YES Problem: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(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: [cons](x0, x1) = 4x0 + x1, [s](x0) = x0, [active](x0) = x0, [incr](x0) = x0, [mark](x0) = x0, [tail](x0) = 4x0 + 2, [nats] = 0, [odds] = 0, [ok](x0) = x0, [proper](x0) = x0, [head](x0) = 4x0, [top](x0) = x0, [pairs] = 0, [0] = 0 orientation: active(nats()) = 0 >= 0 = mark(cons(0(),incr(nats()))) active(pairs()) = 0 >= 0 = mark(cons(0(),incr(odds()))) active(odds()) = 0 >= 0 = mark(incr(pairs())) active(incr(cons(X,XS))) = 4X + XS >= 4X + XS = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = 16X + 4XS >= X = mark(X) active(tail(cons(X,XS))) = 16X + 4XS + 2 >= XS = mark(XS) active(cons(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = cons(active(X1),X2) active(incr(X)) = X >= X = incr(active(X)) active(s(X)) = X >= X = s(active(X)) active(head(X)) = 4X >= 4X = head(active(X)) active(tail(X)) = 4X + 2 >= 4X + 2 = tail(active(X)) cons(mark(X1),X2) = 4X1 + X2 >= 4X1 + X2 = mark(cons(X1,X2)) incr(mark(X)) = X >= X = mark(incr(X)) s(mark(X)) = X >= X = mark(s(X)) head(mark(X)) = 4X >= 4X = mark(head(X)) tail(mark(X)) = 4X + 2 >= 4X + 2 = mark(tail(X)) proper(nats()) = 0 >= 0 = ok(nats()) proper(cons(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(incr(X)) = X >= X = incr(proper(X)) proper(pairs()) = 0 >= 0 = ok(pairs()) proper(odds()) = 0 >= 0 = ok(odds()) proper(s(X)) = X >= X = s(proper(X)) proper(head(X)) = 4X >= 4X = head(proper(X)) proper(tail(X)) = 4X + 2 >= 4X + 2 = tail(proper(X)) cons(ok(X1),ok(X2)) = 4X1 + X2 >= 4X1 + X2 = ok(cons(X1,X2)) incr(ok(X)) = X >= X = ok(incr(X)) s(ok(X)) = X >= X = ok(s(X)) head(ok(X)) = 4X >= 4X = ok(head(X)) tail(ok(X)) = 4X + 2 >= 4X + 2 = ok(tail(X)) top(mark(X)) = X >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(X)) problem: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [active](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [tail](x0) = [0 0 0]x0 [0 0 0] , [0] [nats] = [1] [0], [1] [odds] = [1] [0], [1 0 1] [ok](x0) = [0 0 0]x0 [0 1 0] , [1 0 0] [0] [proper](x0) = [0 0 0]x0 + [1] [0 0 0] [1], [1 0 0] [1] [head](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [0] [top](x0) = [0 0 0]x0 + [0] [1 0 0] [1], [1] [pairs] = [1] [0], [0] [0] = [0] [0] orientation: [0] [0] active(nats()) = [0] >= [0] = mark(cons(0(),incr(nats()))) [0] [0] [1] [1] active(pairs()) = [0] >= [0] = mark(cons(0(),incr(odds()))) [0] [0] [1] [1] active(odds()) = [0] >= [0] = mark(incr(pairs())) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(incr(cons(X,XS))) = [0 0 0]X + [0 0 0]XS >= [0 0 0]X + [0 0 0]XS = mark(cons(s(X),incr(XS))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1] [1 0 0] active(head(cons(X,XS))) = [0 0 0]X + [0 0 0]XS + [0] >= [0 0 0]X = mark(X) [0 0 0] [0 0 0] [0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(incr(X)) = [0 0 0]X >= [0 0 0]X = incr(active(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(s(X)) = [0 0 0]X >= [0 0 0]X = s(active(X)) [0 0 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)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] active(tail(X)) = [0 0 0]X >= [0 0 0]X = tail(active(X)) [0 0 