YES Problem: active(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(head(cons(X,L))) -> mark(X) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 1]x0 + [0 0 0]x1 [0 0 1] [0 0 0] , [1] [zeros] = [0] [0], [1 0 0] [incr](x0) = [0 0 1]x0 [0 1 0] , [1 0 0] [0] [mark](x0) = [0 0 0]x0 + [1] [0 0 0] [1], [1 0 0] [s](x0) = [0 0 1]x0 [0 0 1] , [1 0 0] [1] [head](x0) = [0 0 0]x0 + [1] [0 1 0] [0], [1] [nil] = [1] [0], [1] [nats] = [0] [0], [1 0 0] [top](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [proper](x0) = [0 0 0]x0 + [1] [0 0 0] [1], [1 0 0] [0] [tail](x0) = [0 0 0]x0 + [1] [0 0 0] [1], [0] [0] = [0] [0], [1 0 0] [ok](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [adx](x0) = [0 0 1]x0 + [0] [0 0 0] [1], [1 0 0] [0] [active](x0) = [0 0 0]x0 + [1] [0 0 0] [1] orientation: [1] [1] active(incr(nil())) = [1] >= [1] = mark(nil()) [1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] active(incr(cons(X,L))) = [0 0 0]L + [0 0 0]X + [1] >= [0 0 0]L + [0 0 0]X + [1] = mark(cons(s(X),incr(L))) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1] [1] active(adx(nil())) = [1] >= [1] = mark(nil()) [1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] active(adx(cons(X,L))) = [0 0 0]L + [0 0 0]X + [1] >= [0 0 0]L + [0 0 0]X + [1] = mark(incr(cons(X,adx(L)))) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1] [1] active(nats()) = [1] >= [1] = mark(adx(zeros())) [1] [1] [1] [1] active(zeros()) = [1] >= [1] = mark(cons(0(),zeros())) [1] [1] [1 0 0] [1 0 0] [1] [1 0 0] [0] active(head(cons(X,L))) = [0 0 0]L + [0 0 0]X + [1] >= [0 0 0]X + [1] = mark(X) [0 0 0] [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [0] active(tail(cons(X,L))) = [0 0 0]L + [0 0 0]X + [1] >= [0 0 0]L + [1] = mark(L) [0 0 0] [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] active(incr(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = incr(active(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] active(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = cons(active(X1),X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] active(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(active(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] active(adx(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = adx(active(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1] [1 0 0] [1] active(head(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = head(active(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] active(tail(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = tail(active(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] incr(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = mark(incr(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = mark(cons(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] s(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = mark(s(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] adx(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = mark(adx(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1] [1 0 0] [1] head(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = mark(head(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] tail(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = mark(tail(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] proper(incr(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = incr(proper(X)) [0 0 0] [1] [0 0 0] [1] [1] [1] proper(nil()) = [1] >= [0] = ok(nil()) [1] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] proper(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] proper(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] proper(adx(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = adx(proper(X)) [0 0 0] [1] [0 0 0] [1] [1] [1] proper(nats()) = [1] >= [0] = ok(nats()) [1] [0] [1] [1] proper(zeros()) = [1] >= [0] = ok(zeros()) [1] [0] [0] [0] proper(0()) = [1] >= [0] = ok(0()) [1] [0] [1 