YES Problem: active(filter(cons(X,Y),0(),M)) -> mark(cons(0(),filter(Y,M,M))) active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(sieve(cons(0(),Y))) -> mark(cons(0(),sieve(Y))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(zprimes()) -> mark(sieve(nats(s(s(0()))))) active(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(sieve(X)) -> sieve(active(X)) active(nats(X)) -> nats(active(X)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) sieve(mark(X)) -> mark(sieve(X)) nats(mark(X)) -> mark(nats(X)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(nats(X)) -> nats(proper(X)) proper(zprimes()) -> ok(zprimes()) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) sieve(ok(X)) -> ok(sieve(X)) nats(ok(X)) -> ok(nats(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: Matrix Interpretation Processor: dim=2 interpretation: [1 1] [top](x0) = [0 0]x0, [2 0] [ok](x0) = [0 1]x0, [proper](x0) = x0, [0] [zprimes] = [3], [2 0] [0] [nats](x0) = [0 1]x0 + [1], [0] [sieve](x0) = x0 + [1], [s](x0) = x0, [0] [mark](x0) = x0 + [1], [2 0] [active](x0) = [0 1]x0, [0] [filter](x0, x1, x2) = x0 + x1 + x2 + [2], [0] [0] = [0], [2 0] [1 0] [cons](x0, x1) = [0 1]x0 + [0 0]x1 orientation: [2 0] [4 0] [2 0] [0] [2 0] [1 0] [0] active(filter(cons(X,Y),0(),M)) = [0 1]M + [0 1]X + [0 0]Y + [2] >= [0 0]M + [0 0]Y + [1] = mark(cons(0(),filter(Y,M,M))) [2 0] [2 0] [4 0] [2 0] [0] [1 0] [1 0] [2 0] [1 0] [0] active(filter(cons(X,Y),s(N),M)) = [0 1]M + [0 1]N + [0 1]X + [0 0]Y + [2] >= [0 0]M + [0 0]N + [0 1]X + [0 0]Y + [1] = mark(cons(X,filter(Y,N,M))) [2 0] [0] [1 0] [0] active(sieve(cons(0(),Y))) = [0 0]Y + [1] >= [0 0]Y + [1] = mark(cons(0(),sieve(Y))) [4 0] [2 0] [0] [4 0] [1 0] [0] active(sieve(cons(s(N),Y))) = [0 1]N + [0 0]Y + [1] >= [0 1]N + [0 0]Y + [1] = mark(cons(s(N),sieve(filter(Y,N,N)))) [4 0] [0] [4 0] [0] active(nats(N)) = [0 1]N + [1] >= [0 1]N + [1] = mark(cons(N,nats(s(N)))) [0] [0] active(zprimes()) = [3] >= [3] = mark(sieve(nats(s(s(0()))))) [2 0] [2 0] [2 0] [0] [2 0] [0] active(filter(X1,X2,X3)) = [0 1]X1 + [0 1]X2 + [0 1]X3 + [2] >= [0 1]X1 + X2 + X3 + [2] = filter(active(X1),X2,X3) [2 0] [2 0] [2 0] [0] [2 0] [0] active(filter(X1,X2,X3)) = [0 1]X1 + [0 1]X2 + [0 1]X3 + [2] >= X1 + [0 1]X2 + X3 + [2] = filter(X1,active(X2),X3) [2 0] [2 0] [2 0] [0] [2 0] [0] active(filter(X1,X2,X3)) = [0 1]X1 + [0 1]X2 + [0 1]X3 + [2] >= X1 + X2 + [0 1]X3 + [2] = filter(X1,X2,active(X3)) [4 0] [2 0] [4 0] [1 0] active(cons(X1,X2)) = [0 1]X1 + [0 0]X2 >= [0 1]X1 + [0 0]X2 = cons(active(X1),X2) [2 0] [2 0] active(s(X)) = [0 1]X >= [0 1]X = s(active(X)) [2 0] [0] [2 0] [0] active(sieve(X)) = [0 1]X + [1] >= [0 1]X + [1] = sieve(active(X)) [4 0] [0] [4 0] [0] active(nats(X)) = [0 1]X + [1] >= [0 1]X + [1] = nats(active(X)) [0] [0] filter(mark(X1),X2,X3) = X1 + X2 + X3 + [3] >= X1 + X2 + X3 + [3] = mark(filter(X1,X2,X3)) [0] [0] filter(X1,mark(X2),X3) = X1 + X2 + X3 + [3] >= X1 + X2 + X3 + [3] = mark(filter(X1,X2,X3)) [0] [0] filter(X1,X2,mark(X3)) = X1 + X2 + X3 + [3] >= X1 + X2 + X3 + [3] = mark(filter(X1,X2,X3)) [2 0] [1 0] [0] [2 0] [1 0] [0] cons(mark(X1),X2) = [0 1]X1 + [0 0]X2 + [1] >= [0 1]X1 + [0 0]X2 + [1] = mark(cons(X1,X2)) [0] [0] s(mark(X)) = X + [1] >= X + [1] = mark(s(X)) [0] [0] sieve(mark(X)) = X + [2] >= X + [2] = mark(sieve(X)) [2 0] [0] [2 0] [0] nats(mark(X)) = [0 1]X + [2] >= [0 1]X + [2] = mark(nats(X)) [0] [0] proper(filter(X1,X2,X3)) = X1 + X2 + X3 + [2] >= X1 + X2 + X3 + [2] = filter(proper(X1),proper(X2),proper(X3)) [2 0] [1 0] [2 0] [1 0] proper(cons(X1,X2)) = [0 1]X1 + [0 0]X2 >= [0 1]X1 + [0 0]X2 = cons(proper(X1),proper(X2)) [0] [0] proper(0()) = [0] >= [0] = ok(0()) proper(s(X)) = X >= X = s(proper(X)) [0] [0] proper(sieve(X)) = X + [1] >= X + [1] = sieve(proper(X)) [2 0] [0] [2 0] [0] proper(nats(X)) = [0 1]X + [1] >= [0 1]X + [1] = nats(proper(X)) [0] [0] proper(zprimes()) = [3] >= [3] = ok(zprimes()) [2 0] [2 0] [2 0] [0] [2 0] [2 0] [2 0] [0] filter(ok(X1),ok(X2),ok(X3)) = [0 1]X1 + [0 1]X2 + [0 1]X3 + [2] >= [0 1]X1 + [0 1]X2 + [0 1]X3 + [2] = ok(filter(X1,X2,X3)) [4 0] [2 0] [4 0] [2 0] cons(ok(X1),ok(X2)) = [0 1]X1 + [0 0]X2 >= [0 1]X1 + [0 0]X2 = ok(cons(X1,X2)) [2 0] [2 0] s(ok(X)) = [0 1]X >= [0 1]X = ok(s(X)) [2 0] [0] [2 0] [0] sieve(ok(X)) = [0 1]X + [1] >= [0 1]X + [1] = ok(sieve(X)) [4 0] [0] [4 0] [0] nats(ok(X)) = [0 1]X + [1] >= [0 1]X + [1] = ok(nats(X)) [1 1] [1] [1 1] top(mark(X)) = [0 0]X + [0] >= [0 0]X = top(proper(X)) [2 1] [2 1] top(ok(X)) = [0 0]X >= [0 0]X = top(active(X)) problem: active(filter(cons(X,Y),0(),M)) -> mark(cons(0(),filter(Y,M,M))) active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(sieve(cons(0(),Y))) -> mark(cons(0(),sieve(Y))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(zprimes()) -> mark(sieve(nats(s(s(0()))))) active(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(sieve(X)) -> sieve(active(X)) active(nats(X)) -> nats(active(X)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) sieve(mark(X)) -> mark(sieve(X)) nats(mark(X)) -> mark(nats(X)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(nats(X)) -> nats(proper(X)) proper(zprimes()) -> ok(zprimes()) filter(ok(X1),ok(X2),ok(X3)) -> ok(filter(X1,X2,X3)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) s(ok(X)) -> ok(s(X)) sieve(ok(X)) -> ok(sieve(X)) nats(ok(X)) -> ok(nats(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = x0 + 4, [ok](x0) = 4x0 + 2, [proper](x0) = 6x0, [zprimes] = 4, [nats](x0) = 2x0, [sieve](x0) = x0, [s](x0) = x0, [mark](x0) = x0, [active](x0) = 4x0, [filter](x0, x1, x2) = 2x0 + 2x1 + x2, [0] = 2, [cons](x0, x1) = 2x0 + x1 orientation: active(filter(cons(X,Y),0(),M)) = 4M + 16X + 8Y + 16 >= 3M + 2Y + 4 = mark(cons(0(),filter(Y,M,M))) active(filter(cons(X,Y),s(N),M)) = 4M + 8N + 16X + 8Y >= M + 2N + 2X + 2Y = mark(cons(X,filter(Y,N,M))) active(sieve(cons(0(),Y))) = 4Y + 16 >= Y + 4 = mark(cons(0(),sieve(Y))) active(sieve(cons(s(N),Y))) = 8N + 4Y >= 5N + 2Y = mark(cons(s(N),sieve(filter(Y,N,N)))) active(nats(N)) = 8N >= 4N = mark(cons(N,nats(s(N)))) active(zprimes()) = 16 >= 4 = mark(sieve(nats(s(s(0()))))) active(filter(X1,X2,X3)) = 8X1 + 8X2 + 4X3 >= 8X1 + 2X2 + X3 = filter(active(X1),X2,X3) active(filter(X1,X2,X3)) = 8X1 + 8X2 + 4X3 >= 2X1 + 8X2 + X3 = filter(X1,active(X2),X3) active(filter(X1,X2,X3)) = 8X1 + 8X2 + 4X3 >= 2X1 + 2X2 + 4X3 = filter(X1,X2,active(X3)) active(cons(X1,X2)) = 8X1 + 4X2 >= 8X1 + X2 = cons(active(X1),X2) active(s(X)) = 4X >= 4X = s(active(X)) active(sieve(X)) = 4X >= 4X = sieve(active(X)) active(nats(X)) = 8X >= 8X = nats(active(X)) filter(mark(X1),X2,X3) = 2X1 + 2X2 + X3 >= 2X1 + 2X2 + X3 = mark(filter(X1,X2,X3)) filter(X1,mark(X2),X3) = 2X1 + 2X2 + X3 >= 2X1 + 2X2 + X3 = mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) = 2X1 + 2X2 + X3 >= 2X1 + 2X2 + X3 = mark(filter(X1,X2,X3)) cons(mark(X1),X2) = 2X1 + X2 >= 2X1 + X2 = mark(cons(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) sieve(mark(X)) = X >= X = mark(sieve(X)) nats(mark(X)) = 2X >= 2X = mark(nats(X)) proper(filter(X1,X2,X3)) = 12X1 + 12X2 + 6X3 >= 12X1 + 12X2 + 6X3 = filter(proper(X1),proper(X2),proper(X3)) proper(cons(X1,X2)) = 12X1 + 6X2 >= 12X1 + 6X2 = cons(proper(X1),proper(X2)) proper(0()) = 12 >= 10 = ok(0()) proper(s(X)) = 6X >= 6X = s(proper(X)) proper(sieve(X)) = 6X >= 6X = sieve(proper(X)) proper(nats(X)) = 12X >= 12X = nats(proper(X)) proper(zprimes()) = 24 >= 18 = ok(zprimes()) filter(ok(X1),ok(X2),ok(X3)) = 8X1 + 8X2 + 4X3 + 10 >= 8X1 + 8X2 + 4X3 + 2 = ok(filter(X1,X2,X3)) cons(ok(X1),ok(X2)) = 8X1 + 4X2 + 6 >= 8X1 + 4X2 + 2 = ok(cons(X1,X2)) s(ok(X)) = 4X + 2 >= 4X + 2 = ok(s(X)) sieve(ok(X)) = 4X + 2 >= 4X + 2 = ok(sieve(X)) nats(ok(X)) = 8X + 4 >= 8X + 2 = ok(nats(X)) top(ok(X)) = 4X + 6 >= 4X + 4 = top(active(X)) problem: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(sieve(cons(s(N),Y))) -> mark(cons(s(N),sieve(filter(Y,N,N)))) active(nats(N)) -> mark(cons(N,nats(s(N)))) active(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(sieve(X)) -> sieve(active(X)) active(nats(X)) -> nats(active(X)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) sieve(mark(X)) -> mark(sieve(X)) nats(mark(X)) -> mark(nats(X)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(sieve(X)) -> sieve(proper(X)) proper(nats(X)) -> nats(proper(X)) s(ok(X)) -> ok(s(X)) sieve(ok(X)) -> ok(sieve(X)) Matrix Interpretation Processor: dim=1 interpretation: [ok](x0) = x0 + 7, [proper](x0) = 4x0, [nats](x0) = 2x0 + 4, [sieve](x0) = 2x0 + 4, [s](x0) = 2x0, [mark](x0) = x0, [active](x0) = 4x0, [filter](x0, x1, x2) = x0 + 2x1 + 4x2, [cons](x0, x1) = x0 + x1 orientation: active(filter(cons(X,Y),s(N),M)) = 16M + 16N + 4X + 4Y >= 4M + 2N + X + Y = mark(cons(X,filter(Y,N,M))) active(sieve(cons(s(N),Y))) = 16N + 8Y + 16 >= 14N + 2Y + 4 = mark(cons(s(N),sieve(filter(Y,N,N)))) active(nats(N)) = 