mark(hd(X)) = 4X + 4 >= 4X + 4 = a__hd(mark(X)) mark(tl(X)) = X + 0 >= X + 0 = a__tl(mark(X)) mark(cons(X1,X2)) = 1X1 + X2 + 0 >= 1X1 + X2 = cons(X1,X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = X + 0 >= X = s(X) a__nats() = 3 >= 3 = nats() a__adx(X) = X + 0 >= X = adx(X) a__zeros() = 2 >= 2 = zeros() a__incr(X) = X + 0 >= X = incr(X) a__hd(X) = 4X + 4 >= 4X + 4 = hd(X) a__tl(X) = X + 0 >= X = tl(X) problem: DPs: a__tl#(cons(X,Y)) -> mark#(Y) mark#(adx(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) mark#(hd(X)) -> a__hd#(mark(X)) a__hd#(cons(X,Y)) -> 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) Arctic Interpretation Processor: dimension: 1 interpretation: [a__tl#](x0) = x0 + 0, [mark#](x0) = x0 + 0, [a__hd#](x0) = x0 + 0, [tl](x0) = x0, [hd](x0) = 2x0 + 3, [nats] = 0, [a__tl](x0) = x0 + 0, [mark](x0) = x0 + 0, [a__hd](x0) = 2x0 + 3, [adx](x0) = x0, [incr](x0) = x0 + 0, [s](x0) = x0, [a__incr](x0) = x0 + 0, [cons](x0, x1) = x0 + x1 + 0, [zeros] = 0, [0] = 0, [a__adx](x0) = x0 + 0, [a__zeros] = 0, [a__nats] = 0 orientation: a__tl#(cons(X,Y)) = X + Y + 0 >= Y + 0 = mark#(Y) mark#(adx(X)) = X + 0 >= X + 0 = mark#(X) mark#(incr(X)) = X + 0 >= X + 0 = mark#(X) mark#(hd(X)) = 2X + 3 >= X + 0 = a__hd#(mark(X)) a__hd#(cons(X,Y)) = X + Y + 0 >= X + 0 = mark#(X) mark#(tl(X)) = X + 0 >= X + 0 = mark#(X) mark#(tl(X)) = X + 0 >= X + 0 = a__tl#(mark(X)) a__nats() = 0 >= 0 = a__adx(a__zeros()) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__incr(cons(X,Y)) = X + Y + 0 >= X + Y + 0 = cons(s(X),incr(Y)) a__adx(cons(X,Y)) = X + Y + 0 >= X + Y + 0 = a__incr(cons(X,adx(Y))) a__hd(cons(X,Y)) = 2X + 2Y + 3 >= X + 0 = mark(X) a__tl(cons(X,Y)) = X + Y + 0 >= Y + 0 = mark(Y) mark(nats()) = 0 >= 0 = a__nats() mark(adx(X)) = X + 0 >= X + 0 = a__adx(mark(X)) mark(zeros()) = 0 >= 0 = a__zeros() mark(incr(X)) = X + 0 >= X + 0 = a__incr(mark(X)) mark(hd(X)) = 2X + 3 >= 2X + 3 = a__hd(mark(X)) mark(tl(X)) = X + 0 >= X + 0 = a__tl(mark(X)) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = X + 0 >= X = s(X) a__nats() = 0 >= 0 = nats() a__adx(X) = X + 0 >= X = adx(X) a__zeros() = 0 >= 0 = zeros() a__incr(X) = X + 0 >= X + 0 = incr(X) a__hd(X) = 2X + 3 >= 2X + 3 = hd(X) a__tl(X) = X + 0 >= X = tl(X) problem: DPs: a__tl#(cons(X,Y)) -> mark#(Y) mark#(adx(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) a__hd#(cons(X,Y)) -> 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) SCC Processor: #sccs: 1 #rules: 5 #arcs: 38/36 DPs: a__tl#(cons(X,Y)) -> mark#(Y) mark#(adx(X)) -> mark#(X) mark#(incr(X)) -> 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) Arctic Interpretation Processor: dimension: 1 interpretation: [a__tl#](x0) = x0 + 0, [mark#](x0) = x0, [tl](x0) = x0 + 0, [hd](x0) = x0 + 5, [nats] = 7, [a__tl](x0) = x0 + 0, [mark](x0) = x0, [a__hd](x0) = x0 + 5, [adx](x0) = 2x0 + 0, [incr](x0) = x0 + 1, [s](x0) = 1, [a__incr](x0) = x0 + 1, [cons](x0, x1) = x0 + x1 + 1, [zeros] = 5, [0] = 3, [a__adx](x0) = 2x0 + 0, [a__zeros] = 5, [a__nats] = 7 orientation: