/export/starexec/sandbox/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- 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=1 interpretation: [top](x0) = x0 + 5, [ok](x0) = x0, [proper](x0) = x0, [tail](x0) = 2x0 + 1, [head](x0) = 4x0, [0] = 0, [zeros] = 0, [nats] = 4, [adx](x0) = 2x0, [s](x0) = x0, [cons](x0, x1) = x0 + x1, [mark](x0) = x0, [active](x0) = x0, [incr](x0) = x0, [nil] = 0 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())) = 0 >= 0 = mark(nil()) active(adx(cons(X,L))) = 2L + 2X >= 2L + X = mark(incr(cons(X,adx(L)))) active(nats()) = 4 >= 0 = mark(adx(zeros())) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(head(cons(X,L))) = 4L + 4X >= X = mark(X) active(tail(cons(X,L))) = 2L + 2X + 1 >= L = mark(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 >= 2X = adx(active(X)) active(head(X)) = 4X >= 4X = head(active(X)) active(tail(X)) = 2X + 1 >= 2X + 1 = 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 >= 2X = mark(adx(X)) head(mark(X)) = 4X >= 4X = mark(head(X)) tail(mark(X)) = 2X + 1 >= 2X + 1 = 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 >= 2X = adx(proper(X)) proper(nats()) = 4 >= 4 = 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)) = 2X + 1 >= 2X + 1 = 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 >= 2X = ok(adx(X)) head(ok(X)) = 4X >= 4X = ok(head(X)) tail(ok(X)) = 2X + 1 >= 2X + 1 = ok(tail(X)) top(mark(X)) = X + 5 >= X + 5 = top(proper(X)) top(ok(X)) = X + 5 >= X + 5 = 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(head(cons(X,L))) -> mark(X) 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] [top](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [ok](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [proper](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1] [tail](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [1] [head](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [0] [0] = [0] [0], [1] [zeros] = [0] [0], [0] [nats] = [0] [0], [1 0 0] [1] [adx](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] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [active](x0) = [0 0 0]x0 [1 0 0] , [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [0] [nil] = [0] [0] orientation: [0] [0] active(incr(nil())) = [0] >= [0] = mark(nil()) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] 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))) [1 0 0] [1 0 0] [0 0 0] [0 0 0] [1] [0] active(adx(nil())) = [0] >= [0] = mark(nil()) [1] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] active(adx(cons(X,L))) = [0 0 0]L + [0 0 0]X + [0] >= [0 0 0]L + [0 0 0]X + [0] = mark(incr(cons(X,adx(L)))) [1 0 0] [1 0 0] [1] [0 0 0] [0 0 0] [0] [1] [1] active(zeros()) = [0] >= [0] = mark(cons(0(),zeros())) [1] [0] [1 0 0] [1 0 0] [1] [1 0 0] active(head(cons(X,L))) = [0 0 0]L + [0 0 0]X + [0] >= [0 0 0]X = mark(X) [1 0 0] [1 0 0] [1] [0 0 0] [1 0 0] [1 0 0] active(incr(X)) = [0 0 0]X >= [0 0 0]X = incr(active(X)) [1 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) [1 0 0] [1 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)) [1 0 0] [0 0 0] [1 0 0] [1] [1 0 0] [1] active(adx(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = adx(active(X)) [1 0 0] [1] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] active(head(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = head(active(X)) [1 0 0] [1] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] active(tail(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = tail(active(X)) [1 0 0] [1] [0 0 0] [0] [1 0 0] [1 0 0] incr(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(incr(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [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 