8N + 16 >= 5N + 4 = mark(cons(N,nats(s(N)))) active(filter(X1,X2,X3)) = 4X1 + 8X2 + 16X3 >= 4X1 + 2X2 + 4X3 = filter(active(X1),X2,X3) active(filter(X1,X2,X3)) = 4X1 + 8X2 + 16X3 >= X1 + 8X2 + 4X3 = filter(X1,active(X2),X3) active(filter(X1,X2,X3)) = 4X1 + 8X2 + 16X3 >= X1 + 2X2 + 16X3 = filter(X1,X2,active(X3)) active(cons(X1,X2)) = 4X1 + 4X2 >= 4X1 + X2 = cons(active(X1),X2) active(s(X)) = 8X >= 8X = s(active(X)) active(sieve(X)) = 8X + 16 >= 8X + 4 = sieve(active(X)) active(nats(X)) = 8X + 16 >= 8X + 4 = nats(active(X)) filter(mark(X1),X2,X3) = X1 + 2X2 + 4X3 >= X1 + 2X2 + 4X3 = mark(filter(X1,X2,X3)) filter(X1,mark(X2),X3) = X1 + 2X2 + 4X3 >= X1 + 2X2 + 4X3 = mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) = X1 + 2X2 + 4X3 >= X1 + 2X2 + 4X3 = mark(filter(X1,X2,X3)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(cons(X1,X2)) s(mark(X)) = 2X >= 2X = mark(s(X)) sieve(mark(X)) = 2X + 4 >= 2X + 4 = mark(sieve(X)) nats(mark(X)) = 2X + 4 >= 2X + 4 = mark(nats(X)) proper(filter(X1,X2,X3)) = 4X1 + 8X2 + 16X3 >= 4X1 + 8X2 + 16X3 = filter(proper(X1),proper(X2),proper(X3)) proper(cons(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = cons(proper(X1),proper(X2)) proper(s(X)) = 8X >= 8X = s(proper(X)) proper(sieve(X)) = 8X + 16 >= 8X + 4 = sieve(proper(X)) proper(nats(X)) = 8X + 16 >= 8X + 4 = nats(proper(X)) s(ok(X)) = 2X + 14 >= 2X + 7 = ok(s(X)) sieve(ok(X)) = 2X + 18 >= 2X + 11 = ok(sieve(X)) problem: active(filter(cons(X,Y),s(N),M)) -> mark(cons(X,filter(Y,N,M))) active(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) sieve(mark(X)) -> mark(sieve(X)) nats(mark(X)) -> mark(nats(X)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) Matrix Interpretation Processor: dim=1 interpretation: [proper](x0) = x0, [nats](x0) = x0, [sieve](x0) = x0 + 4, [s](x0) = x0 + 1, [mark](x0) = x0, [active](x0) = x0, [filter](x0, x1, x2) = x0 + 4x1 + x2, [cons](x0, x1) = x0 + x1 + 6 orientation: active(filter(cons(X,Y),s(N),M)) = M + 4N + X + Y + 10 >= M + 4N + X + Y + 6 = mark(cons(X,filter(Y,N,M))) active(filter(X1,X2,X3)) = X1 + 4X2 + X3 >= X1 + 4X2 + X3 = filter(active(X1),X2,X3) active(filter(X1,X2,X3)) = X1 + 4X2 + X3 >= X1 + 4X2 + X3 = filter(X1,active(X2),X3) active(filter(X1,X2,X3)) = X1 + 4X2 + X3 >= X1 + 4X2 + X3 = filter(X1,X2,active(X3)) active(cons(X1,X2)) = X1 + X2 + 6 >= X1 + X2 + 6 = cons(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) filter(mark(X1),X2,X3) = X1 + 4X2 + X3 >= X1 + 4X2 + X3 = mark(filter(X1,X2,X3)) filter(X1,mark(X2),X3) = X1 + 4X2 + X3 >= X1 + 4X2 + X3 = mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) = X1 + 4X2 + X3 >= X1 + 4X2 + X3 = mark(filter(X1,X2,X3)) cons(mark(X1),X2) = X1 + X2 + 6 >= X1 + X2 + 6 = mark(cons(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) sieve(mark(X)) = X + 4 >= X + 4 = mark(sieve(X)) nats(mark(X)) = X >= X = mark(nats(X)) proper(filter(X1,X2,X3)) = X1 + 4X2 + X3 >= X1 + 