/export/starexec/sandbox2/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem: active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = 4x0, [ok](x0) = x0, [proper](x0) = x0, [tl](x0) = x0, [hd](x0) = x0, [s](x0) = x0, [incr](x0) = x0, [cons](x0, x1) = x0 + 4x1, [0] = 0, [mark](x0) = x0, [adx](x0) = x0, [zeros] = 0, [active](x0) = x0, [nats] = 2 orientation: active(nats()) = 2 >= 0 = mark(adx(zeros())) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(incr(cons(X,Y))) = X + 4Y >= X + 4Y = mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) = X + 4Y >= X + 4Y = mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) = X + 4Y >= X = mark(X) active(tl(cons(X,Y))) = X + 4Y >= Y = mark(Y) active(adx(X)) = X >= X = adx(active(X)) active(incr(X)) = X >= X = incr(active(X)) active(hd(X)) = X >= X = hd(active(X)) active(tl(X)) = X >= X = tl(active(X)) adx(mark(X)) = X >= X = mark(adx(X)) incr(mark(X)) = X >= X = mark(incr(X)) hd(mark(X)) = X >= X = mark(hd(X)) tl(mark(X)) = X >= X = mark(tl(X)) proper(nats()) = 2 >= 2 = ok(nats()) proper(adx(X)) = X >= X = adx(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(incr(X)) = X >= X = incr(proper(X)) proper(s(X)) = X >= X = s(proper(X)) proper(hd(X)) = X >= X = hd(proper(X)) proper(tl(X)) = X >= X = tl(proper(X)) adx(ok(X)) = X >= X = ok(adx(X)) cons(ok(X1),ok(X2)) = X1 + 4X2 >= X1 + 4X2 = ok(cons(X1,X2)) incr(ok(X)) = X >= X = ok(incr(X)) s(ok(X)) = X >= X = ok(s(X)) hd(ok(X)) = X >= X = ok(hd(X)) tl(ok(X)) = X >= X = ok(tl(X)) top(mark(X)) = 4X >= 4X = top(proper(X)) top(ok(X)) = 4X >= 4X = top(active(X)) problem: active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [top](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 1 1] [ok](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [proper](x0) = [0 1 0]x0 [0 0 0] , [1 1 0] [tl](x0) = [0 0 0]x0 [0 0 1] , [1] [hd](x0) = x0 + [0] [0], [1 1 0] [s](x0) = [0 0 0]x0 [0 0 0] , [incr](x0) = x0 , [1 1 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [0] [0] = [0] [0], [1 1 0] [0] [mark](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [adx](x0) = x0 , [0] [zeros] = [0] [0], [1 1 0] [0] [active](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1] [nats] = [0] [0] orientation: [0] [0] active(zeros()) = [0] >= [0] = mark(cons(0(),zeros())) [1] [1] [1 1 0] [1 1 0] [0] [1 1 0] [1 1 0] [0] active(incr(cons(X,Y))) = [0 0 0]X + [0 0 0]Y + [0] >= [0 0 0]X + [0 0 0]Y + [0] = mark(cons(s(X),incr(Y))) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 0] [1 1 0] [0] [1 1 0] [1 1 0] [0] active(adx(cons(X,Y))) = [0 0 0]X + [0 0 0]Y + [0] >= [0 0 0]X + [0 0 0]Y + [0] = mark(incr(cons(X,adx(Y)))) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 0] [1 1 0] [1] [1 1 0] [0] active(hd(cons(X,Y))) = [0 0 0]X + [0 0 0]Y + [0] >= [0 0 0]X + [0] = mark(X) [0 0 0] [0 0 0] [1] [0 0 0] [1] [1 1 0] [1 1 0] [0] [1 1 0] [0] active(tl(cons(X,Y))) = [0 0 0]X + [0 0 0]Y + [0] >= [0 0 0]Y + [0] = mark(Y) [0 0 0] [0 0 0] [1] [0 0 0] [1] [1 1 0] [0] [1 1 0] [0] active(adx(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = adx(active(X)) [0 0 0] [1] [0 0 0] [1] [1 1 0] [0] [1 1 0] [0] active(incr(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = incr(active(X)) [0 0 0] [1] [0 0 0] [1] [1 1 0] [1] [1 1 0] [1] active(hd(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = hd(active(X)) [0 0 0] [1] [0 0 0] [1] [1 1 0] [0] [1 1 0] [0] active(tl(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = tl(active(X)) [0 0 0] [1] [0 0 0] [1] [1 1 0] [0] [1 1 0] [0] adx(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(adx(X)) [0 0 0] [1] [0 0 0] [1] [1 1 0] [0] [1 1 0] [0] incr(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(incr(X)) [0 0 0] [1] [0 0 0] [1] [1 1 0] [1] [1 1 0] [1] hd(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(hd(X)) [0 0 0] [1] [0 0 0] [1] [1 1 0] [0] [1 1 0] [0] tl(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(tl(X)) [0 