0] [1 0 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(s(X)) [0 0 0] [0 0 0] [1 0 0] [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 0 0] [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 0 0] [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 0] [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] [1 0 0] [1] proper(adx(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = adx(proper(X)) [0 0 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] [1 0 0] [1] proper(head(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = head(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] proper(tail(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = tail(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 0 1] [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 1] [1 0 1] [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 1] [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 1] [1] [1 0 0] [1] adx(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(adx(X)) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1] [1 0 0] [1] head(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(head(X)) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1] [1 0 0] [1] tail(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(tail(X)) [0 0 0] [0] [0 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 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(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(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 1 0] [top](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [ok](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [proper](x0) = [0 1 0]x0 [1 0 0] , [1 0 0] [tail](x0) = [0 1 0]x0 [0 0 0] , [1 1 0] [head](x0) = [0 0 0]x0 [1 0 0] , [0] [0] = [0] [1], [0] [zeros] = [1] [0], [1] [nats] = [0] [0], [1 0 0] [adx](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [s](x0) = [0 1 0]x0 [0 0 0] , [1 1 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [0 0 0] , [1 0 0] [mark](x0) = [0 1 0]x0 [0 0 0] , [1 1 0] [active](x0) = [0 0 0]x0 [1 0 0] , [1 0 0] [incr](x0) = [0 1 0]x0 [0 0 0] , [0] [nil] = [0] [0] orientation: [0] [0] active(incr(nil())) = [0] >= [0] = mark(nil()) [0] [0] [1 0 0] [1 1 0] [1 0 0] [1 1 0] 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))) [1 0 0] [1 1 0] [0 0 0] [0 0 0] [1 0 0] [1 1 0] [1 0 0] [1 1 0] 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)))) [1 0 0] [1 1 0] [0 0 0] [0 0 0] [1] [0] active(zeros()) = [0] >= [0] = mark(cons(0(),zeros())) [0] [0] [1 1 0] [1 1 0] active(incr(X)) = [0 0 0]X >= [0 0 0]X = incr(active(X)) [1 0 0] [0 0 0] [1 1 0] [1 0 0] [1 1 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) [1 1 0] [1 0 0] [1 0 0] [0 0 0] [1 1 0] [1 1 0] active(s(X)) = [0 0 0]X >= [0 0 0]X = s(active(X)) [1 0 0] [0 0 0] [1 1 0] [1 1 0] active(adx(X)) = [0 0 0]X >= [0 0 0]X = adx(active(X)) [1 0 0] [0 0 0] [1 1 0] [1 1 0] active(head(X)) = [0 0 0]X >= [0 0 0]X = head(active(X)) [1 1 0] [1 1 0] [1 1 0] [1 1 0] active(tail(X)) = [0 0 0]X >= [0 0 0]X = tail(active(X)) [1 0 0] [0 0 0] [1 0 0] [1 0 0] incr(mark(X)) = [0 1 0]X >= [0 1 0]X = mark(incr(X)) [0 0 0] [0 0 0] [1 1 0] [1 0 0] [1 1 0] [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 0] [1 0 0] s(mark(X)) = [0 1 0]X >= [0 1 0]X = mark(s(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] adx(mark(X)) = [0 1 0]X >= [0 1 0]X = mark(adx(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] head(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(head(X)) [1 0 0] [0 0 0] [1 0 0] [1 0 0] tail(mark(X)) = [0 1 0]X >= [0 1 0]X = mark(tail(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] proper(incr(X)) = [0 1 0]X >= [0 1 0]X = incr(proper(X)) [1 0 0] [0 0 0] [0] [0] proper(nil()) = [0] >= [0] = ok(nil()) [0] [0] [1 1 0] [1 0 0] [1 1 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)) [1 1 0] [1 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] proper(s(X)) = [0 1 0]X >= [0 1 0]X = s(proper(X)) [1 0 0] [0 0 0] [1 0 0] [1 0 0] proper(adx(X)) = [0 1 0]X >= [0 1 0]X = adx(proper(X)) [1 0 0] [0 0 0] [1] [1] proper(nats()) = [0] >= [0] = ok(nats()) [1] [0] [0] [0] proper(zeros()) = [1] >= [1] = ok(zeros()) [0] [0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 1 0] [1 1 0] proper(head(X)) = [0 0 0]X >= [0 0 0]X = head(proper(X)) [1 1 0] [1 0 0] [1 0 0] [1 0 0] proper(tail(X)) = [0 1 0]X >= [0 1 0]X = tail(proper(X)) [1 0 0] [0 0 0] [1 0 0] [1 0 0] incr(ok(X)) = [0 1 0]X >= [0 1 0]X = ok(incr(X)) [0 0 0] [0 0 0] [1 1 0] [1 0 0] [1 1 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 1 0]X >= [0 1 0]X = ok(s(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] adx(ok(X)) = [0 1 0]X >= [0 1 0]X = ok(adx(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] head(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(head(X)) [1 0 0] [0 0 0] [1 0 0] [1 0 0] tail(ok(X)) = [0 1 0]X >= [0 1 0]X = ok(tail(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] top(mark(X)) = [0 0 0]X >= [0 0 0]X = top(proper(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 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(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: [top](x0) = 4x0, [ok](x0) = x0, [proper](x0) = x0, [tail](x0) = x0, [head](x0) = 2x0, [0] = 0, [zeros] = 0, [nats] = 0, [adx](x0) = 3x0 + 4, [s](x0) = x0, [cons](x0, x1) = x0 + 2x1 + 4, [mark](x0) = x0, [active](x0) = x0, [incr](x0) = x0, [nil] = 0 orientation: active(incr(nil())) = 0 >= 0 = mark(nil()) active(incr(cons(X,L))) = 2L + X + 4 >= 2L + X + 4 = mark(cons(s(X),incr(L))) active(adx(cons(X,L))) = 6L + 3X + 16 >= 6L + X + 12 = mark(incr(cons(X,adx(L)))) active(incr(X)) = X >= X = incr(active(X)) active(cons(X1,X2)) = X1 + 2X2 + 4 >= X1 + 2X2 + 4 = cons(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(adx(X)) = 3X + 4 >= 3X + 4 = adx(active(X)) active(head(X)) = 2X >= 2X = head(active(X)) active(tail(X)) = X >= X = tail(active(X)) incr(mark(X)) = X >= X = mark(incr(X)) cons(mark(X1),X2) = X1 + 2X2 + 4 >= X1 + 2X2 + 4 = mark(cons(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) adx(mark(X)) = 3X + 4 >= 3X + 4 = mark(adx(X)) head(mark(X)) = 2X >= 2X = 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 + 2X2 + 4 >= X1 + 2X2 + 4 = cons(proper(X1),proper(X2)) proper(s(X)) = X >= X = s(proper(X)) proper(adx(X)) = 3X + 4 >= 3X + 4 = adx(proper(X)) proper(nats()) = 0 >= 0 = ok(nats()) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(0()) = 0 >= 0 = ok(0()) proper(head(X)) = 2X >= 2X = 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 + 2X2 + 4 >= X1 + 2X2 + 4 = ok(cons(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) adx(ok(X)) = 3X + 4 >= 3X + 4 = ok(adx(X)) head(ok(X)) = 2X >= 2X = ok(head(X)) tail(ok(X)) = X >= X = ok(tail(X)) top(mark(X)) = 4X >= 4X = top(proper(X)) top(ok(X)) = 4X >= 4X = top(active(X)) problem: active(incr(nil())) -> mark(nil()) 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: [top](x0) = x0 + 1, [ok](x0) = x0, [proper](x0) = x0, [tail](x0) = 4x0, [head](x0) = x0, [0] = 2, [zeros] = 4, [nats] = 1, [adx](x0) = 2x0, [s](x0) = x0, [cons](x0, x1) = x0 + 4x1, [mark](x0) = x0, [active](x0) = x0, [incr](x0) = 4x0, [nil] = 2 orientation: active(incr(nil())) = 8 >= 2 = mark(nil()) active(incr(cons(X,L))) = 16L + 4X >= 16L + X = mark(cons(s(X),incr(L))) active(incr(X)) = 4X >= 4X = 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)) = 