4X2 + X3 = filter(proper(X1),proper(X2),proper(X3)) proper(cons(X1,X2)) = X1 + X2 + 6 >= X1 + X2 + 6 = cons(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) problem: active(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) sieve(mark(X)) -> mark(sieve(X)) nats(mark(X)) -> mark(nats(X)) proper(filter(X1,X2,X3)) -> filter(proper(X1),proper(X2),proper(X3)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) Matrix Interpretation Processor: dim=1 interpretation: [proper](x0) = 6x0, [nats](x0) = 4x0, [sieve](x0) = 4x0 + 2, [s](x0) = 4x0 + 3, [mark](x0) = x0, [active](x0) = x0, [filter](x0, x1, x2) = x0 + 3x1 + 2x2 + 2, [cons](x0, x1) = x0 + x1 orientation: active(filter(X1,X2,X3)) = X1 + 3X2 + 2X3 + 2 >= X1 + 3X2 + 2X3 + 2 = filter(active(X1),X2,X3) active(filter(X1,X2,X3)) = X1 + 3X2 + 2X3 + 2 >= X1 + 3X2 + 2X3 + 2 = filter(X1,active(X2),X3) active(filter(X1,X2,X3)) = X1 + 3X2 + 2X3 + 2 >= X1 + 3X2 + 2X3 + 2 = filter(X1,X2,active(X3)) active(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(active(X1),X2) active(s(X)) = 4X + 3 >= 4X + 3 = s(active(X)) filter(mark(X1),X2,X3) = X1 + 3X2 + 2X3 + 2 >= X1 + 3X2 + 2X3 + 2 = mark(filter(X1,X2,X3)) filter(X1,mark(X2),X3) = X1 + 3X2 + 2X3 + 2 >= X1 + 3X2 + 2X3 + 2 = mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) = X1 + 3X2 + 2X3 + 2 >= X1 + 3X2 + 2X3 + 2 = mark(filter(X1,X2,X3)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(cons(X1,X2)) s(mark(X)) = 4X + 3 >= 4X + 3 = mark(s(X)) sieve(mark(X)) = 4X + 2 >= 4X + 2 = mark(sieve(X)) nats(mark(X)) = 4X >= 4X = mark(nats(X)) proper(filter(X1,X2,X3)) = 6X1 + 18X2 + 12X3 + 12 >= 6X1 + 18X2 + 12X3 + 2 = filter(proper(X1),proper(X2),proper(X3)) proper(cons(X1,X2)) = 6X1 + 6X2 >= 6X1 + 6X2 = cons(proper(X1),proper(X2)) proper(s(X)) = 24X + 18 >= 24X + 3 = s(proper(X)) problem: active(filter(X1,X2,X3)) -> filter(active(X1),X2,X3) active(filter(X1,X2,X3)) -> filter(X1,active(X2),X3) active(filter(X1,X2,X3)) -> filter(X1,X2,active(X3)) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) sieve(mark(X)) -> mark(sieve(X)) nats(mark(X)) -> mark(nats(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) Matrix Interpretation Processor: dim=1 interpretation: [proper](x0) = 4x0, [nats](x0) = 4x0, [sieve](x0) = 2x0, [s](x0) = x0, [mark](x0) = x0, [active](x0) = 4x0 + 2, [filter](x0, x1, x2) = 4x0 + x1 + 5x2 + 4, [cons](x0, x1) = 2x0 + x1 + 1 orientation: active(filter(X1,X2,X3)) = 16X1 + 4X2 + 20X3 + 18 >= 16X1 + X2 + 5X3 + 12 = filter(active(X1),X2,X3) active(filter(X1,X2,X3)) = 16X1 + 4X2 + 20X3 + 18 >= 4X1 + 4X2 + 5X3 + 6 = filter(X1,active(X2),X3) active(filter(X1,X2,X3)) = 16X1 + 4X2 + 20X3 + 18 >= 4X1 + X2 + 20X3 + 14 = filter(X1,X2,active(X3)) active(cons(X1,X2)) = 8X1 + 4X2 + 6 >= 8X1 + X2 + 5 = cons(active(X1),X2) active(s(X)) = 4X + 2 >= 4X + 2 = s(active(X)) filter(mark(X1),X2,X3) = 4X1 + X2 + 5X3 + 4 >= 4X1 + X2 + 5X3 + 4 = mark(filter(X1,X2,X3)) filter(X1,mark(X2),X3) = 