0] [0 0 0] [1 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] 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] 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] 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 0 0] tail(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(tail(X)) [0 0 0] [0 0 0] [0] [0] proper(nats()) = [1] >= [0] = ok(nats()) [1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] proper(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0] [0] proper(0()) = [1] >= [0] = ok(0()) [1] [0] [1 0 0] [0] [1 0 0] proper(incr(X)) = [0 0 0]X + [1] >= [0 0 0]X = incr(proper(X)) [0 0 0] [1] [0 0 0] [1] [1] proper(pairs()) = [1] >= [0] = ok(pairs()) [1] [1] [1] [1] proper(odds()) = [1] >= [0] = ok(odds()) [1] [1] [1 0 0] [0] [1 0 0] proper(s(X)) = [0 0 0]X + [1] >= [0 0 0]X = s(proper(X)) [0 0 0] [1] [0 0 0] [1 0 0] [1] [1 0 0] [1] proper(head(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = head(proper(X)) [0 0 0] [1] [0 0 0] [0] [1 0 0] [0] [1 0 0] proper(tail(X)) = [0 0 0]X + [1] >= [0 0 0]X = tail(proper(X)) [0 0 0] [1] [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] 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 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] 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 0 0] tail(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(tail(X)) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] top(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = top(proper(X)) [1 0 0] [1] [1 0 0] [1] [1 0 1] [0] [1 0 0] [0] top(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = top(active(X)) [1 0 1] [1] [1 0 0] [1] problem: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [cons](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [1 0 0] [1 0 0] , [1 1 0] [s](x0) = [0 0 0]x0 [0 1 1] , [1 1 0] [active](x0) = [0 0 0]x0 [1 1 0] , [1 0 0] [incr](x0) = [0 1 0]x0 [1 0 0] , [1 0 0] [mark](x0) = [0 1 0]x0 [1 0 0] , [tail](x0) = x0 , [0] [nats] = [1] [0], [0] [odds] = [0] [0], [1 0 0] [ok](x0) = [0 1 0]x0 [1 0 0] , [proper](x0) = x0 , [1 0 0] [head](x0) = [0 1 0]x0 [1 0 0] , [1 1 0] [top](x0) = [0 0 0]x0 [0 1 0] , [0] [pairs] = [0] [0], [0] [0] = [0] [0] orientation: [1] [0] active(nats()) = [0] >= [0] = mark(cons(0(),incr(nats()))) [1] [0] [0] [0] active(pairs()) = [0] >= [0] = mark(cons(0(),incr(odds()))) [0] [0] [0] [0] active(odds()) = [0] >= [0] = mark(incr(pairs())) [0] [0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] active(incr(cons(X,XS))) = [0 0 0]X + [0 0 0]XS >= [0 0 0]X + [0 0 0]XS = mark(cons(s(X),incr(XS))) [1 1 0] [1 0 0] [1 1 0] [1 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 1 0] [1 0 0] [1 1 0] [1 1 0] active(incr(X)) = [0 0 0]X >= [0 0 0]X = incr(active(X)) [1 1 0] [1 1 0] [1 1 0] [1 1 0] active(s(X)) = [0 0 0]X >= [0 0 0]X = s(active(X)) [1 1 0] [1 1 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 1 0] [1 1 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(mark(X1),X2) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = mark(cons(X1,X2)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] incr(mark(X)) = [0 1 0]X >= [0 1 0]X = mark(incr(X)) [1 0 0] [1 0 0] [1 1 0] [1 1 