0 0] [1] [0 0 0] [1] [1] [1] proper(nats()) = [0] >= [0] = ok(nats()) [0] [0] [1 0 0] [1 0 0] proper(adx(X)) = [0 1 0]X >= [0 1 0]X = adx(proper(X)) [0 0 0] [0 0 0] [0] [0] proper(zeros()) = [0] >= [0] = ok(zeros()) [0] [0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] proper(cons(X1,X2)) = [0 0 0]X1 + [0 1 0]X2 >= [0 0 0]X1 + [0 1 0]X2 = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 0 0] [1 0 0] proper(incr(X)) = [0 1 0]X >= [0 1 0]X = incr(proper(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 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(hd(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = hd(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 1 0] proper(tl(X)) = [0 0 0]X >= [0 0 0]X = tl(proper(X)) [0 0 0] [0 0 0] [1 1 1] [1 1 1] adx(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(adx(X)) [0 0 0] [0 0 0] [1 1 1] [1 1 1] [1 1 0] [1 1 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 1 1] [1 1 1] incr(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(incr(X)) [0 0 0] [0 0 0] [1 1 1] [1 1 0] s(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(s(X)) [0 0 0] [0 0 0] [1 1 1] [1] [1 1 1] [1] hd(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(hd(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1 1 1] tl(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(tl(X)) [0 0 0] [0 0 0] [1 1 0] [0] [1 0 0] [0] top(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = top(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 1 1] [0] [1 1 0] [0] top(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = top(active(X)) [0 0 0] [1] [0 0 0] [1] problem: active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(tl(cons(X,Y))) -> mark(Y) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = 2x0, [ok](x0) = x0, [proper](x0) = x0, [tl](x0) = 6x0 + 4, [hd](x0) = 4x0, [s](x0) = x0, [incr](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [mark](x0) = x0, [adx](x0) = 4x0, [zeros] = 0, [active](x0) = x0, [nats] = 0 orientation: active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(incr(cons(X,Y))) = X + Y >= X + Y = mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) = 4X + 4Y >= X + 4Y = mark(incr(cons(X,adx(Y)))) active(tl(cons(X,Y))) = 6X + 6Y + 4 >= Y = mark(Y) active(adx(X)) = 4X >= 4X = adx(active(X)) active(incr(X)) = X >= X = incr(active(X)) active(hd(X)) = 4X >= 4X = hd(active(X)) active(tl(X)) = 6X + 4 >= 6X + 4 = tl(active(X)) adx(mark(X)) = 4X >= 4X = mark(adx(X)) incr(mark(X)) = X >= X = mark(incr(X)) hd(mark(X)) = 4X >= 4X = mark(hd(X)) tl(mark(X)) = 6X + 4 >= 6X + 4 = mark(tl(X)) proper(nats()) = 0 >= 0 = ok(nats()) proper(adx(X)) = 4X >= 4X = adx(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(incr(X)) = X >= X = incr(proper(X)) proper(s(X)) = X >= X = s(proper(X)) proper(hd(X)) = 4X >= 4X = hd(proper(X)) proper(tl(X)) = 6X + 4 >= 6X + 4 = tl(proper(X)) adx(ok(X)) = 4X >= 4X = ok(adx(X)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) incr(ok(X)) = X >= X = ok(incr(X)) s(ok(X)) = X >= X = ok(s(X)) hd(ok(X)) = 4X >= 4X = ok(hd(X)) tl(ok(X)) = 6X + 4 >= 6X + 4 = ok(tl(X)) top(mark(X)) = 2X >= 2X = top(proper(X)) top(ok(X)) = 2X >= 2X = top(active(X)) problem: active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [top](x0) = [1 0 1]x0 [0 0 0] , [1 1 0] [ok](x0) = [0 1 0]x0 [0 0 1] , [1 0 0] [proper](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [tl](x0) = [0 0 0]x0 [0 1 1] , [1 0 0] [hd](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [0] = [0] [1], [mark](x0) = x0 , [1 0 0] [adx](x0) = [0 0 0]x0 [0 0 1] , [0] [zeros] = [0] [1], [1 0 1] [active](x0) = [0 0 0]x0 [0 0 0] , [0] [nats] = [0] [0] orientation: [1] [0] active(zeros()) = [0] >= [0] = mark(cons(0(),zeros())) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(incr(cons(X,Y))) = [0 0 0]X + [0 0 0]Y >= [0 0 0]X + [0 0 0]Y = mark(cons(s(X),incr(Y))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(adx(cons(X,Y))) = [0 0 0]X + [0 0 0]Y >= [0 0 0]X + [0 0 0]Y = mark(incr(cons(X,adx(Y)))) [0 0 0] [0 0 0] [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(incr(X)) = [0 0 0]X >= [0 0 0]X = incr(active(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] active(hd(X)) = [0 0 0]X >= [0 0 0]X = hd(active(X)) [0 0 0] [0 0 0] [1 1 1] [1 0 1] active(tl(X)) = [0 0 0]X >= [0 0 0]X = tl(active(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] adx(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(adx(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] incr(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(incr(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] hd(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(hd(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] tl(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(tl(X)) [0 1 1] [0 1 1] [0] [0] proper(nats()) = [0] >= [0] = ok(nats()) [0] [0] [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(zeros()) = [0] >= [0] = ok(zeros()) [1] [1] [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] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [1] [1] [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] [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(hd(X)) = [0 0 0]X >= [0 0 0]X = hd(proper(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] proper(tl(X)) = [0 0 0]X >= [0 0 0]X = tl(proper(X)) [0 1 1] [0 0 1] [1 1 0] [1 0 0] adx(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(adx(X)) [0 0 1] [0 0 1] [1 1 0] [1 1 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 1 0] [1 0 0] incr(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(incr(X)) [0 0 1] [0 0 1] [1 1 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 1 0] [1 0 0] hd(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(hd(X)) [0 0 1] [0 0 1] [1 1 0] [1 0 0] tl(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(tl(X)) [0 1 1] [0 1 1] [1 0 1] [1 0 1] top(mark(X)) = [1 0 1]X >= [1 0 1]X = top(proper(X)) [0 0 0] [0 0 0] [1 1 1] [1 0 1] top(ok(X)) = [1 1 1]X >= [1 0 1]X = top(active(X)) [0 0 0] [0 0 0] problem: active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = x0, [ok](x0) = x0, [proper](x0) = x0, [tl](x0) = 4x0, [hd](x0) = 4x0, [s](x0) = x0, [incr](x0) = x0, [cons](x0, x1) = x0 + x1 + 1, [0] = 0, [mark](x0) = x0, [adx](x0) = 2x0, [zeros] = 0, [active](x0) = x0, [nats] = 0 orientation: active(incr(cons(X,Y))) = X + Y + 1 >= X + Y + 1 = mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) = 2X + 2Y + 2 >= X + 2Y + 1 = mark(incr(cons(X,adx(Y)))) active(adx(X)) = 2X >= 2X = adx(active(X)) active(incr(X)) = X >= X = incr(active(X)) active(hd(X)) = 4X >= 4X = hd(active(X)) active(tl(X)) = 4X >= 4X = tl(active(X)) adx(mark(X)) = 2X >= 2X = mark(adx(X)) incr(mark(X)) = X >= X = mark(incr(X)) hd(mark(X)) = 4X >= 4X = mark(hd(X)) tl(mark(X)) = 4X >= 4X = mark(tl(X)) proper(nats()) = 0 >= 0 = ok(nats()) proper(adx(X)) = 2X >= 2X = adx(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(incr(X)) = X >= X = incr(proper(X)) proper(s(X)) = X >= X = s(proper(X)) proper(hd(X)) = 4X >= 4X = hd(proper(X)) proper(tl(X)) = 4X >= 4X = tl(proper(X)) adx(ok(X)) = 2X >= 2X = ok(adx(X)) cons(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(cons(X1,X2)) incr(ok(X)) = X >= X = ok(incr(X)) s(ok(X)) = X >= X = ok(s(X)) hd(ok(X)) = 4X >= 4X = ok(hd(X)) tl(ok(X)) = 4X >= 4X = ok(tl(X)) top(mark(X)) = X >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(X)) problem: active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(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 0] [ok](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [proper](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [tl](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [0] [hd](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [incr](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [1 