2X >= 2X = adx(active(X)) active(head(X)) = X >= X = head(active(X)) active(tail(X)) = 4X >= 4X = tail(active(X)) incr(mark(X)) = 4X >= 4X = 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)) = 2X >= 2X = mark(adx(X)) head(mark(X)) = X >= X = mark(head(X)) tail(mark(X)) = 4X >= 4X = mark(tail(X)) proper(incr(X)) = 4X >= 4X = incr(proper(X)) proper(nil()) = 2 >= 2 = 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)) = 2X >= 2X = adx(proper(X)) proper(nats()) = 1 >= 1 = ok(nats()) proper(zeros()) = 4 >= 4 = ok(zeros()) proper(0()) = 2 >= 2 = ok(0()) proper(head(X)) = X >= X = head(proper(X)) proper(tail(X)) = 4X >= 4X = tail(proper(X)) incr(ok(X)) = 4X >= 4X = 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)) = 2X >= 2X = ok(adx(X)) head(ok(X)) = X >= X = ok(head(X)) tail(ok(X)) = 4X >= 4X = ok(tail(X)) top(mark(X)) = X + 1 >= X + 1 = top(proper(X)) top(ok(X)) = X + 1 >= X + 1 = 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: [top](x0) = x0 + 2, [ok](x0) = x0, [proper](x0) = x0, [tail](x0) = 2x0, [head](x0) = x0 + 2, [0] = 0, [zeros] = 0, [nats] = 0, [adx](x0) = x0, [s](x0) = 3x0 + 5, [cons](x0, x1) = x0 + 4x1 + 7, [mark](x0) = x0 + 4, [active](x0) = x0, [incr](x0) = 3x0, [nil] = 7 orientation: active(incr(cons(X,L))) = 12L + 3X + 21 >= 12L + 3X + 16 = mark(cons(s(X),incr(L))) active(incr(X)) = 3X >= 3X = incr(active(X)) active(cons(X1,X2)) = X1 + 4X2 + 7 >= X1 + 4X2 + 7 = cons(active(X1),X2) active(s(X)) = 3X + 5 >= 3X + 5 = s(active(X)) active(adx(X)) = X >= X = adx(active(X)) active(head(X)) = X + 2 >= X + 2 = head(active(X)) active(tail(X)) = 2X >= 2X = tail(active(X)) incr(mark(X)) = 3X + 12 >= 3X + 4 = mark(incr(X)) cons(mark(X1),X2) = X1 + 4X2 + 11 >= X1 + 4X2 + 11 = mark(cons(X1,X2)) s(mark(X)) = 3X + 17 >= 3X + 9 = mark(s(X)) adx(mark(X)) = X + 4 >= X + 4 = mark(adx(X)) head(mark(X)) = X + 6 >= X + 6 = mark(head(X)) tail(mark(X)) = 2X + 8 >= 2X + 4 = mark(tail(X)) proper(incr(X)) = 3X >= 3X = incr(proper(X)) proper(nil()) = 7 >= 7 = ok(nil()) proper(cons(X1,X2)) = X1 + 4X2 + 7 >= X1 + 4X2 + 7 = cons(proper(X1),proper(X2)) proper(s(X)) = 3X + 5 >= 3X + 5 = 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)) = 2X >= 2X = tail(proper(X)) incr(ok(X)) = 3X >= 3X = ok(incr(X)) cons(ok(X1),ok(X2)) = X1 + 4X2 + 7 >= X1 + 4X2 + 7 = ok(cons(X1,X2)) s(ok(X)) = 3X + 5 >= 3X + 5 = ok(s(X)) adx(ok(X)) = X >= X = ok(adx(X)) head(ok(X)) = X + 2 >= X + 2 = ok(head(X)) tail(ok(X)) = 2X >= 2X = ok(tail(X)) top(mark(X)) = X + 6 >= X + 2 = top(proper(X)) top(ok(X)) = X + 2 >= X + 2 = 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)) cons(mark(X1),X2) -> mark(cons(X1,X2)) adx(mark(X)) -> mark(adx(X)) head(mark(X)) -> mark(head(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: [top](x0) = 2x0 + 7, [ok](x0) = 2x0 + 1, [proper](x0) = 6x0 + 1, [tail](x0) = 5x0 + 3, [head](x0) = x0, [0] = 0, [zeros] = 0, [nats] = 0, [adx](x0) = 4x0 + 2, [s](x0) = x0, [cons](x0, x1) = 2x0 + 2x1 + 1, [mark](x0) = x0 + 2, [active](x0) = 2x0, [incr](x0) = x0, [nil] = 1 orientation: active(incr(X)) = 2X >= 2X = incr(active(X)) active(cons(X1,X2)) = 4X1 + 4X2 + 2 >= 4X1 + 2X2 + 1 = cons(active(X1),X2) active(s(X)) = 2X >= 2X = s(active(X)) active(adx(X)) = 8X + 4 >= 8X + 2 = adx(active(X)) active(head(X)) = 2X >= 2X = head(active(X)) active(tail(X)) = 10X + 6 >= 10X + 3 = tail(active(X)) cons(mark(X1),X2) = 2X1 + 2X2 + 5 >= 2X1 + 2X2 + 3 = mark(cons(X1,X2)) adx(mark(X)) = 4X + 10 >= 4X + 4 = mark(adx(X)) head(mark(X)) = X + 2 >= X + 2 = mark(head(X)) proper(incr(X)) = 6X + 1 >= 6X + 1 = incr(proper(X)) proper(nil()) = 7 >= 3 = ok(nil()) proper(cons(X1,X2)) = 12X1 + 12X2 + 7 >= 12X1 + 12X2 + 5 = cons(proper(X1),proper(X2)) proper(s(X)) = 6X + 1 >= 6X + 1 = s(proper(X)) proper(adx(X)) = 24X + 13 >= 24X + 6 = adx(proper(X)) proper(nats()) = 1 >= 1 = ok(nats()) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(0()) = 1 >= 1 = ok(0()) proper(head(X)) = 6X + 1 >= 6X + 1 = head(proper(X)) proper(tail(X)) = 30X + 19 >= 30X + 8 = tail(proper(X)) incr(ok(X)) = 2X + 1 >= 2X + 1 = ok(incr(X)) cons(ok(X1),ok(X2)) = 4X1 + 4X2 + 5 >= 4X1 + 4X2 + 3 = ok(cons(X1,X2)) s(ok(X)) = 2X + 1 >= 2X + 1 = ok(s(X)) adx(ok(X)) = 8X + 6 >= 8X + 5 = ok(adx(X)) head(ok(X)) = 2X + 1 >= 2X + 1 = ok(head(X)) tail(ok(X)) = 10X + 8 >= 10X + 7 = ok(tail(X)) top(ok(X)) = 4X + 9 >= 4X + 7 = top(active(X)) problem: active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) head(mark(X)) -> mark(head(X)) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) Matrix Interpretation Processor: dim=3 interpretation: [ok](x0) = x0 , [1 0 0] [proper](x0) = [0 1 0]x0 [0 0 0] , [1 0 1] [0] [head](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [0] [0] = [0] [0], [0] [zeros] = [0] [0], [0] [nats] = [0] [0], [1 0 0] [1] [s](x0) = [0 1 0]x0 + [0] [0 0 0] [0], [1 0 1] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [0] [active](x0) = [1 1 0]x0 + [0] [0 0 0] [1], [1 0 0] [0] [incr](x0) = [0 1 0]x0 + [1] [0 0 0] [0] orientation: [1 1 0] [1] [1 1 0] [0] active(incr(X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = incr(active(X)) [0 0 0] [1] [0 0 0] [0] [1 1 0] [1] [1 1 0] [1] active(s(X)) = [1 1 0]X + [1] >= [1 1 0]X + [0] = s(active(X)) [0 0 0] [1] [0 0 0] [0] [1 1 1] [1] [1 1 0] [1] active(head(X)) = [1 1 1]X + [1] >= [1 1 0]X + [1] = head(active(X)) [0 0 0] [1] [0 0 0] [0] [1 0 1] [0] [1 0 1] head(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X = mark(head(X)) [0 0 0] [0] [0 0 0] [1 0 0] [0] [1 0 0] [0] proper(incr(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = incr(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] proper(s(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = s(proper(X)) [0 0 0] [0] [0 0 0] [0] [0] [0] proper(nats()) = [0] >= [0] = ok(nats()) [0] [0] [0] [0] proper(zeros()) = [0] >= [0] = ok(zeros()) [0] [0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 0 1] [0] [1 0 0] [0] proper(head(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = head(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] incr(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(incr(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] s(ok(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = ok(s(X)) [0 0 0] [0] [0 0 0] [0] [1 0 1] [0] [1 0 1] [0] head(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(head(X)) [0 0 0] [0] [0 0 0] [0] problem: active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) head(mark(X)) -> mark(head(X)) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) Matrix Interpretation Processor: dim=1 interpretation: [ok](x0) = 5x0 + 7, [proper](x0) = 6x0 + 5, [head](x0) = 4x0 + 3, [0] = 2, [zeros] = 2, [nats] = 2, [s](x0) = x0, [mark](x0) = x0, [active](x0) = x0, [incr](x0) = x0 orientation: active(s(X)) = X >= X = s(active(X)) active(head(X)) = 4X + 3 >= 4X + 3 = head(active(X)) head(mark(X)) = 4X + 3 >= 4X + 3 = mark(head(X)) proper(incr(X)) = 6X + 5 >= 6X + 5 = incr(proper(X)) proper(s(X)) = 6X + 5 >= 6X + 5 = s(proper(X)) proper(nats()) = 17 >= 17 = ok(nats()) proper(zeros()) = 17 >= 17 = ok(zeros()) proper(0()) = 17 >= 17 = ok(0()) proper(head(X)) = 24X + 23 >= 24X + 23 = head(proper(X)) incr(ok(X)) = 5X + 7 >= 5X + 7 = ok(incr(X)) s(ok(X)) = 5X + 7 >= 5X + 7 = ok(s(X)) head(ok(X)) = 20X + 31 >= 20X + 22 = ok(head(X)) problem: active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) head(mark(X)) -> mark(head(X)) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) Matrix Interpretation Processor: dim=3 interpretation: [0] [ok](x0) = x0 + [1] [0], [1 0 1] [0] [proper](x0) = [0 0 0]x0 + [1] [0 0 1] [0], [1 1 0] [0] [head](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [0] [0] = [0] [0], [0] [zeros] = [0] [0], [0] [nats] = [0] [0], [1 1 0] [0] [s](x0) = [0 1 0]x0 + [0] [0 0 1] [1], [1] [mark](x0) = x0 + [0] [0], [1 0 1] [active](x0) = [0 0 0]x0 [1 0 1] , [1 1 0] [0] [incr](x0) = [0 1 0]x0 + [0] [0 0 1] [1] orientation: [1 1 1] [1] [1 0 1] [0] active(s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(active(X)) [1 1 1] [1] [1 0 1] [1] [1 1 1] [1] [1 0 1] [0] active(head(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = head(active(X)) [1 1 1] [1] [1 0 1] [1] [1 1 0] [1] [1 1 0] [1] head(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(head(X)) [0 0 1] [1] [0 0 1] [1] [1 1 1] [1] [1 0 1] [1] proper(incr(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = incr(proper(X)) [0 0 1] [1] [0 0 1] [1] [1 1 1] [1] [1 0 1] [1] proper(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(proper(X)) [0 0 1] [1] [0 0 1] [1] [0] [0] proper(nats()) = [1] >= [1] = ok(nats()) [0] [0] [0] [0] proper(zeros()) = [1] >= [1] = ok(zeros()) [0] [0] [0] [0] proper(0()) = [1] >= [1] = ok(0()) [0] [0] [1 1 1] [1] [1 0 1] [1] proper(head(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = head(proper(X)) [0 0 1] [1] [0 0 1] [1] [1 1 0] [1] [1 1 0] [0] incr(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(incr(X)) [0 0 1] [1] [0 0 1] [1] [1 1 0] [1] [1 1 0] [0] s(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(s(X)) [0 0 1] [1] [0 0 1] [1] problem: head(mark(X)) -> mark(head(X)) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) Matrix Interpretation Processor: dim=1 interpretation: [ok](x0) = x0 + 4, [proper](x0) = 4x0 + 1, [head](x0) = 4x0 + 1, [0] = 4, [zeros] = 1, [nats] = 1, [s](x0) = x0, [mark](x0) = x0 + 2, [incr](x0) = 4x0 + 1 orientation: head(mark(X)) = 4X + 9 >= 4X + 3 = mark(head(X)) proper(incr(X)) = 16X + 5 >= 16X + 5 = incr(proper(X)) proper(s(X)) = 4X + 1 >= 4X + 1 = s(proper(X)) proper(nats()) = 5 >= 5 = ok(nats()) proper(zeros()) = 5 >= 5 = ok(zeros()) proper(0()) = 17 >= 8 = ok(0()) proper(head(X)) = 16X + 5 >= 16X + 5 = head(proper(X)) problem: proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(head(X)) -> head(proper(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [ok](x0) = [0 0 0]x0 [0 0 1] , [1 1 1] [proper](x0) = [0 1 0]x0 [0 1 1] , [head](x0) = x0 , [0] [zeros] = [0] [0], [0] [nats] = [0] [1], [0] [s](x0) = x0 + [1] [0], [0] [incr](x0) = x0 + [0] [1] orientation: [1 1 1] [1] [1 1 1] [0] proper(incr(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = incr(proper(X)) [0 1 1] [1] [0 1 1] [1] [1 1 1] [1] [1 1 1] [0] proper(s(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = s(proper(X)) [0 1 1] [1] [0 1 1] [0] [1] [1] proper(nats()) = [0] >= [0] = ok(nats()) [1] [1] [0] [0] proper(zeros()) = [0] >= [0] = ok(zeros()) [0] [0] [1 1 1] [1 1 1] proper(head(X)) = [0 1 0]X >= [0 1 0]X = head(proper(X)) [0 1 1] [0 1 1] problem: proper(nats()) -> ok(nats()) proper(zeros()) -> ok(zeros()) proper(head(X)) -> head(proper(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [ok](x0) = [0 0 0]x0 [0 1 0] , [1 1 0] [proper](x0) = [1 0 1]x0 [0 1 0] , [0] [head](x0) = x0 + [1] [1], [0] [zeros] = [1] [0], [0] [nats] = [1] [0] orientation: [1] [0] proper(nats()) = [0] >= [0] = ok(nats()) [1] [1] [1] [0] proper(zeros()) = [0] >= [0] = ok(zeros()) [1] [1] [1 1 0] [1] [1 1 0] [0] proper(head(X)) = [1 0 1]X + [1] >= [1 0 1]X + [1] = head(proper(X)) [0 1 0] [1] [0 1 0] [1] problem: Qed