0 0] [1] [1 0 0] [1] proper(head(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = head(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] proper(tail(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = tail(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] incr(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(incr(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(cons(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] s(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(s(X)) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] adx(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X = ok(adx(X)) [0 0 0] [1] [0 0 0] [1 0 0] [1] [1 0 0] [1] head(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = ok(head(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [0] [1 0 0] tail(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X = ok(tail(X)) [0 0 0] [1] [0 0 0] [1 0 0] [1 0 0] top(mark(X)) = [0 0 0]X >= [0 0 0]X = top(proper(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] top(ok(X)) = [0 0 0]X >= [0 0 0]X = top(active(X)) [0 0 0] [0 0 0] problem: active(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(tail(cons(X,L))) -> mark(L) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [cons](x0, x1) = 2x0 + 2x1, [zeros] = 0, [incr](x0) = x0, [mark](x0) = x0, [s](x0) = x0, [head](x0) = x0 + 2, [nil] = 0, [nats] = 0, [top](x0) = x0, [proper](x0) = x0, [tail](x0) = x0 + 2, [0] = 0, [ok](x0) = x0, [adx](x0) = x0, [active](x0) = x0 orientation: active(incr(nil())) = 0 >= 0 = mark(nil()) active(incr(cons(X,L))) = 2L + 2X >= 2L + 2X = mark(cons(s(X),incr(L))) active(adx(nil())) = 0 >= 0 = mark(nil()) active(adx(cons(X,L))) = 2L + 2X >= 2L + 2X = mark(incr(cons(X,adx(L)))) active(nats()) = 0 >= 0 = mark(adx(zeros())) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(tail(cons(X,L))) = 2L + 2X + 2 >= L = mark(L) active(incr(X)) = X >= X = incr(active(X)) active(cons(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = cons(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(adx(X)) = X >= X = adx(active(X)) active(head(X)) = X + 2 >= X + 2 = head(active(X)) active(tail(X)) = X + 2 >= X + 2 = tail(active(X)) incr(mark(X)) = X >= X = mark(incr(X)) cons(mark(X1),X2) = 2X1 + 2X2 >= 2X1 + 2X2 = mark(cons(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) adx(mark(X)) = X >= X = mark(adx(X)) head(mark(X)) = X + 2 >= X + 2 = mark(head(X)) tail(mark(X)) = X + 2 >= X + 2 = mark(tail(X)) proper(incr(X)) = X >= X = incr(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(cons(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = cons(proper(X1),proper(X2)) proper(s(X)) = X >= X = s(proper(X)) proper(adx(X)) = X >= X = adx(proper(X)) proper(nats()) = 0 >= 0 = ok(nats()) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(0()) = 0 >= 0 = ok(0()) proper(head(X)) = X + 2 >= X + 2 = head(proper(X)) proper(tail(X)) = X + 2 >= X + 2 = tail(proper(X)) incr(ok(X)) = X >= X = ok(incr(X)) cons(ok(X1),ok(X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = ok(cons(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) adx(ok(X)) = X >= X = ok(adx(X)) head(ok(X)) = X + 2 >= X + 2 = ok(head(X)) tail(ok(X)) = X + 2 >= X + 2 = ok(tail(X)) top(mark(X)) = X >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(X)) problem: active(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [cons](x0, x1) = x0 + 4x1, [zeros] = 0, [incr](x0) = x0, [mark](x0) = x0, [s](x0) = x0, [head](x0) = 4x0, [nil] = 0, [nats] = 2, [top](x0) = 2x0, [proper](x0) = x0, [tail](x0) = x0, [0] = 0, [ok](x0) = x0, [adx](x0) = x0, [active](x0) = x0 orientation: active(incr(nil())) = 0 >= 0 = mark(nil()) active(incr(cons(X,L))) = 4L + X >= 4L + X = mark(cons(s(X),incr(L))) active(adx(nil())) = 0 >= 0 = mark(nil()) active(adx(cons(X,L))) = 4L + X >= 4L + X = mark(incr(cons(X,adx(L)))) active(nats()) = 