0 0] [0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 1] [1], [0] [0] = [0] [0], [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [adx](x0) = [0 0 0]x0 [0 0 0] , [0] [zeros] = [0] [0], [1 0 0] [0] [active](x0) = [0 0 0]x0 + [1] [0 0 1] [0], [1] [nats] = [0] [1] orientation: [1 0 0] [1 0 1] [1] [1 0 0] [1 0 1] [0] active(incr(cons(X,Y))) = [0 0 0]X + [0 0 0]Y + [1] >= [0 0 0]X + [0 0 0]Y + [0] = mark(cons(s(X),incr(Y))) [0 0 0] [0 0 1] [1] [0 0 0] [0 0 1] [1] [1 0 0] [0] [1 0 0] active(adx(X)) = [0 0 0]X + [1] >= [0 0 0]X = adx(active(X)) [0 0 0] [0] [0 0 0] [1 0 1] [0] [1 0 1] active(incr(X)) = [0 0 0]X + [1] >= [0 0 0]X = incr(active(X)) [0 0 1] [0] [0 0 1] [1 0 0] [0] [1 0 0] [0] active(hd(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = hd(active(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] active(tl(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = tl(active(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] 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] incr(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(incr(X)) [0 0 1] [0 0 1] [1 0 0] [0] [1 0 0] [0] hd(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(hd(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] tl(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(tl(X)) [0 0 0] [0 0 0] [1] [1] proper(nats()) = [0] >= [0] = ok(nats()) [1] [1] [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(zeros()) = [0] >= [0] = ok(zeros()) [0] [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 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = cons(proper(X1),proper(X2)) [0 0 0] [0 0 1] [1] [0 0 0] [0 0 1] [1] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 0 1] [1 0 1] proper(incr(X)) = [0 0 0]X >= [0 0 0]X = incr(proper(X)) [0 0 1] [0 0 1] [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] [0] [1 0 0] [0] proper(hd(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = hd(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] proper(tl(X)) = [0 0 0]X >= [0 0 0]X = tl(proper(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] [0] [1 0 0] [1 0 0] [0] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = ok(cons(X1,X2)) [0 0 0] [0 0 1] [1] [0 0 0] [0 0 1] [1] [1 0 1] [1 0 1] incr(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(incr(X)) [0 0 1] [0 0 1] [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] [0] hd(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(hd(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] tl(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(tl(X)) [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 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(adx(X)) -> adx(active(X)) active(incr(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(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 1 0] [0] [ok](x0) = [0 1 0]x0 + [0] [0 0 1] [1], [1 1 0] [0] [proper](x0) = [0 1 0]x0 + [0] [0 0 1] [1], [tl](x0) = x0 , [1 0 0] [hd](x0) = [0 1 0]x0 [0 1 1] , [s](x0) = x0 , [incr](x0) = x0 , [1 0 0] [1] [cons](x0, x1) = x0 + [0 1 0]x1 + [0] [0 0 0] [0], [0] [0] = [0] [0], [1 1 0] [0] [mark](x0) = [0 1 0]x0 + [0] [0 0 1] [1], [1 0 1] [0] [adx](x0) = [0 1 0]x0 + [1] [0 0 1] [0], [0] [zeros] = [0] [0], [1 1 0] [active](x0) = [0 1 0]x0 [0 0 1] , [0] [nats] = [0] [0] orientation: [1 1 1] [1] [1 1 1] [0] active(adx(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = adx(active(X)) [0 0 1] [0] [0 0 1] [0] [1 1 0] [1 1 0] active(incr(X)) = [0 1 0]X >= [0 1 0]X = incr(active(X)) [0 0 1] [0 0 1] [1 1 0] [1 1 0] active(hd(X)) = [0 1 0]X >= [0 1 0]X = hd(active(X)) [0 1 1] [0 1 1] [1 1 0] [1 1 0] active(tl(X)) = [0 1 0]X >= [0 1 0]X = tl(active(X)) [0 0 1] [0 0 1] [1 1 1] [1] [1 1 1] [1] adx(mark(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = mark(adx(X)) [0 0 1] [1] [0 0 1] [1] [1 1 0] [0] [1 