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(s(X)) [1 1 0] [1 1 0] [1 0 0] [1 0 0] head(mark(X)) = [0 1 0]X >= [0 1 0]X = mark(head(X)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] tail(mark(X)) = [0 1 0]X >= [0 1 0]X = mark(tail(X)) [1 0 0] [1 0 0] [0] [0] proper(nats()) = [1] >= [1] = ok(nats()) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(cons(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = cons(proper(X1),proper(X2)) [1 0 0] [1 0 0] [1 0 0] [1 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)) [1 0 0] [1 0 0] [0] [0] proper(pairs()) = [0] >= [0] = ok(pairs()) [0] [0] [0] [0] proper(odds()) = [0] >= [0] = ok(odds()) [0] [0] [1 1 0] [1 1 0] proper(s(X)) = [0 0 0]X >= [0 0 0]X = s(proper(X)) [0 1 1] [0 1 1] [1 0 0] [1 0 0] proper(head(X)) = [0 1 0]X >= [0 1 0]X = head(proper(X)) [1 0 0] [1 0 0] proper(tail(X)) = X >= X = tail(proper(X)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(ok(X1),ok(X2)) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = ok(cons(X1,X2)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] incr(ok(X)) = [0 1 0]X >= [0 1 0]X = ok(incr(X)) [1 0 0] [1 0 0] [1 1 0] [1 1 0] s(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(s(X)) [1 1 0] [1 1 0] [1 0 0] [1 0 0] head(ok(X)) = [0 1 0]X >= [0 1 0]X = ok(head(X)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] tail(ok(X)) = [0 1 0]X >= [0 1 0]X = ok(tail(X)) [1 0 0] [1 0 0] [1 1 0] [1 1 0] top(mark(X)) = [0 0 0]X >= [0 0 0]X = top(proper(X)) [0 1 0] [0 1 0] [1 1 0] [1 1 0] top(ok(X)) = [0 0 0]X >= [0 0 0]X = top(active(X)) [0 1 0] [0 0 0] problem: active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [cons](x0, x1) = x0 + [0 0 0]x1 [0 0 0] , [1 1 1] [s](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [active](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [incr](x0) = [0 1 0]x0 [0 0 0] , [1 1 0] [mark](x0) = [0 0 0]x0 [0 0 1] , [1 1 0] [tail](x0) = [0 0 0]x0 [0 0 0] , [0] [nats] = [1] [0], [0] [odds] = [1] [0], [ok](x0) = x0 , [proper](x0) = x0 , [1 1 0] [head](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [top](x0) = [0 0 0]x0 [0 0 1] , [0] [pairs] = [1] [0], [0] [0] = [0] [0] orientation: [1] [0] active(pairs()) = [0] >= [0] = mark(cons(0(),incr(odds()))) [0] [0] [1] [1] active(odds()) = [0] >= [0] = mark(incr(pairs())) [0] [0] [1 1 1] [1 0 1] [1 1 1] [1 0 1] active(incr(cons(X,XS))) = [0 0 0]X + [0 0 0]XS >= [0 0 0]X + [0 0 0]XS = mark(cons(s(X),incr(XS))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 0 1] [1 1 0] [1 0 1] active(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 1 0] active(incr(X)) = [0 0 0]X >= [0 0 0]X = incr(active(X)) [0 0 0] [0 0 0] [1 1 1] [1 1 0] active(s(X)) = [0 0 0]X >= [0 0 0]X = s(active(X)) [0 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)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] active(tail(X)) = [0 0 0]X >= [0 0 0]X = tail(active(X)) [0 0 0] [0 0 0] [1 1 0] [1 0 1] [1 1 0] [1 0 1] 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 1] [0 0 0] [0 0 1] [0 0 0] [1 1 1] [1 1 1] incr(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(incr(X)) [0 0 0] [0 0 0] [1 1 1] [1 1 