a__tl#(cons(X,Y)) = X + Y + 1 >= Y = mark#(Y) mark#(adx(X)) = 2X + 0 >= X = mark#(X) mark#(incr(X)) = X + 1 >= X = mark#(X) mark#(tl(X)) = X + 0 >= X = mark#(X) mark#(tl(X)) = X + 0 >= X + 0 = a__tl#(mark(X)) a__nats() = 7 >= 7 = a__adx(a__zeros()) a__zeros() = 5 >= 5 = cons(0(),zeros()) a__incr(cons(X,Y)) = X + Y + 1 >= Y + 1 = cons(s(X),incr(Y)) a__adx(cons(X,Y)) = 2X + 2Y + 3 >= X + 2Y + 1 = a__incr(cons(X,adx(Y))) a__hd(cons(X,Y)) = X + Y + 5 >= X = mark(X) a__tl(cons(X,Y)) = X + Y + 1 >= Y = mark(Y) mark(nats()) = 7 >= 7 = a__nats() mark(adx(X)) = 2X + 0 >= 2X + 0 = a__adx(mark(X)) mark(zeros()) = 5 >= 5 = a__zeros() mark(incr(X)) = X + 1 >= X + 1 = a__incr(mark(X)) mark(hd(X)) = X + 5 >= X + 5 = a__hd(mark(X)) mark(tl(X)) = X + 0 >= X + 0 = a__tl(mark(X)) mark(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(X1,X2) mark(0()) = 3 >= 3 = 0() mark(s(X)) = 1 >= 1 = s(X) a__nats() = 7 >= 7 = nats() a__adx(X) = 2X + 0 >= 2X + 0 = adx(X) a__zeros() = 5 >= 5 = zeros() a__incr(X) = X + 1 >= X + 1 = incr(X) a__hd(X) = X + 5 >= X + 5 = hd(X) a__tl(X) = X + 0 >= X + 0 = tl(X) problem: DPs: a__tl#(cons(X,Y)) -> mark#(Y) mark#(incr(X)) -> 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) Arctic Interpretation Processor: dimension: 1 interpretation: [a__tl#](x0) = x0, [mark#](x0) = x0 + 0, [tl](x0) = 4x0 + 0, [hd](x0) = 1x0, [nats] = 0, [a__tl](x0) = 4x0 + 0, [mark](x0) = x0, [a__hd](x0) = 1x0, [adx](x0) = x0 + 0, [incr](x0) = x0 + 0, [s](x0) = x0, [a__incr](x0) = x0 + 0, [cons](x0, x1) = x0 + x1 + 0, [zeros] = 0, [0] = 0, [a__adx](x0) = x0 + 0, [a__zeros] = 0, [a__nats] = 0 orientation: a__tl#(cons(X,Y)) = X + Y + 0 >= Y + 0 = mark#(Y) mark#(incr(X)) = X + 0 >= X + 0 = mark#(X) mark#(tl(X)) = 4X + 0 >= X + 0 = mark#(X) mark#(tl(X)) = 4X + 0 >= X = a__tl#(mark(X)) a__nats() = 0 >= 0 = a__adx(a__zeros()) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__incr(cons(X,Y)) = X + Y + 0 >= X + Y + 0 = cons(s(X),incr(Y)) a__adx(cons(X,Y)) = X + Y + 0 >= X + Y + 0 = a__incr(cons(X,adx(Y))) a__hd(cons(X,Y)) = 1X + 1Y + 1 >= X = mark(X) a__tl(cons(X,Y)) = 4X + 4Y + 4 >= Y = mark(Y) mark(nats()) = 0 >= 0 = a__nats() mark(adx(X)) = X + 0 >= X + 0 = a__adx(mark(X)) mark(zeros()) = 0 >= 0 = a__zeros() mark(incr(X)) = X + 0 >= X + 0 = a__incr(mark(X)) mark(hd(X)) = 1X >= 1X = a__hd(mark(X)) mark(tl(X)) = 4X + 0 >= 4X + 0 = a__tl(mark(X)) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = X >= X = s(X) a__nats() = 0 >= 0 = nats() a__adx(X) = X + 0 >= X + 0 = adx(X) a__zeros() = 0 >= 0 = zeros() a__incr(X) = X + 0 >= X + 0 = incr(X) a__hd(X) = 1X >= 1X = hd(X) a__tl(X) = 4X + 0 >= 4X + 0 = tl(X) problem: DPs: a__tl#(cons(X,Y)) -> mark#(Y) mark#(incr(X)) -> mark#(X) mark#(tl(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) SCC Processor: #sccs: 1 #rules: 2 #arcs: 17/9 DPs: mark#(incr(X)) -> mark#(X) mark#(tl(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) Subterm Criterion Processor: simple projection: pi(mark#) = 0 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