1 0] [0] incr(mark(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = mark(incr(X)) [0 0 1] [1] [0 0 1] [1] [1 1 0] [0] [1 1 0] [0] hd(mark(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = mark(hd(X)) [0 1 1] [1] [0 1 1] [1] [1 1 0] [0] [1 1 0] [0] tl(mark(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = mark(tl(X)) [0 0 1] [1] [0 0 1] [1] [0] [0] proper(nats()) = [0] >= [0] = ok(nats()) [1] [1] [1 1 1] [1] [1 1 1] [1] proper(adx(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = adx(proper(X)) [0 0 1] [1] [0 0 1] [1] [0] [0] proper(zeros()) = [0] >= [0] = ok(zeros()) [1] [1] [1 1 0] [1 1 0] [1] [1 1 0] [1 1 0] [1] proper(cons(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0] = cons(proper(X1),proper(X2)) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [1] [1] [1 1 0] [0] [1 1 0] [0] proper(incr(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = incr(proper(X)) [0 0 1] [1] [0 0 1] [1] [1 1 0] [0] [1 1 0] [0] proper(s(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = s(proper(X)) [0 0 1] [1] [0 0 1] [1] [1 1 0] [0] [1 1 0] [0] proper(hd(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = hd(proper(X)) [0 1 1] [1] [0 1 1] [1] [1 1 0] [0] [1 1 0] [0] proper(tl(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = tl(proper(X)) [0 0 1] [1] [0 0 1] [1] [1 1 1] [1] [1 1 1] [1] adx(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(adx(X)) [0 0 1] [1] [0 0 1] [1] [1 1 0] [1 1 0] [1] [1 1 0] [1 1 0] [1] cons(ok(X1),ok(X2)) = [0 1 0]X1 + [0 1 0]X2 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0] = ok(cons(X1,X2)) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 1 0] [0] [1 1 0] [0] incr(ok(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = ok(incr(X)) [0 0 1] [1] [0 0 1] [1] [1 1 0] [0] [1 1 0] [0] s(ok(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = ok(s(X)) [0 0 1] [1] [0 0 1] [1] [1 1 0] [0] [1 1 0] [0] hd(ok(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = ok(hd(X)) [0 1 1] [1] [0 1 1] [1] [1 1 0] [0] [1 1 0] [0] tl(ok(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = ok(tl(X)) [0 0 1] [1] [0 0 1] [1] [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(X)) -> incr(active(X)) active(hd(X)) -> hd(active(X)) active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) hd(ok(X)) -> ok(hd(X)) tl(ok(X)) -> ok(tl(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = 2x0 + 4, [ok](x0) = 5x0 + 7, [proper](x0) = 6x0 + 5, [tl](x0) = x0, [hd](x0) = 3x0 + 2, [s](x0) = 4x0 + 3, [incr](x0) = 4x0 + 3, [cons](x0, x1) = 2x0 + x1 + 2, [0] = 2, [mark](x0) = 6x0 + 5, [adx](x0) = x0, [zeros] = 2, [active](x0) = 4x0 + 2, [nats] = 2 orientation: active(incr(X)) = 16X + 14 >= 16X + 11 = incr(active(X)) active(hd(X)) = 12X + 10 >= 12X + 8 = hd(active(X)) active(tl(X)) = 4X + 2 >= 4X + 2 = tl(active(X)) adx(mark(X)) = 6X + 5 >= 6X + 5 = mark(adx(X)) incr(mark(X)) = 24X + 23 >= 24X + 23 = mark(incr(X)) hd(mark(X)) = 18X + 17 >= 18X + 17 = mark(hd(X)) tl(mark(X)) = 6X + 5 >= 6X + 5 = mark(tl(X)) proper(nats()) = 17 >= 17 = ok(nats()) proper(adx(X)) = 6X + 5 >= 6X + 5 = adx(proper(X)) proper(zeros()) = 17 >= 17 = ok(zeros()) proper(cons(X1,X2)) = 12X1 + 6X2 + 17 >= 12X1 + 6X2 + 17 = cons(proper(X1),proper(X2)) proper(0()) = 17 >= 17 = ok(0()) proper(incr(X)) = 24X + 23 >= 24X + 23 = incr(proper(X)) proper(s(X)) = 24X + 23 >= 24X + 23 = s(proper(X)) proper(hd(X)) = 18X + 17 >= 18X + 17 = hd(proper(X)) proper(tl(X)) = 6X + 5 >= 6X + 5 = tl(proper(X)) adx(ok(X)) = 5X + 7 >= 5X + 7 = ok(adx(X)) cons(ok(X1),ok(X2)) = 10X1 + 5X2 + 23 >= 10X1 + 5X2 + 17 = ok(cons(X1,X2)) incr(ok(X)) = 20X + 31 >= 20X + 22 = ok(incr(X)) s(ok(X)) = 20X + 31 >= 20X + 22 = ok(s(X)) hd(ok(X)) = 15X + 23 >= 15X + 17 = ok(hd(X)) tl(ok(X)) = 5X + 7 >= 5X + 7 = ok(tl(X)) top(mark(X)) = 12X + 14 >= 12X + 14 = top(proper(X)) top(ok(X)) = 10X + 18 >= 8X + 8 = top(active(X)) problem: active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) incr(mark(X)) -> mark(incr(X)) hd(mark(X)) -> mark(hd(X)) tl(mark(X)) -> mark(tl(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) tl(ok(X)) -> ok(tl(X)) top(mark(X)) -> top(proper(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = 2x0 + 5, [ok](x0) = 4x0, [proper](x0) = 4x0, [tl](x0) = 4x0, [hd](x0) = 2x0, [s](x0) = 2x0, [incr](x0) = 2x0, [cons](x0, x1) = 4x0 + x1, [0] = 0, [mark](x0) = 4x0 + 2, [adx](x0) = x0, [zeros] = 0, [active](x0) = 4x0, [nats] = 4 orientation: active(tl(X)) = 16X >= 16X = tl(active(X)) adx(mark(X)) = 4X + 2 >= 4X + 2 = mark(adx(X)) incr(mark(X)) = 8X + 4 >= 8X + 2 = mark(incr(X)) hd(mark(X)) = 8X + 4 >= 8X + 2 = mark(hd(X)) tl(mark(X)) = 16X + 8 >= 16X + 2 = mark(tl(X)) proper(nats()) = 16 >= 16 = ok(nats()) proper(adx(X)) = 4X >= 4X = adx(proper(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 16X1 + 4X2 >= 16X1 + 4X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(incr(X)) = 8X >= 8X = incr(proper(X)) proper(s(X)) = 8X >= 8X = s(proper(X)) proper(hd(X)) = 8X >= 8X = hd(proper(X)) proper(tl(X)) = 16X >= 16X = tl(proper(X)) adx(ok(X)) = 4X >= 4X = ok(adx(X)) tl(ok(X)) = 16X >= 16X = ok(tl(X)) top(mark(X)) = 8X + 9 >= 8X + 5 = top(proper(X)) problem: active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) tl(ok(X)) -> ok(tl(X)) Matrix Interpretation Processor: dim=1 interpretation: [ok](x0) = 2x0 + 2, [proper](x0) = 3x0 + 2, [tl](x0) = 2x0 + 1, [hd](x0) = x0, [s](x0) = 6x0 + 5, [incr](x0) = x0, [cons](x0, x1) = x0 + x1 + 1, [0] = 0, [mark](x0) = x0, [adx](x0) = x0, [zeros] = 0, [active](x0) = x0, [nats] = 0 orientation: active(tl(X)) = 2X + 1 >= 2X + 1 = tl(active(X)) adx(mark(X)) = X >= X = mark(adx(X)) proper(nats()) = 2 >= 2 = ok(nats()) proper(adx(X)) = 3X + 2 >= 3X + 2 = adx(proper(X)) proper(zeros()) = 2 >= 2 = ok(zeros()) proper(cons(X1,X2)) = 3X1 + 3X2 + 5 >= 3X1 + 3X2 + 5 = cons(proper(X1),proper(X2)) proper(0()) = 2 >= 2 = ok(0()) proper(incr(X)) = 3X + 2 >= 3X + 2 = incr(proper(X)) proper(s(X)) = 18X + 17 >= 18X + 17 = s(proper(X)) proper(hd(X)) = 3X + 2 >= 3X + 2 = hd(proper(X)) proper(tl(X)) = 6X + 5 >= 6X + 5 = tl(proper(X)) adx(ok(X)) = 2X + 2 >= 2X + 2 = ok(adx(X)) tl(ok(X)) = 4X + 5 >= 4X + 4 = ok(tl(X)) problem: active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) adx(ok(X)) -> ok(adx(X)) Matrix Interpretation Processor: dim=1 interpretation: [ok](x0) = 2x0 + 2, [proper](x0) = 3x0 + 2, [tl](x0) = 4x0 + 3, [hd](x0) = x0, [s](x0) = 5x0 + 4, [incr](x0) = x0, [cons](x0, x1) = 4x0 + x1 + 4, [0] = 0, [mark](x0) = x0, [adx](x0) = 4x0 + 3, [zeros] = 0, [active](x0) = 4x0 + 3, [nats] = 0 orientation: active(tl(X)) = 16X + 15 >= 16X + 15 = tl(active(X)) adx(mark(X)) = 4X + 3 >= 4X + 3 = mark(adx(X)) proper(nats()) = 2 >= 2 = ok(nats()) proper(adx(X)) = 12X + 11 >= 12X + 11 = adx(proper(X)) proper(zeros()) = 2 >= 2 = ok(zeros()) proper(cons(X1,X2)) = 12X1 + 3X2 + 14 >= 12X1 + 3X2 + 14 = cons(proper(X1),proper(X2)) proper(0()) = 2 >= 2 = ok(0()) proper(incr(X)) = 3X + 2 >= 3X + 2 = incr(proper(X)) proper(s(X)) = 15X + 14 >= 15X + 14 = s(proper(X)) proper(hd(X)) = 3X + 2 >= 3X + 2 = hd(proper(X)) proper(tl(X)) = 12X + 11 >= 12X + 11 = tl(proper(X)) adx(ok(X)) = 8X + 11 >= 8X + 8 = ok(adx(X)) problem: active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) proper(nats()) -> ok(nats()) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [ok](x0) = [0 1 0]x0 [0 0 0] , [1 1 0] [0] [proper](x0) = [1 1 0]x0 + [0] [0 0 0] [1], [1 0 0] [tl](x0) = [0 1 0]x0 [0 0 0] , [1 0 1] [0] [hd](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 0] [s](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [incr](x0) = [0 1 0]x0 [0 0 0] , [1 0 1] [1 0 0] [0] [cons](x0, x1) = [0 1 0]x0 + [0 1 0]x1 + [1] [0 0 0] [0 0 0] [0], [0] [0] = [0] [0], [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 1] , [1 0 1] [0] [adx](x0) = [0 1 0]x0 + [1] [0 0 