2 >= 0 = mark(adx(zeros())) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(incr(X)) = X >= X = incr(active(X)) active(cons(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = cons(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(adx(X)) = X >= X = adx(active(X)) active(head(X)) = 4X >= 4X = head(active(X)) active(tail(X)) = X >= X = tail(active(X)) incr(mark(X)) = X >= X = mark(incr(X)) cons(mark(X1),X2) = X1 + 4X2 >= X1 + 4X2 = mark(cons(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) adx(mark(X)) = X >= X = mark(adx(X)) head(mark(X)) = 4X >= 4X = mark(head(X)) tail(mark(X)) = X >= X = mark(tail(X)) proper(incr(X)) = X >= X = incr(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(cons(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = cons(proper(X1),proper(X2)) proper(s(X)) = X >= X = s(proper(X)) proper(adx(X)) = X >= X = adx(proper(X)) proper(nats()) = 2 >= 2 = ok(nats()) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(0()) = 0 >= 0 = ok(0()) proper(head(X)) = 4X >= 4X = head(proper(X)) proper(tail(X)) = X >= X = tail(proper(X)) incr(ok(X)) = X >= X = ok(incr(X)) cons(ok(X1),ok(X2)) = X1 + 4X2 >= X1 + 4X2 = ok(cons(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) adx(ok(X)) = X >= X = ok(adx(X)) head(ok(X)) = 4X >= 4X = ok(head(X)) tail(ok(X)) = X >= X = ok(tail(X)) top(mark(X)) = 2X >= 2X = top(proper(X)) top(ok(X)) = 2X >= 2X = top(active(X)) problem: active(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(zeros()) -> mark(cons(0(),zeros())) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [zeros] = [0] [1], [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 1] , [1 0 1] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [head](x0) = [0 0 0]x0 [0 0 1] , [0] [nil] = [0] [0], [0] [nats] = [0] [0], [1 0 1] [top](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [proper](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [tail](x0) = [0 0 0]x0 [0 0 1] , [0] [0] = [0] [0], [1 0 0] [ok](x0) = [0 0 0]x0 [0 1 1] , [1 0 0] [adx](x0) = [0 0 0]x0 [0 0 1] , [1 0 1] [active](x0) = [0 0 0]x0 [0 0 0] orientation: [0] [0] active(incr(nil())) = [0] >= [0] = mark(nil()) [0] [0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] active(incr(cons(X,L))) = [0 0 0]L + [0 0 0]X >= [0 0 0]L + [0 0 0]X = mark(cons(s(X),incr(L))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] active(adx(nil())) = [0] >= [0] = mark(nil()) [0] [0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] active(adx(cons(X,L))) = [0 0 0]L + [0 0 0]X >= [0 0 0]L + [0 0 0]X = mark(incr(cons(X,adx(L)))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1] [0] active(zeros()) = [0] >= [0] = mark(cons(0(),zeros())) [0] [0] [1 0 1] [1 0 1] active(incr(X)) = [0 0 0]X >= [0 0 0]X = incr(active(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1 0 1] [1 0 0] active(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] active(s(X)) = [0 0 0]X >= [0 0 0]X = s(active(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] active(adx(X)) = [0 0 0]X >= [0 0 0]X = adx(active(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] active(head(X)) = [0 0 0]X >= [0 0 0]X = head(active(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] active(tail(X)) = [0 0 0]X >= [0 0 0]X = tail(active(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] incr(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(incr(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1 0 1] [1 0 0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(cons(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] s(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(s(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] adx(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(adx(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] head(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(head(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] tail(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(tail(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] proper(incr(X)) = [0 