YES Problem: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) active(and(X1,X2)) -> and(active(X1),X2) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(s(X)) -> s(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) s(mark(X)) -> mark(s(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: Matrix Interpretation Processor: dim=1 interpretation: [0] = 0, [ok](x0) = x0, [and](x0, x1) = 2x0 + 2x1 + 5, [mark](x0) = x0, [plus](x0, x1) = 4x0 + x1, [tt] = 0, [proper](x0) = x0, [top](x0) = x0, [s](x0) = x0 + 2, [active](x0) = x0 orientation: active(and(tt(),X)) = 2X + 5 >= X = mark(X) active(plus(N,0())) = 4N >= N = mark(N) active(plus(N,s(M))) = M + 4N + 2 >= M + 4N + 2 = mark(s(plus(N,M))) active(and(X1,X2)) = 2X1 + 2X2 + 5 >= 2X1 + 2X2 + 5 = and(active(X1),X2) active(plus(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = plus(active(X1),X2) active(plus(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = plus(X1,active(X2)) active(s(X)) = X + 2 >= X + 2 = s(active(X)) and(mark(X1),X2) = 2X1 + 2X2 + 5 >= 2X1 + 2X2 + 5 = mark(and(X1,X2)) plus(mark(X1),X2) = 4X1 + X2 >= 4X1 + X2 = mark(plus(X1,X2)) plus(X1,mark(X2)) = 4X1 + X2 >= 4X1 + X2 = mark(plus(X1,X2)) s(mark(X)) = X + 2 >= X + 2 = mark(s(X)) proper(and(X1,X2)) = 2X1 + 2X2 + 5 >= 2X1 + 2X2 + 5 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(plus(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = plus(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = X + 2 >= X + 2 = s(proper(X)) and(ok(X1),ok(X2)) = 2X1 + 2X2 + 5 >= 2X1 + 2X2 + 5 = ok(and(X1,X2)) plus(ok(X1),ok(X2)) = 4X1 + X2 >= 4X1 + X2 = ok(plus(X1,X2)) s(ok(X)) = X + 2 >= X + 2 = ok(s(X)) top(mark(X)) = X >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(X)) problem: active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) active(and(X1,X2)) -> and(active(X1),X2) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(s(X)) -> s(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) s(mark(X)) -> mark(s(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [0] = 0, [ok](x0) = x0, [and](x0, x1) = x0 + x1 + 1, [mark](x0) = x0, [plus](x0, x1) = x0 + 2x1, [tt] = 0, [proper](x0) = x0, [top](x0) = 2x0 + 3, [s](x0) = x0 + 1, [active](x0) = x0 orientation: active(plus(N,0())) = N >= N = mark(N) active(plus(N,s(M))) = 2M + N + 2 >= 2M + N + 1 = mark(s(plus(N,M))) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(plus(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = plus(X1,active(X2)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) plus(mark(X1),X2) = X1 + 2X2 >= X1 + 2X2 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + 2X2 >= X1 + 2X2 = mark(plus(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) proper(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(plus(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = plus(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) and(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(and(X1,X2)) plus(ok(X1),ok(X2)) = X1 + 2X2 >= X1 + 2X2 = ok(plus(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) top(mark(X)) = 2X + 3 >= 2X + 3 = top(proper(X)) top(ok(X)) = 2X + 3 >= 2X + 3 = top(active(X)) problem: active(plus(N,0())) -> mark(N) active(and(X1,X2)) -> and(active(X1),X2) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(s(X)) -> s(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) s(mark(X)) -> mark(s(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [0] [0] = [1] [0], [1 0 0] [0] [ok](x0) = [0 1 1]x0 + [0] [0 0 0] [1], [1 0 0] [and](x0, x1) = [0 1 0]x0 + x1 [0 0 0] , [1 0 0] [0] [mark](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 0] [plus](x0, x1) = [0 1 0]x0 + x1 [0 0 0] , [0] [tt] = [0] [0], [1 0 0] [0] [proper](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [1 1 1] [top](x0) = [0 0 0]x0 [0 0 0] , [s](x0) = x0 , [1] [active](x0) = x0 + [0] [0] orientation: [1 0 0] [1] [1 0 0] [0] active(plus(N,0())) = [0 1 0]N + [1] >= [0 1 0]N + [1] = mark(N) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] active(and(X1,X2)) = [0 1 0]X1 + X2 + [0] >= [0 1 0]X1 + X2 + [0] = and(active(X1),X2) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] active(plus(X1,X2)) = [0 1 0]X1 + X2 + [0] >= [0 1 0]X1 + X2 + [0] = plus(active(X1),X2) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] active(plus(X1,X2)) = [0 1 0]X1 + X2 + [0] >= [0 1 0]X1 + X2 + [0] = plus(X1,active(X2)) [0 0 0] [0] [0 0 0] [0] [1] [1] active(s(X)) = X + [0] >= X + [0] = s(active(X)) [0] [0] [1 0 0] [0] [1 0 0] [1 0 0] [0] and(mark(X1),X2) = [0 1 0]X1 + X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = mark(and(X1,X2)) [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [1 0 0] [0] plus(mark(X1),X2) = [0 1 0]X1 + X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = mark(plus(X1,X2)) [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] plus(X1,mark(X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = mark(plus(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [0] [1 0 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] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] proper(and(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0] = and(proper(X1),proper(X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [0] [0] proper(tt()) = [0] >= [0] = ok(tt()) [1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] proper(plus(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0] = plus(proper(X1),proper(X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [0] [0] proper(0()) = [1] >= [1] = ok(0()) [1] [1] [1 0 0] [0] [1 0 0] [0] proper(s(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = s(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] and(ok(X1),ok(X2)) = [0 1 1]X1 + [0 1 1]X2 + [0] >= [0 1 0]X1 + [0 1 1]X2 + [0] = ok(and(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] plus(ok(X1),ok(X2)) = [0 1 1]X1 + [0 1 1]X2 + [0] >= [0 1 0]X1 + [0 1 1]X2 + [0] = ok(plus(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] s(ok(X)) = [0 1 1]X + [0] >= [0 1 1]X + [0] = ok(s(X)) [0 0 0] [1] [0 0 0] [1] [1 1 0] [1] [1 1 0] [1] top(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = top(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1] [1 1 1] [1] top(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = top(active(X)) [0 0 0] [0] [0 0 0] [0] problem: active(and(X1,X2)) -> and(active(X1),X2) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(s(X)) -> s(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) s(mark(X)) -> mark(s(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [0] [0] = [0] [1], [1 0 0] [0] [ok](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [and](x0, x1) = x0 + [0 0 0]x1 [0 0 0] , [1 0 0] [1] [mark](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [1 0 0] [plus](x0, x1) = [0 1 1]x0 + [0 0 0]x1 [0 1 1] [0 0 0] , [1] [tt] = [0] [1], [proper](x0) = x0 , [1 0 0] [top](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [s](x0) = [0 0 0]x0 + [1] [0 0 0] [1], [1 0 0] [active](x0) = [0 0 1]x0 [0 1 0] orientation: [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(and(X1,X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 1]X1 + [0 0 0]X2 = and(active(X1),X2) [0 1 0] [0 0 0] [0 1 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(plus(X1,X2)) = [0 1 1]X1 + [0 0 0]X2 >= [0 1 1]X1 + [0 0 0]X2 = plus(active(X1),X2) [0 1 1] [0 0 0] [0 1 1] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(plus(X1,X2)) = [0 1 1]X1 + [0 0 0]X2 >= [0 1 1]X1 + [0 0 0]X2 = plus(X1,active(X2)) [0 1 1] [0 0 0] [0 1 1] [0 0 0] [1 0 0] [0] [1 0 0] [0] active(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(active(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] and(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = mark(and(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] plus(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = mark(plus(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] plus(X1,mark(X2)) = [0 1 1]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = mark(plus(X1,X2)) [0 1 1] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] s(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = mark(s(X)) [0 0 0] [1] [0 0 0] [0] [1 0 0] [1 0 0] proper(and(X1,X2)) = X1 + [0 0 0]X2 >= X1 + [0 0 0]X2 = and(proper(X1),proper(X2)) [0 0 0] [0 0 0] [1] [1] proper(tt()) = [0] >= [0] = ok(tt()) [1] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(plus(X1,X2)) = [0 1 1]X1 + [0 0 0]X2 >= [0 1 1]X1 + [0 0 0]X2 = plus(proper(X1),proper(X2)) [0 1 1] [0 0 0] [0 1 1] [0 0 0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [1] [1] [1 0 0] [0] [1 0 0] [0] proper(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] and(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = ok(and(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] plus(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0] = ok(plus(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] s(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = ok(s(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1] [1 0 0] top(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X = top(proper(X)) [0 0 0] [0] [0 0 0] [1 