4X1 + X2 + 5X3 + 4 >= 4X1 + X2 + 5X3 + 4 = mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) = 4X1 + X2 + 5X3 + 4 >= 4X1 + X2 + 5X3 + 4 = mark(filter(X1,X2,X3)) cons(mark(X1),X2) = 2X1 + X2 + 1 >= 2X1 + X2 + 1 = mark(cons(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) sieve(mark(X)) = 2X >= 2X = mark(sieve(X)) nats(mark(X)) = 4X >= 4X = mark(nats(X)) proper(cons(X1,X2)) = 8X1 + 4X2 + 4 >= 8X1 + 4X2 + 1 = cons(proper(X1),proper(X2)) problem: active(s(X)) -> s(active(X)) filter(mark(X1),X2,X3) -> mark(filter(X1,X2,X3)) filter(X1,mark(X2),X3) -> mark(filter(X1,X2,X3)) filter(X1,X2,mark(X3)) -> mark(filter(X1,X2,X3)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) sieve(mark(X)) -> mark(sieve(X)) nats(mark(X)) -> mark(nats(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [nats](x0) = [0 0 1]x0 [0 0 1] , [1 0 1] [sieve](x0) = [0 0 0]x0 [0 0 1] , [1 0 1] [0] [s](x0) = [0 1 0]x0 + [1] [0 0 1] [0], [0] [mark](x0) = x0 + [0] [1], [1 1 0] [active](x0) = [0 1 0]x0 [0 0 1] , [1 0 1] [1 0 1] [1 0 1] [0] [filter](x0, x1, x2) = [0 0 0]x0 + [1 0 0]x1 + [0 0 0]x2 + [1] [0 0 1] [0 1 1] [0 0 1] [0], [1 0 1] [1 0 0] [cons](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 0 1] [0 0 0] orientation: [1 1 1] [1] [1 1 1] [0] active(s(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = s(active(X)) [0 0 1] [0] [0 0 1] [0] [1 0 1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1 0 1] [0] filter(mark(X1),X2,X3) = [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [1] = mark(filter(X1,X2,X3)) [0 0 1] [0 1 1] [0 0 1] [1] [0 0 1] [0 1 1] [0 0 1] [1] [1 0 1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1 0 1] [0] filter(X1,mark(X2),X3) = [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [1] = mark(filter(X1,X2,X3)) [0 0 1] [0 1 1] [0 0 1] [1] [0 0 1] [0 1 1] [0 0 1] [1] [1 0 1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1 0 1] [0] filter(X1,X2,mark(X3)) = [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [1] = mark(filter(X1,X2,X3)) [0 0 1] [0 1 1] [0 0 1] [1] [0 0 1] [0 1 1] [0 0 1] [1] [1 0 1] [1 0 0] [1] [1 0 1] [1 0 0] [0] cons(mark(X1),X2) = [0 1 0]X1 + [0 0 0]X2 + [0] >= [0 1 0]X1 + [0 0 0]X2 + [0] = mark(cons(X1,X2)) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 0 1] [1] [1 0 1] [0] s(mark(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = mark(s(X)) [0 0 1] [1] [0 0 1] [1] [1 0 1] [1] [1 0 1] [0] sieve(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(sieve(X)) [0 0 1] [1] [0 0 1] [1] [1 0 0] [0] [1 0 0] [0] nats(mark(X)) = [0 0 1]X + [1] >= [0 0 1]X + [0] = mark(nats(X)) [0 0 1] [1] [0 0 1] [1] problem: nats(mark(X)) -> mark(nats(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [nats](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [0] [mark](x0) = [0 0 0]x0 + [0] [0 0 1] [1] orientation: [1 0 1] [1] [1 0 1] [0] nats(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(nats(X)) [0 0 1] [1] [0 0 1] [1] problem: Qed