1] s(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(s(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)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] tail(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(tail(X)) [0 0 0] [0 0 0] [0] [0] proper(nats()) = [1] >= [1] = ok(nats()) [0] [0] [1 0 1] [1 0 1] proper(cons(X1,X2)) = X1 + [0 0 0]X2 >= X1 + [0 0 0]X2 = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 0 1] [1 0 1] proper(incr(X)) = [0 1 0]X >= [0 1 0]X = incr(proper(X)) [0 0 0] [0 0 0] [0] [0] proper(pairs()) = [1] >= [1] = ok(pairs()) [0] [0] [0] [0] proper(odds()) = [1] >= [1] = ok(odds()) [0] [0] [1 1 1] [1 1 1] 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(head(X)) = [0 0 0]X >= [0 0 0]X = head(proper(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] proper(tail(X)) = [0 0 0]X >= [0 0 0]X = tail(proper(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] cons(ok(X1),ok(X2)) = X1 + [0 0 0]X2 >= X1 + [0 0 0]X2 = ok(cons(X1,X2)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] incr(ok(X)) = [0 1 0]X >= [0 1 0]X = ok(incr(X)) [0 0 0] [0 0 0] [1 1 1] [1 1 1] s(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(s(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)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] tail(ok(X)) = [0 0 0]X >= [0 0 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 1] [0 0 1] [1 1 0] [1 1 0] top(ok(X)) = [0 0 0]X >= [0 0 0]X = top(active(X)) [0 0 1] [0 0 0] problem: active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [cons](x0, x1) = x0 + x1 + 3, [s](x0) = 3x0, [active](x0) = x0, [incr](x0) = 3x0, [mark](x0) = x0, [tail](x0) = 2x0 + 4, [nats] = 1, [odds] = 0, [ok](x0) = x0, [proper](x0) = x0, [head](x0) = 4x0, [top](x0) = x0, [pairs] = 0, [0] = 2 orientation: active(odds()) = 0 >= 0 = mark(incr(pairs())) active(incr(cons(X,XS))) = 3X + 3XS + 9 >= 3X + 3XS + 3 = mark(cons(s(X),incr(XS))) active(cons(X1,X2)) = X1 + X2 + 3 >= X1 + X2 + 3 = cons(active(X1),X2) active(incr(X)) = 3X >= 3X = incr(active(X)) active(s(X)) = 3X >= 3X = s(active(X)) active(head(X)) = 4X >= 4X = head(active(X)) active(tail(X)) = 2X + 4 >= 2X + 4 = tail(active(X)) cons(mark(X1),X2) = X1 + X2 + 3 >= X1 + X2 + 3 = mark(cons(X1,X2)) incr(mark(X)) = 3X >= 3X = mark(incr(X)) s(mark(X)) = 3X >= 3X = mark(s(X)) head(mark(X)) = 4X >= 4X = mark(head(X)) tail(mark(X)) = 2X + 4 >= 2X + 4 = mark(tail(X)) proper(nats()) = 1 >= 1 = ok(nats()) proper(cons(X1,X2)) = X1 + X2 + 3 >= X1 + X2 + 3 = cons(proper(X1),proper(X2)) proper(0()) = 2 >= 2 = ok(0()) proper(incr(X)) = 3X >= 3X = incr(proper(X)) proper(pairs()) = 0 >= 0 = ok(pairs()) proper(odds()) = 0 >= 0 = ok(odds()) proper(s(X)) = 3X >= 3X = s(proper(X)) proper(head(X)) = 4X >= 4X = head(proper(X)) proper(tail(X)) = 2X + 4 >= 2X + 4 = tail(proper(X)) cons(ok(X1),ok(X2)) = X1 + X2 + 3 >= X1 + X2 + 3 = ok(cons(X1,X2)) incr(ok(X)) = 3X >= 3X = ok(incr(X)) s(ok(X)) = 3X >= 3X = ok(s(X)) head(ok(X)) = 4X >= 4X = ok(head(X)) tail(ok(X)) = 2X + 4 >= 2X + 4 = ok(tail(X)) top(mark(X)) = X >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(X)) problem: active(odds()) -> mark(incr(pairs())) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [cons](x0, x1) = x0 + x1, [s](x0) = 4x0, [active](x0) = x0, [incr](x0) = 2x0, [mark](x0) = x0, [tail](x0) = x0, [nats] = 0, [odds] = 1, [ok](x0) = x0, [proper](x0) = x0, [head](x0) = 2x0, [top](x0) = x0, [pairs] = 0, [0] = 0 orientation: active(odds()) = 1 >= 0 = mark(incr(pairs())) active(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(active(X1),X2) active(incr(X)) = 2X >= 2X = incr(active(X)) active(s(X)) = 4X >= 4X = s(active(X)) active(head(X)) = 2X >= 2X = head(active(X)) active(tail(X)) = X >= X = tail(active(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(cons(X1,X2)) incr(mark(X)) = 2X >= 2X = mark(incr(X)) s(mark(X)) = 4X >= 4X = mark(s(X)) head(mark(X)) = 2X >= 2X = mark(head(X)) tail(mark(X)) = X >= X = mark(tail(X)) proper(nats()) = 0 >= 0 = ok(nats()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(incr(X)) = 2X >= 2X = incr(proper(X)) proper(pairs()) = 0 >= 0 = ok(pairs()) proper(odds()) = 1 >= 1 = ok(odds()) proper(s(X)) = 4X >= 4X = s(proper(X)) proper(head(X)) = 2X >= 2X = head(proper(X)) proper(tail(X)) = X >= X = tail(proper(X)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) incr(ok(X)) = 2X >= 2X = ok(incr(X)) s(ok(X)) = 4X >= 4X = ok(s(X)) head(ok(X)) = 2X >= 2X = ok(head(X)) tail(ok(X)) = X >= X = ok(tail(X)) top(mark(X)) = X >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(X)) problem: active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) head(mark(X)) -> mark(head(X)) tail(mark(X)) -> mark(tail(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [cons](x0, x1) = 2x0 + x1, [s](x0) = x0 + 4, [active](x0) = x0, [incr](x0) = 4x0, [mark](x0) = x0 + 4, [tail](x0) = 2x0, [nats] = 0, [odds] = 0, [ok](x0) = x0, [proper](x0) = x0, [head](x0) = 2x0, [top](x0) = 2x0 + 3, [pairs] = 0, [0] = 0 orientation: active(cons(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = cons(active(X1),X2) active(incr(X)) = 4X >= 4X = incr(active(X)) active(s(X)) = X + 4 >= X + 4 = s(active(X)) active(head(X)) = 2X >= 2X = head(active(X)) active(tail(X)) = 2X >= 2X = tail(active(X)) cons(mark(X1),X2) = 2X1 + X2 + 8 >= 2X1 + X2 + 4 = mark(cons(X1,X2)) incr(mark(X)) = 4X + 16 >= 4X + 4 = mark(incr(X)) s(mark(X)) = X + 8 >= X + 8 = mark(s(X)) head(mark(X)) = 2X + 8 >= 2X + 4 = mark(head(X)) tail(mark(X)) = 2X + 8 >= 2X + 4 = mark(tail(X)) proper(nats()) = 0 >= 0 = ok(nats()) proper(cons(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(incr(X)) = 4X >= 4X = incr(proper(X)) proper(pairs()) = 0 >= 0 = ok(pairs()) proper(odds()) = 0 >= 0 = ok(odds()) proper(s(X)) = X + 4 >= X + 4 = s(proper(X)) proper(head(X)) = 2X >= 2X = head(proper(X)) proper(tail(X)) = 2X >= 2X = tail(proper(X)) cons(ok(X1),ok(X2)) = 2X1 + X2 >= 2X1 + X2 = ok(cons(X1,X2)) incr(ok(X)) = 4X >= 4X = ok(incr(X)) s(ok(X)) = X + 4 >= X + 4 = ok(s(X)) head(ok(X)) = 2X >= 2X = ok(head(X)) tail(ok(X)) = 2X >= 2X = ok(tail(X)) top(mark(X)) = 2X + 11 >= 2X + 3 = top(proper(X)) top(ok(X)) = 2X + 3 >= 2X + 3 = top(active(X)) problem: active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) s(mark(X)) -> mark(s(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) proper(head(X)) -> head(proper(X)) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) head(ok(X)) -> ok(head(X)) tail(ok(X)) -> ok(tail(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [cons](x0, x1) = 