0] [1], [0] [zeros] = [1] [1], [1 0 0] [0] [active](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [0] [nats] = [1] [0] orientation: [1 0 0] [0] [1 0 0] active(tl(X)) = [0 0 0]X + [0] >= [0 0 0]X = tl(active(X)) [0 0 0] [1] [0 0 0] [1 0 1] [0] [1 0 1] [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] proper(nats()) = [1] >= [1] = ok(nats()) [1] [0] [1 1 1] [1] [1 1 0] [1] proper(adx(X)) = [1 1 1]X + [1] >= [1 1 0]X + [1] = adx(proper(X)) [0 0 0] [1] [0 0 0] [1] [1] [1] proper(zeros()) = [1] >= [1] = ok(zeros()) [1] [0] [1 1 1] [1 1 0] [1] [1 1 0] [1 1 0] [1] proper(cons(X1,X2)) = [1 1 1]X1 + [1 1 0]X2 + [1] >= [1 1 0]X1 + [1 1 0]X2 + [1] = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [1] [0] [1 1 0] [0] [1 1 0] proper(incr(X)) = [1 1 0]X + [0] >= [1 1 0]X = incr(proper(X)) [0 0 0] [1] [0 0 0] [1 1 0] [0] [1 1 0] proper(s(X)) = [1 1 0]X + [0] >= [1 1 0]X = s(proper(X)) [0 0 0] [1] [0 0 0] [1 1 1] [1] [1 1 0] [1] proper(hd(X)) = [1 1 1]X + [1] >= [1 1 0]X + [1] = hd(proper(X)) [0 0 0] [1] [0 0 0] [0] [1 1 0] [0] [1 1 0] proper(tl(X)) = [1 1 0]X + [0] >= [1 1 0]X = tl(proper(X)) [0 0 0] [1] [0 0 0] problem: active(tl(X)) -> tl(active(X)) adx(mark(X)) -> mark(adx(X)) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [ok](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [0] [proper](x0) = [0 0 0]x0 + [1] [0 0 1] [0], [1 1 0] [0] [tl](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 0] [1] [hd](x0) = [0 0 0]x0 + [0] [0 0 1] [0], [1 1 0] [0] [s](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [1 0 0] [1] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 1] [0 0 1] [0], [0] [0] = [0] [0], [1 0 0] [1] [mark](x0) = [0 1 0]x0 + [0] [0 0 0] [0], [1 1 0] [0] [adx](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [0] [zeros] = [0] [0], [1 1 1] [active](x0) = [0 0 0]x0 [0 0 1] orientation: [1 1 1] [1] [1 1 1] [0] active(tl(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = tl(active(X)) [0 0 1] [1] [0 0 1] [1] [1 1 0] [1] [1 1 0] [1] adx(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(adx(X)) [0 0 0] [1] [0 0 0] [0] [1 1 1] [1] [1 0 1] [1] proper(adx(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = adx(proper(X)) [0 0 1] [1] [0 0 1] [1] [0] [0] proper(zeros()) = [1] >= [0] = ok(zeros()) [0] [0] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] proper(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0] = cons(proper(X1),proper(X2)) [0 0 1] [0 0 1] [0] [0 0 1] [0 0 1] [0] [0] [0] proper(0()) = [1] >= [0] = ok(0()) [0] [0] [1 0 1] [0] [1 0 1] proper(incr(X)) = [0 0 0]X + [1] >= [0 0 0]X = incr(proper(X)) [0 0 1] [0] [0 0 1] [1 1 1] [1] [1 0 1] [1] proper(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = s(proper(X)) [0 0 1] [1] [0 0 1] [1] [1 0 1] [1] [1 0 1] [1] proper(hd(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = hd(proper(X)) [0 0 1] [0] [0 0 1] [0] [1 1 1] [1] [1 0 1] [1] proper(tl(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = tl(proper(X)) [0 0 1] [1] [0 0 1] [1] problem: adx(mark(X)) -> mark(adx(X)) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) proper(tl(X)) -> tl(proper(X)) Matrix Interpretation Processor: dim=1 interpretation: [ok](x0) = 3x0 + 2, [proper](x0) = 2x0 + 3, [tl](x0) = 2x0 + 4, [hd](x0) = 2x0 + 3, [s](x0) = x0, [incr](x0) = 3x0 + 6, [cons](x0, x1) = 2x0 + x1 + 6, [0] = 1, [mark](x0) = x0, [adx](x0) = 3x0 + 6, [zeros] = 1 orientation: adx(mark(X)) = 3X + 6 >= 3X + 6 = mark(adx(X)) proper(adx(X)) = 6X + 15 >= 6X + 15 = adx(proper(X)) proper(zeros()) = 5 >= 5 = ok(zeros()) proper(cons(X1,X2)) = 4X1 + 2X2 + 15 >= 4X1 + 2X2 + 15 = cons(proper(X1),proper(X2)) proper(0()) = 5 >= 5 = ok(0()) proper(incr(X)) = 6X + 15 >= 6X + 15 = incr(proper(X)) proper(s(X)) = 2X + 3 >= 2X + 3 = s(proper(X)) proper(hd(X)) = 4X + 9 >= 4X + 9 = hd(proper(X)) proper(tl(X)) = 4X + 11 >= 4X + 10 = tl(proper(X)) problem: adx(mark(X)) -> mark(adx(X)) proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [ok](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [proper](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [hd](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [s](x0) = [0 1 0]x0 [0 0 0] , [1 1 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 1 0] [cons](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [0] = [0] [0], [0] [mark](x0) = x0 + [1] [1], [1 1 0] [adx](x0) = [0 0 1]x0 [0 1 0] , [0] [zeros] = [0] [0] orientation: [1 1 0] [1] [1 1 0] [0] adx(mark(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = mark(adx(X)) [0 1 0] [1] [0 1 0] [1] [1 1 1] [1 1 0] proper(adx(X)) = [0 0 0]X >= [0 0 0]X = adx(proper(X)) [0 0 0] [0 0 0] [0] [0] proper(zeros()) = [0] >= [0] = ok(zeros()) [0] [0] [1 1 0] [1 1 0] [1 1 0] [1 1 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] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 1 0] [1 1 0] proper(incr(X)) = [0 0 0]X >= [0 0 0]X = incr(proper(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] proper(s(X)) = [0 0 0]X >= [0 0 0]X = s(proper(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] proper(hd(X)) = [0 0 0]X >= [0 0 0]X = hd(proper(X)) [0 0 0] [0 0 0] problem: proper(adx(X)) -> adx(proper(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [ok](x0) = [0 0 0]x0 + [1] [0 0 1] [0], [1 0 1] [0] [proper](x0) = [0 1 0]x0 + [1] [0 0 1] [0], [hd](x0) = x0 , [1 1 0] [0] [s](x0) = [0 1 0]x0 + [0] [0 0 1] [1], [incr](x0) = x0 , [1 0 0] [cons](x0, x1) = [0 0 0]x0 + x1 [0 0 1] , [0] [0] = [0] [1], [1 0 0] [0] [adx](x0) = [0 0 1]x0 + [0] [0 0 1] [1], [0] [zeros] = [0] [0] orientation: [1 0 1] [1] [1 0 1] [0] proper(adx(X)) = [0 0 1]X + [1] >= [0 0 1]X + [0] = adx(proper(X)) [0 0 1] [1] [0 0 1] [1] [0] [0] proper(zeros()) = [1] >= [1] = ok(zeros()) [0] [0] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] [0] proper(cons(X1,X2)) = [0 0 0]X1 + [0 1 0]X2 + [1] >= [0 0 0]X1 + [0 1 0]X2 + [1] = cons(proper(X1),proper(X2)) [0 0 1] [0 0 1] [0] [0 0 1] [0 0 1] [0] [1] [0] proper(0()) = [1] >= [1] = ok(0()) [1] [1] [1 0 1] [0] [1 0 1] [0] proper(incr(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = incr(proper(X)) [0 0 1] [0] [0 0 1] [0] [1 1 1] [1] [1 1 1] [1] proper(s(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = s(proper(X)) [0 0 1] [1] [0 0 1] [1] [1 0 1] [0] [1 0 1] [0] proper(hd(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = hd(proper(X)) [0 0 1] [0] [0 0 1] [0] problem: proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(hd(X)) -> hd(proper(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [ok](x0) = [0 0 0]x0 + [1] [0 0 0] [1], [1 1 0] [0] [proper](x0) = [1 1 0]x0 + [0] [0 0 0] [1], [1 0 0] [0] [hd](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1] [s](x0) = x0 + [0] [0], [1 0 0] [0] [incr](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 0] [0] [cons](x0, x1) = x0 + [0 1 0]x1 + [1] [0 0 0] [0], [0] [zeros] = [1] [0] orientation: [1] [0] proper(zeros()) = [1] >= [1] = ok(zeros()) [1] [1] [1 1 0] [1 1 0] [1] [1 1 0] [1 1 0] [0] proper(cons(X1,X2)) = [1 1 0]X1 + [1 1 0]X2 + [1] >= [1 1 0]X1 + [1 1 0]X2 + [1] = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 0] [1] [1 1 0] [0] proper(incr(X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = incr(proper(X)) [0 0 0] [1] [0 0 0] [0] [1 1 0] [1] [1 1 0] [1] proper(s(X)) = [1 1 0]X + [1] >= [1 1 0]X + [0] = s(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 1 0] [1] [1 1 0] [0] proper(hd(X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = hd(proper(X)) [0 0 0] [1] [0 0 0] [0] problem: proper(s(X)) -> s(proper(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [proper](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [0] [s](x0) = [0 0 0]x0 + [0] [0 0 1] [1] orientation: [1 0 1] [1] [1 0 1] [0] proper(s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(proper(X)) [0 0 1] [1] [0 0 1] [1] problem: Qed