0 0]X >= [0 0 0]X = incr(proper(X)) [0 0 1] [0 0 1] [0] [0] proper(nil()) = [0] >= [0] = ok(nil()) [0] [0] [1 0 1] [1 0 0] [1 0 1] [1 0 0] proper(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] proper(s(X)) = [0 0 0]X >= [0 0 0]X = s(proper(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] proper(adx(X)) = [0 0 0]X >= [0 0 0]X = adx(proper(X)) [0 0 1] [0 0 1] [0] [0] proper(nats()) = [0] >= [0] = ok(nats()) [0] [0] [0] [0] proper(zeros()) = [0] >= [0] = ok(zeros()) [1] [1] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 0 0] [1 0 0] proper(head(X)) = [0 0 0]X >= [0 0 0]X = head(proper(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] proper(tail(X)) = [0 0 0]X >= [0 0 0]X = tail(proper(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] incr(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(incr(X)) [0 1 1] [0 0 1] [1 1 1] [1 0 0] [1 0 1] [1 0 0] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(cons(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] s(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(s(X)) [0 1 1] [0 0 1] [1 0 0] [1 0 0] adx(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(adx(X)) [0 1 1] [0 0 1] [1 0 0] [1 0 0] head(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(head(X)) [0 1 1] [0 0 1] [1 0 0] [1 0 0] tail(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(tail(X)) [0 1 1] [0 0 1] [1 0 1] [1 0 1] top(mark(X)) = [0 0 0]X >= [0 0 0]X = top(proper(X)) [0 0 0] [0 0 0] [1 1 1] [1 0 1] top(ok(X)) = [0 0 0]X >= [0 0 0]X = top(active(X)) [0 0 0] [0 0 0] problem: active(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(nil())) -> mark(nil()) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [cons](x0, x1) = x0 + x1, [zeros] = 4, [incr](x0) = x0, [mark](x0) = x0, [s](x0) = x0, [head](x0) = 4x0, [nil] = 0, [nats] = 1, [top](x0) = x0, [proper](x0) = x0, [tail](x0) = x0 + 4, [0] = 0, [ok](x0) = x0, [adx](x0) = 2x0 + 4, [active](x0) = x0 orientation: active(incr(nil())) = 0 >= 0 = mark(nil()) active(incr(cons(X,L))) = L + X >= L + X = mark(cons(s(X),incr(L))) active(adx(nil())) = 4 >= 0 = mark(nil()) active(adx(cons(X,L))) = 2L + 2X + 4 >= 2L + X + 4 = mark(incr(cons(X,adx(L)))) active(incr(X)) = X >= X = incr(active(X)) active(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(adx(X)) = 2X + 4 >= 2X + 4 = adx(active(X)) active(head(X)) = 4X >= 4X = head(active(X)) active(tail(X)) = X + 4 >= X + 4 = tail(active(X)) incr(mark(X)) = X >= X = mark(incr(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(cons(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) adx(mark(X)) = 2X + 4 >= 2X + 4 = mark(adx(X)) head(mark(X)) = 4X >= 4X = mark(head(X)) tail(mark(X)) = X + 4 >= X + 4 = mark(tail(X)) proper(incr(X)) = X >= X = incr(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(s(X)) = X >= X = s(proper(X)) proper(adx(X)) = 2X + 4 >= 2X + 4 = adx(proper(X)) proper(nats()) = 1 >= 1 = ok(nats()) proper(zeros()) = 4 >= 4 = ok(zeros()) proper(0()) = 0 >= 0 = ok(0()) proper(head(X)) = 4X >= 4X = head(proper(X)) proper(tail(X)) = X + 4 >= X + 4 = tail(proper(X)) incr(ok(X)) = X >= X = ok(incr(X)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) adx(ok(X)) = 2X + 4 >= 2X + 4 = ok(adx(X)) head(ok(X)) = 4X >= 4X = ok(head(X)) tail(ok(X)) = X + 4 >= X + 4 = ok(tail(X)) top(mark(X)) = X >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(X)) problem: active(incr(nil())) -> mark(nil()) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [cons](x0, x1) = x0 + x1 + 1, [zeros] = 0, [incr](x0) = x0 + 4, [mark](x0) = x0, [s](x0) = x0, [head](x0) = x0, [nil] = 0, [nats] = 0, [top](x0) = 2x0, [proper](x0) = x0, [tail](x0) = 4x0, [0] = 0, [ok](x0) = x0, [adx](x0) = 