0 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(and(X1,X2)) -> and(active(X1),X2) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(s(X)) -> s(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) s(mark(X)) -> mark(s(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) s(ok(X)) -> ok(s(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [0] = 1, [ok](x0) = 2x0 + 2, [and](x0, x1) = x0 + 2x1 + 1, [mark](x0) = x0 + 4, [plus](x0, x1) = 6x0 + 2x1 + 4, [tt] = 2, [proper](x0) = 3x0 + 1, [top](x0) = 4x0 + 5, [s](x0) = x0, [active](x0) = x0 orientation: active(and(X1,X2)) = X1 + 2X2 + 1 >= X1 + 2X2 + 1 = and(active(X1),X2) active(plus(X1,X2)) = 6X1 + 2X2 + 4 >= 6X1 + 2X2 + 4 = plus(active(X1),X2) active(plus(X1,X2)) = 6X1 + 2X2 + 4 >= 6X1 + 2X2 + 4 = plus(X1,active(X2)) active(s(X)) = X >= X = s(active(X)) and(mark(X1),X2) = X1 + 2X2 + 5 >= X1 + 2X2 + 5 = mark(and(X1,X2)) plus(mark(X1),X2) = 6X1 + 2X2 + 28 >= 6X1 + 2X2 + 8 = mark(plus(X1,X2)) plus(X1,mark(X2)) = 6X1 + 2X2 + 12 >= 6X1 + 2X2 + 8 = mark(plus(X1,X2)) s(mark(X)) = X + 4 >= X + 4 = mark(s(X)) proper(and(X1,X2)) = 3X1 + 6X2 + 4 >= 3X1 + 6X2 + 4 = and(proper(X1),proper(X2)) proper(tt()) = 7 >= 6 = ok(tt()) proper(plus(X1,X2)) = 18X1 + 6X2 + 13 >= 18X1 + 6X2 + 12 = plus(proper(X1),proper(X2)) proper(0()) = 4 >= 4 = ok(0()) proper(s(X)) = 3X + 1 >= 3X + 1 = s(proper(X)) and(ok(X1),ok(X2)) = 2X1 + 4X2 + 7 >= 2X1 + 4X2 + 4 = ok(and(X1,X2)) plus(ok(X1),ok(X2)) = 12X1 + 4X2 + 20 >= 12X1 + 4X2 + 10 = ok(plus(X1,X2)) s(ok(X)) = 2X + 2 >= 2X + 2 = ok(s(X)) top(ok(X)) = 8X + 13 >= 4X + 5 = top(active(X)) problem: active(and(X1,X2)) -> and(active(X1),X2) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(s(X)) -> s(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) s(mark(X)) -> mark(s(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) s(ok(X)) -> ok(s(X)) Matrix Interpretation Processor: dim=1 interpretation: [0] = 0, [ok](x0) = 3x0 + 6, [and](x0, x1) = 2x0 + x1 + 3, [mark](x0) = x0 + 6, [plus](x0, x1) = x0 + x1, [proper](x0) = 7x0 + 6, [s](x0) = 4x0 + 3, [active](x0) = 7x0 + 6 orientation: active(and(X1,X2)) = 14X1 + 7X2 + 27 >= 14X1 + X2 + 15 = and(active(X1),X2) active(plus(X1,X2)) = 7X1 + 7X2 + 6 >= 7X1 + X2 + 6 = plus(active(X1),X2) active(plus(X1,X2)) = 7X1 + 7X2 + 6 >= X1 + 7X2 + 6 = plus(X1,active(X2)) active(s(X)) = 28X + 27 >= 28X + 27 = s(active(X)) and(mark(X1),X2) = 2X1 + X2 + 15 >= 2X1 + X2 + 9 = mark(and(X1,X2)) s(mark(X)) = 4X + 27 >= 4X + 9 = mark(s(X)) proper(and(X1,X2)) = 14X1 + 7X2 + 27 >= 14X1 + 7X2 + 21 = and(proper(X1),proper(X2)) proper(0()) = 6 >= 6 = ok(0()) proper(s(X)) = 28X + 27 >= 28X + 27 = s(proper(X)) s(ok(X)) = 12X + 27 >= 12X + 15 = ok(s(X)) problem: active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(s(X)) -> s(active(X)) proper(0()) -> ok(0()) proper(s(X)) -> s(proper(X)) Matrix Interpretation Processor: dim=1 interpretation: [0] = 3, [ok](x0) = 6x0 + 5, [plus](x0, x1) = 4x0 + x1 + 1, [proper](x0) = 6x0 + 5, [s](x0) = 4x0 + 4, [active](x0) = x0 orientation: active(plus(X1,X2)) = 4X1 + X2 + 1 >= 4X1 + X2 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = 4X1 + X2 + 1 >= 4X1 + X2 + 1 = plus(X1,active(X2)) active(s(X)) = 4X + 4 >= 4X + 4 = s(active(X)) proper(0()) = 23 >= 23 = ok(0()) proper(s(X)) = 24X + 29 >= 24X + 24 = s(proper(X)) problem: active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(s(X)) -> s(active(X)) proper(0()) -> ok(0()) Matrix Interpretation Processor: dim=1 interpretation: [0] = 2, [ok](x0) = x0, [plus](x0, x1) = x0 + x1, [proper](x0) = x0 + 1, [s](x0) = x0 + 2, [active](x0) = x0 + 6 orientation: active(plus(X1,X2)) = X1 + X2 + 6 >= X1 + X2 + 6 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + X2 + 6 >= X1 + X2 + 6 = plus(X1,active(X2)) active(s(X)) = X + 8 >= X + 8 = s(active(X)) proper(0()) = 3 >= 2 = ok(0()) problem: active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) active(s(X)) -> s(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [plus](x0, x1) = x0 + 4x1 + 2, [s](x0) = 2x0, [active](x0) = 2x0 orientation: active(plus(X1,X2)) = 2X1 + 8X2 + 4 >= 2X1 + 4X2 + 2 = plus(active(X1),X2) active(plus(X1,X2)) = 2X1 + 8X2 + 4 >= X1 + 8X2 + 2 = plus(X1,active(X2)) active(s(X)) = 4X >= 4X = s(active(X)) problem: active(s(X)) -> s(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [s](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 1 0] [active](x0) = [0 1 0]x0 [0 0 0] orientation: [1 1 0] [1] [1 1 0] [0] active(s(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = s(active(X)) [0 0 0] [0] [0 0 0] [0] problem: Qed