2x0 + 4x1 + 4, [s](x0) = x0, [active](x0) = 4x0 + 3, [incr](x0) = x0, [mark](x0) = x0, [tail](x0) = 2x0 + 1, [nats] = 0, [odds] = 0, [ok](x0) = 4x0 + 4, [proper](x0) = 6x0 + 4, [head](x0) = x0, [top](x0) = x0, [pairs] = 0, [0] = 0 orientation: active(cons(X1,X2)) = 8X1 + 16X2 + 19 >= 8X1 + 4X2 + 10 = cons(active(X1),X2) active(incr(X)) = 4X + 3 >= 4X + 3 = incr(active(X)) active(s(X)) = 4X + 3 >= 4X + 3 = s(active(X)) active(head(X)) = 4X + 3 >= 4X + 3 = head(active(X)) active(tail(X)) = 8X + 7 >= 8X + 7 = tail(active(X)) s(mark(X)) = X >= X = mark(s(X)) proper(nats()) = 4 >= 4 = ok(nats()) proper(cons(X1,X2)) = 12X1 + 24X2 + 28 >= 12X1 + 24X2 + 28 = cons(proper(X1),proper(X2)) proper(0()) = 4 >= 4 = ok(0()) proper(incr(X)) = 6X + 4 >= 6X + 4 = incr(proper(X)) proper(pairs()) = 4 >= 4 = ok(pairs()) proper(odds()) = 4 >= 4 = ok(odds()) proper(s(X)) = 6X + 4 >= 6X + 4 = s(proper(X)) proper(head(X)) = 6X + 4 >= 6X + 4 = head(proper(X)) proper(tail(X)) = 12X + 10 >= 12X + 9 = tail(proper(X)) cons(ok(X1),ok(X2)) = 8X1 + 16X2 + 28 >= 8X1 + 16X2 + 20 = ok(cons(X1,X2)) incr(ok(X)) = 4X + 4 >= 4X + 4 = ok(incr(X)) s(ok(X)) = 4X + 4 >= 4X + 4 = ok(s(X)) head(ok(X)) = 4X + 4 >= 4X + 4 = ok(head(X)) tail(ok(X)) = 8X + 9 >= 8X + 8 = ok(tail(X)) top(ok(X)) = 4X + 4 >= 4X + 3 = top(active(X)) problem: active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) s(mark(X)) -> mark(s(X)) proper(nats()) -> ok(nats()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(pairs()) -> ok(pairs()) proper(odds()) -> ok(odds()) proper(s(X)) -> s(proper(X)) 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: [cons](x0, x1) = x0 + 2x1 + 1, [s](x0) = 2x0 + 1, [active](x0) = 4x0 + 3, [incr](x0) = 2x0 + 1, [mark](x0) = 4x0 + 3, [tail](x0) = 7x0 + 6, [nats] = 1, [odds] = 2, [ok](x0) = x0, [proper](x0) = 4x0, [head](x0) = x0, [pairs] = 4, [0] = 0 orientation: active(incr(X)) = 8X + 7 >= 8X + 7 = incr(active(X)) active(s(X)) = 8X + 7 >= 8X + 7 = s(active(X)) active(head(X)) = 4X + 3 >= 4X + 3 = head(active(X)) active(tail(X)) = 28X + 27 >= 28X + 27 = tail(active(X)) s(mark(X)) = 8X + 7 >= 8X + 7 = mark(s(X)) proper(nats()) = 4 >= 1 = ok(nats()) proper(cons(X1,X2)) = 4X1 + 8X2 + 4 >= 4X1 + 8X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(incr(X)) = 8X + 4 >= 8X + 1 = incr(proper(X)) proper(pairs()) = 16 >= 4 = ok(pairs()) proper(odds()) = 8 >= 2 = ok(odds()) proper(s(X)) = 8X + 4 >= 8X + 1 = s(proper(X)) proper(head(X)) = 4X >= 4X = head(proper(X)) incr(ok(X)) = 2X + 1 >= 2X + 1 = ok(incr(X)) s(ok(X)) = 2X + 1 >= 2X + 1 = ok(s(X)) head(ok(X)) = X >= X = ok(head(X)) problem: active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) s(mark(X)) -> mark(s(X)) 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: [s](x0) = x0, [active](x0) = 3x0 + 1, [incr](x0) = x0, [mark](x0) = 2x0, [tail](x0) = 5x0 + 2, [ok](x0) = 2x0 + 2, [proper](x0) = 3x0 + 1, [head](x0) = 3x0 + 1, [0] = 1 orientation: active(incr(X)) = 3X + 1 >= 3X + 1 = incr(active(X)) active(s(X)) = 3X + 1 >= 3X + 1 = s(active(X)) active(head(X)) = 9X + 4 >= 9X + 4 = head(active(X)) active(tail(X)) = 15X + 7 >= 15X + 7 = tail(active(X)) s(mark(X)) = 2X >= 2X = mark(s(X)) proper(0()) = 4 >= 4 = ok(0()) proper(head(X)) = 9X + 4 >= 9X + 4 = head(proper(X)) incr(ok(X)) = 2X + 2 >= 2X + 2 = ok(incr(X)) s(ok(X)) = 2X + 2 >= 2X + 2 = ok(s(X)) head(ok(X)) = 6X + 7 >= 6X + 4 = ok(head(X)) problem: active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) s(mark(X)) -> mark(s(X)) 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=1 interpretation: [s](x0) = 2x0 + 1, [active](x0) = x0, [incr](x0) = x0 + 4, [mark](x0) = 3x0 + 2, [tail](x0) = 4x0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [head](x0) = x0 + 2, [0] = 4 orientation: active(incr(X)) = X + 4 >= X + 4 = incr(active(X)) active(s(X)) = 2X + 1 >= 2X + 1 = s(active(X)) active(head(X)) = X + 2 >= X + 2 = head(active(X)) active(tail(X)) = 4X >= 4X = tail(active(X)) s(mark(X)) = 6X + 5 >= 6X + 5 = mark(s(X)) proper(0()) = 5 >= 5 = ok(0()) proper(head(X)) = X + 3 >= X + 3 = head(proper(X)) incr(ok(X)) = X + 5 >= X + 5 = ok(incr(X)) s(ok(X)) = 2X + 3 >= 2X + 2 = ok(s(X)) problem: active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) active(tail(X)) -> tail(active(X)) s(mark(X)) -> mark(s(X)) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) incr(ok(X)) -> ok(incr(X)) Matrix Interpretation Processor: dim=3 interpretation: [s](x0) = x0 , [1 1 0] [active](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [0] [incr](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [0] [tail](x0) = [0 1 1]x0 + [1] [0 0 0] [0], [1 0 0] [ok](x0) = [1 1 0]x0 [0 0 0] , [1 0 0] [0] [proper](x0) = [1 0 0]x0 + [1] [1 0 0] [1], [1 0 0] [head](x0) = [0 1 0]x0 [0 0 0] , [0] [0] = [0] [0] orientation: [1 1 0] [1] [1 1 0] [0] active(incr(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = incr(active(X)) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 1 0] active(s(X)) = [0 1 0]X >= [0 1 0]X = s(active(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] active(head(X)) = [0 1 0]X >= [0 1 0]X = head(active(X)) [0 0 0] [0 0 0] [1 1 1] [1] [1 1 0] [0] active(tail(X)) = [0 1 1]X + [1] >= [0 1 0]X + [1] = tail(active(X)) [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 1] [0 0 1] [0] [0] proper(0()) = [1] >= [0] = ok(0()) [1] [0] [1 0 0] [0] [1 0 0] [0] proper(head(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = head(proper(X)) [1 0 0] [1] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] incr(ok(X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = ok(incr(X)) [0 0 0] [0] [0 0 0] [0] problem: active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) s(mark(X)) -> mark(s(X)) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) incr(ok(X)) -> ok(incr(X)) Matrix Interpretation Processor: dim=1 interpretation: [s](x0) = x0 + 2, [active](x0) = x0, [incr](x0) = 2x0 + 5, [mark](x0) = x0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [head](x0) = x0, [0] = 2 orientation: active(s(X)) = X + 2 >= X + 2 = s(active(X)) active(head(X)) = X >= X = head(active(X)) s(mark(X)) = X + 2 >= X + 2 = mark(s(X)) proper(0()) = 3 >= 3 = ok(0()) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) incr(ok(X)) = 2X + 7 >= 2X + 6 = ok(incr(X)) problem: active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) s(mark(X)) -> mark(s(X)) proper(0()) -> ok(0()) proper(head(X)) -> head(proper(X)) Matrix Interpretation Processor: dim=1 interpretation: [s](x0) = x0 + 2, [active](x0) = x0 + 7, [mark](x0) = x0, [ok](x0) = 4x0 + 3, [proper](x0) = 4x0 + 3, [head](x0) = x0 + 1, [0] = 0 orientation: active(s(X)) = X + 9 >= X + 9 = s(active(X)) active(head(X)) = X + 8 >= X + 8 = head(active(X)) s(mark(X)) = X + 2 >= X + 2 = mark(s(X)) proper(0()) = 3 >= 3 = ok(0()) proper(head(X)) = 4X + 7 >= 4X + 4 = head(proper(X)) problem: active(s(X)) -> s(active(X)) active(head(X)) -> head(active(X)) s(mark(X)) -> mark(s(X)) proper(0()) -> ok(0()) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [s](x0) = [0 1 0]x0 + [0] [0 1 1] [1], [1 0 1] [active](x0) = [0 1 0]x0 [1 0 1] , [1 0 0] [mark](x0) = [1 0 0]x0 [0 0 0] , [1 0 0] [0] [ok](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [proper](x0) = [0 0 0]x0 [0 1 0] , [1 1 0] [1] [head](x0) = [0 0 0]x0 + [1] [0 0 1] [0], [0] [0] = [1] [0] orientation: [1 1 1] [1] [1 0 1] [0] active(s(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = s(active(X)) [1 1 1] [1] [1 1 1] [1] [1 1 1] [1] [1 1 1] [1] active(head(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = head(active(X)) [1 1 1] [1] [1 0 1] [0] [1 0 0] [0] [1 0 0] s(mark(X)) = [1 0 0]X + [0] >= [1 0 0]X = mark(s(X)) [1 0 0] [1] [0 0 0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [1] [1] problem: active(head(X)) -> head(active(X)) s(mark(X)) -> mark(s(X)) proper(0()) -> ok(0()) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [0] [s](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [0] [active](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 1] [0] [mark](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [ok](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1] [proper](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] orientation: [1 0 0] [1] [1 0 0] [1] active(head(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = head(active(X)) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1] [1 0 1] [1] s(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(s(X)) [0 0 0] [1] [0 0 0] [1] [1] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] problem: active(head(X)) -> head(active(X)) s(mark(X)) -> mark(s(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [s](x0) = [0 0 0]x0 [0 1 1] , [1 1 0] [active](x0) = [0 1 0]x0 [0 0 0] , [1 1 1] [0] [mark](x0) = [0 1 0]x0 + [0] [1 0 1] [1], [1 0 0] [0] [head](x0) = [0 1 0]x0 + [1] [0 0 0] [0] orientation: [1 1 0] [1] [1 1 0] [0] active(head(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = head(active(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [0] [1 1 1] [0] s(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(s(X)) [1 1 1] [1] [1 1 1] [1] problem: s(mark(X)) -> mark(s(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [s](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [0] [mark](x0) = [0 1 0]x0 + [1] [0 0 0] [0] orientation: [1 1 0] [1] [1 1 0] [0] s(mark(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = mark(s(X)) [0 0 0] [0] [0 0 0] [0] problem: Qed