6x0, [active](x0) = x0 orientation: active(incr(nil())) = 4 >= 0 = mark(nil()) active(incr(cons(X,L))) = L + X + 5 >= L + X + 5 = mark(cons(s(X),incr(L))) active(adx(cons(X,L))) = 6L + 6X + 6 >= 6L + X + 5 = mark(incr(cons(X,adx(L)))) active(incr(X)) = X + 4 >= X + 4 = incr(active(X)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(adx(X)) = 6X >= 6X = adx(active(X)) active(head(X)) = X >= X = head(active(X)) active(tail(X)) = 4X >= 4X = tail(active(X)) incr(mark(X)) = X + 4 >= X + 4 = mark(incr(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) adx(mark(X)) = 6X >= 6X = mark(adx(X)) head(mark(X)) = X >= X = mark(head(X)) tail(mark(X)) = 4X >= 4X = mark(tail(X)) proper(incr(X)) = X + 4 >= X + 4 = incr(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(proper(X1),proper(X2)) proper(s(X)) = X >= X = s(proper(X)) proper(adx(X)) = 6X >= 6X = adx(proper(X)) proper(nats()) = 0 >= 0 = ok(nats()) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(0()) = 0 >= 0 = ok(0()) proper(head(X)) = X >= X = head(proper(X)) proper(tail(X)) = 4X >= 4X = tail(proper(X)) incr(ok(X)) = X + 4 >= X + 4 = ok(incr(X)) cons(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(cons(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) adx(ok(X)) = 6X >= 6X = ok(adx(X)) head(ok(X)) = X >= X = ok(head(X)) tail(ok(X)) = 4X >= 4X = ok(tail(X)) top(mark(X)) = 2X >= 2X = top(proper(X)) top(ok(X)) = 2X >= 2X = top(active(X)) problem: active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [cons](x0, x1) = 2x0 + x1 + 1, [zeros] = 0, [incr](x0) = 4x0, [mark](x0) = x0, [s](x0) = 4x0, [head](x0) = 4x0, [nil] = 0, [nats] = 0, [top](x0) = x0, [proper](x0) = x0, [tail](x0) = x0, [0] = 0, [ok](x0) = x0, [adx](x0) = x0, [active](x0) = x0 orientation: active(incr(cons(X,L))) = 4L + 8X + 4 >= 4L + 8X + 1 = mark(cons(s(X),incr(L))) active(incr(X)) = 4X >= 4X = incr(active(X)) active(cons(X1,X2)) = 2X1 + X2 + 1 >= 2X1 + X2 + 1 = cons(active(X1),X2) active(s(X)) = 4X >= 4X = s(active(X)) active(adx(X)) = X >= X = adx(active(X)) active(head(X)) = 4X >= 4X = head(active(X)) active(tail(X)) = X >= X = tail(active(X)) incr(mark(X)) = 4X >= 4X = mark(incr(X)) cons(mark(X1),X2) = 2X1 + X2 + 1 >= 2X1 + X2 + 1 = mark(cons(X1,X2)) s(mark(X)) = 4X >= 4X = mark(s(X)) adx(mark(X)) = X >= X = mark(adx(X)) head(mark(X)) = 4X >= 4X = mark(head(X)) tail(mark(X)) = X >= X = mark(tail(X)) proper(incr(X)) = 4X >= 4X = incr(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) proper(cons(X1,X2)) = 2X1 + X2 + 1 >= 2X1 + X2 + 1 = cons(proper(X1),proper(X2)) proper(s(X)) = 4X >= 4X = s(proper(X)) proper(adx(X)) = X >= X = adx(proper(X)) proper(nats()) = 0 >= 0 = ok(nats()) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(0()) = 0 >= 0 = ok(0()) proper(head(X)) = 4X >= 4X = head(proper(X)) proper(tail(X)) = X >= X = tail(proper(X)) incr(ok(X)) = 4X >= 4X = ok(incr(X)) cons(ok(X1),ok(X2)) = 2X1 + X2 + 1 >= 2X1 + X2 + 1 = ok(cons(X1,X2)) s(ok(X)) = 4X >= 4X = ok(s(X)) adx(ok(X)) = X >= X = ok(adx(X)) head(ok(X)) = 4X >= 4X = ok(head(X)) tail(ok(X)) = X >= X = ok(tail(X)) top(mark(X)) = X >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(X)) problem: active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1] [zeros] = [0] [0], [1 0 1] [incr](x0) = [0 0 0]x0 [0 0 0] , [1 1 1] [1] [mark](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [head](x0) = [0 0 0]x0 [0 0 0] , [0] [nil] = [0] [0], [0] [nats] = [0] [0], [1 1 0] [top](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [proper](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [tail](x0) = [0 0 0]x0 [0 0 0] , [0] [0] = [0] [0], [1 0 0] [ok](x0) = [0 0 1]x0 [0 1 1] , [1 0 0] [adx](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [active](x0) = [0 0 0]x0 [0 0 0] orientation: [1 0 1] [1 0 0] active(incr(X)) = [0 0 0]X >= [0 0 0]X = incr(active(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(s(X)) = [0 0 0]X >= [0 0 0]X = s(active(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(adx(X)) = [0 0 0]X >= [0 0 0]X = adx(active(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(head(X)) = [0 0 0]X >= [0 0 0]X = head(active(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(tail(X)) = [0 0 0]X >= [0 0 0]X = tail(active(X)) [0 0 0] [0 0 0] [1 1 1] [1] [1 0 1] [1] incr(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(incr(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1 0 0] [1] [1 0 0] [1 0 0] [1] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = mark(cons(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 1] [1] [1 0 0] [1] s(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(s(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1] [1 0 0] [1] adx(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(adx(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1] [1 0 0] [1] head(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(head(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1] [1 0 0] [1] tail(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(tail(X)) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1 0 0] proper(incr(X)) = [0 0 0]X >= [0 0 0]X = incr(proper(X)) [0 0 0] [0 0 0] [0] [0] proper(nil()) = [0] >= [0] = ok(nil()) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] proper(s(X)) = [0 0 0]X >= [0 0 0]X = s(proper(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] proper(adx(X)) = [0 0 0]X >= [0 0 0]X = adx(proper(X)) [0 0 0] [0 0 0] [0] [0] proper(nats()) = [0] >= [0] = ok(nats()) [0] [0] [1] [1] proper(zeros()) = [0] >= [0] = ok(zeros()) [0] [0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 0 0] [1 0 0] proper(head(X)) = [0 0 0]X >= [0 0 0]X = head(proper(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] proper(tail(X)) = [0 0 0]X >= [0 0 0]X = tail(proper(X)) [0 0 0] [0 0 0] [1 1 1] [1 0 1] incr(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(incr(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(cons(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] s(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(s(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] adx(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(adx(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] head(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(head(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] tail(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(tail(X)) [0 0 0] [0 0 0] [1 1 1] [1] [1 0 0] top(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X = top(proper(X)) [0 0 0] [0] [0 0 0] [1 0 1] [1 0 0] top(ok(X)) = [0 0 0]X >= [0 0 0]X = top(active(X)) [0 0 0] [0 0 0] problem: active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(adx(X)) -> adx(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) adx(ok(X)) -> ok(adx(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [cons](x0, x1) = 5x0 + x1 + 2, [zeros] = 6, [incr](x0) = 2x0, [mark](x0) = x0, [s](x0) = 4x0 + 6, [head](x0) = 2x0 + 2, [nil] = 2, [nats] = 3, [top](x0) = x0 + 6, [proper](x0) = 4x0, [tail](x0) = 4x0, [0] = 2, [ok](x0) = 2x0 + 4, [adx](x0) = x0, [active](x0) = x0 orientation: active(incr(X)) = 2X >= 2X = incr(active(X)) active(cons(X1,X2)) = 5X1 + X2 + 2 >= 5X1 + X2 + 2 = cons(active(X1),X2) active(s(X)) = 4X + 6 >= 4X + 6 = s(active(X)) active(adx(X)) = X >= X = adx(active(X)) active(head(X)) = 2X + 2 >= 2X + 2 = head(active(X)) active(tail(X)) = 4X >= 4X = tail(active(X)) incr(mark(X)) = 2X >= 2X = mark(incr(X)) cons(mark(X1),X2) = 5X1 + X2 + 2 >= 5X1 + X2 + 2 = mark(cons(X1,X2)) s(mark(X)) = 4X + 6 >= 4X + 6 = mark(s(X)) adx(mark(X)) = X >= X = mark(adx(X)) head(mark(X)) = 2X + 2 >= 2X + 2 = mark(head(X)) tail(mark(X)) = 4X >= 4X = mark(tail(X)) proper(incr(X)) = 8X >= 8X = incr(proper(X)) proper(nil()) = 8 >= 8 = ok(nil()) proper(cons(X1,X2)) = 20X1 + 4X2 + 8 >= 20X1 + 4X2 + 2 = cons(proper(X1),proper(X2)) proper(s(X)) = 16X + 24 >= 16X + 6 = s(proper(X)) proper(adx(X)) = 4X >= 4X = adx(proper(X)) proper(nats()) = 12 >= 10 = ok(nats()) proper(zeros()) = 24 >= 16 = ok(zeros()) proper(0()) = 8 >= 8 = ok(0()) proper(head(X)) = 8X + 8 >= 8X + 2 = head(proper(X)) proper(tail(X)) = 16X >= 16X = tail(proper(X)) incr(ok(X)) = 4X + 8 >= 4X + 4 = ok(incr(X)) cons(ok(X1),ok(X2)) = 10X1 + 2X2 + 26 >= 10X1 + 2X2 + 8 = ok(cons(X1,X2)) s(ok(X)) = 8X + 22 >= 8X + 16 = ok(s(X)) adx(ok(X)) = 2X + 4 >= 2X + 4 = ok(adx(X)) head(ok(X)) = 4X + 10 >= 4X + 8 = ok(head(X)) tail(ok(X)) = 8X + 16 >= 8X + 4 = ok(tail(X)) top(ok(X)) = 2X + 10 >= X + 6 = top(active(X)) problem: active(incr(X)) -> incr(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(adx(X)) -> adx(proper(X)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) adx(ok(X)) -> ok(adx(X)) Matrix Interpretation Processor: dim=1 interpretation: [cons](x0, x1) = 4x0 + x1 + 3, [incr](x0) = x0, [mark](x0) = x0 + 2, [s](x0) = 5x0 + 3, [head](x0) = x0, [nil] = 1, [proper](x0) = 4x0 + 2, [tail](x0) = x0, [0] = 1, [ok](x0) = 2x0 + 4, [adx](x0) = x0, [active](x0) = 6x0 + 3 orientation: active(incr(X)) = 6X + 3 >= 6X + 3 = incr(active(X)) active(cons(X1,X2)) = 24X1 + 6X2 + 21 >= 24X1 + X2 + 15 = cons(active(X1),X2) active(s(X)) = 30X + 21 >= 30X + 18 = s(active(X)) active(adx(X)) = 6X + 3 >= 6X + 3 = adx(active(X)) active(head(X)) = 6X + 3 >= 6X + 3 = head(active(X)) active(tail(X)) = 6X + 3 >= 6X + 3 = tail(active(X)) incr(mark(X)) = X + 2 >= X + 2 = mark(incr(X)) cons(mark(X1),X2) = 4X1 + X2 + 11 >= 4X1 + X2 + 5 = mark(cons(X1,X2)) s(mark(X)) = 5X + 13 >= 5X + 5 = mark(s(X)) adx(mark(X)) = X + 2 >= X + 2 = mark(adx(X)) head(mark(X)) = X + 2 >= X + 2 = mark(head(X)) tail(mark(X)) = X + 2 >= X + 2 = mark(tail(X)) proper(incr(X)) = 4X + 2 >= 4X + 2 = incr(proper(X)) proper(nil()) = 6 >= 6 = ok(nil()) proper(adx(X)) = 4X + 2 >= 4X + 2 = adx(proper(X)) proper(0()) = 6 >= 6 = ok(0()) proper(tail(X)) = 4X + 2 >= 4X + 2 = tail(proper(X)) adx(ok(X)) = 2X + 4 >= 2X + 4 = ok(adx(X)) problem: active(incr(X)) -> incr(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(adx(X)) -> adx(proper(X)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) adx(ok(X)) -> ok(adx(X)) Matrix Interpretation Processor: dim=1 interpretation: [incr](x0) = 2x0 + 2, [mark](x0) = 2x0 + 2, [head](x0) = x0, [nil] = 0, [proper](x0) = 4x0 + 6, [tail](x0) = 2x0 + 2, [0] = 0, [ok](x0) = x0 + 6, [adx](x0) = 2x0 + 2, [active](x0) = x0 orientation: active(incr(X)) = 2X + 2 >= 2X + 2 = incr(active(X)) active(adx(X)) = 2X + 2 >= 2X + 2 = adx(active(X)) active(head(X)) = X >= X = head(active(X)) active(tail(X)) = 2X + 2 >= 2X + 2 = tail(active(X)) incr(mark(X)) = 4X + 6 >= 4X + 6 = mark(incr(X)) adx(mark(X)) = 4X + 6 >= 4X + 6 = mark(adx(X)) head(mark(X)) = 2X + 2 >= 2X + 2 = mark(head(X)) tail(mark(X)) = 4X + 6 >= 4X + 6 = mark(tail(X)) proper(incr(X)) = 8X + 14 >= 8X + 14 = incr(proper(X)) proper(nil()) = 6 >= 6 = ok(nil()) proper(adx(X)) = 8X + 14 >= 8X + 14 = adx(proper(X)) proper(0()) = 6 >= 6 = ok(0()) proper(tail(X)) = 8X + 14 >= 8X + 14 = tail(proper(X)) adx(ok(X)) = 2X + 14 >= 2X + 8 = ok(adx(X)) problem: active(incr(X)) -> incr(active(X)) active(adx(X)) -> adx(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) incr(mark(X)) -> mark(incr(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(adx(X)) -> adx(proper(X)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [incr](x0) = [0 0 1]x0 + [0] [0 1 0] [1], [1 0 0] [0] [mark](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [head](x0) = [0 0 0]x0 [0 1 1] , [0] [nil] = [0] [1], [1 1 1] [proper](x0) = [0 1 0]x0 [0 0 1] , [1 0 0] [0] [tail](x0) = [0 0 0]x0 + [1] [0 1 1] [0], [0] [0] = [0] [1], [1 0 0] [0] [ok](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [0] [adx](x0) = [0 0 0]x0 + [1] [0 1 1] [0], [1 1 1] [active](x0) = [0 1 0]x0 [0 0 1] orientation: [1 1 1] [1] [1 1 1] [0] active(incr(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = incr(active(X)) [0 1 0] [1] [0 1 0] [1] [1 1 1] [1] [1 1 1] [0] active(adx(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = adx(active(X)) [0 1 1] [0] [0 1 1] [0] [1 1 1] [1 1 1] active(head(X)) = [0 0 0]X >= [0 0 0]X = head(active(X)) [0 1 1] [0 1 1] [1 1 1] [1] [1 1 1] [0] active(tail(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = tail(active(X)) [0 1 1] [0] [0 1 1] [0] [1 0 0] [0] [1 0 0] [0] incr(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = mark(incr(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] adx(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = mark(adx(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] head(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(head(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] tail(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = mark(tail(X)) [0 0 0] [1] [0 0 0] [1] [1 1 1] [1] [1 1 1] [0] proper(incr(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = incr(proper(X)) [0 1 0] [1] [0 1 0] [1] [1] [0] proper(nil()) = [0] >= [0] = ok(nil()) [1] [1] [1 1 1] [1] [1 1 1] [0] proper(adx(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = adx(proper(X)) [0 1 1] [0] [0 1 1] [0] [1] [0] proper(0()) = [0] >= [0] = ok(0()) [1] [1] [1 1 1] [1] [1 1 1] [0] proper(tail(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = tail(proper(X)) [0 1 1] [0] [0 1 1] [0] problem: active(head(X)) -> head(active(X)) incr(mark(X)) -> mark(incr(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) Matrix Interpretation Processor: dim=1 interpretation: [incr](x0) = x0, [mark](x0) = 4x0 + 4, [head](x0) = x0, [tail](x0) = 2x0 + 1, [adx](x0) = x0, [active](x0) = x0 orientation: active(head(X)) = X >= X = head(active(X)) incr(mark(X)) = 4X + 4 >= 4X + 4 = mark(incr(X)) adx(mark(X)) = 4X + 4 >= 4X + 4 = mark(adx(X)) head(mark(X)) = 4X + 4 >= 4X + 4 = mark(head(X)) tail(mark(X)) = 8X + 9 >= 8X + 8 = mark(tail(X)) problem: active(head(X)) -> head(active(X)) incr(mark(X)) -> mark(incr(X)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(X)) Matrix Interpretation Processor: dim=1 interpretation: [incr](x0) = 4x0 + 2, [mark](x0) = 4x0 + 2, [head](x0) = 6x0 + 3, [adx](x0) = 4x0 + 2, [active](x0) = x0 orientation: active(head(X)) = 6X + 3 >= 6X + 3 = head(active(X)) incr(mark(X)) = 16X + 10 >= 16X + 10 = mark(incr(X)) adx(mark(X)) = 16X + 10 >= 16X + 10 = mark(adx(X)) head(mark(X)) = 24X + 15 >= 24X + 14 = mark(head(X)) problem: active(head(X)) -> head(active(X)) incr(mark(X)) -> mark(incr(X)) adx(mark(X)) -> mark(adx(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [incr](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 1] [mark](x0) = [1 1 0]x0 [0 0 0] , [1 0 0] [0] [head](x0) = [0 0 1]x0 + [0] [0 1 0] [1], [1 0 0] [adx](x0) = [0 1 0]x0 [0 0 0] , [1 1 1] [0] [active](x0) = [0 0 1]x0 + [0] [0 1 0] [1] orientation: [1 1 1] [1] [1 1 1] [0] active(head(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = head(active(X)) [0 0 1] [1] [0 0 1] [1] [1 0 1] [0] [1 0 0] [0] incr(mark(X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = mark(incr(X)) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1 0 0] adx(mark(X)) = [1 1 0]X >= [1 1 0]X = mark(adx(X)) [0 0 0] [0 0 0] problem: incr(mark(X)) -> mark(incr(X)) adx(mark(X)) -> mark(adx(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 1 1] [incr](x0) = [0 1 0]x0 [0 0 1] , [1 0 0] [0] [mark](x0) = [0 0 1]x0 + [1] [0 1 0] [0], [1 1 1] [adx](x0) = [1 1 0]x0 [1 0 1] orientation: [1 1 1] [1] [1 1 1] [0] incr(mark(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = mark(incr(X)) [0 1 0] [0] [0 1 0] [0] [1 1 1] [1] [1 1 1] [0] adx(mark(X)) = [1 0 1]X + [1] >= [1 0 1]X + [1] = mark(adx(X)) [1 1 0] [0] [1 1 0] [0] problem: Qed