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: [top](x0) = x0, [ok](x0) = x0, [proper](x0) = x0, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + 4x1 + 1, [0] = 2, [mark](x0) = x0, [active](x0) = x0, [and](x0, x1) = 5x0 + 2x1, [tt] = 1 orientation: active(and(tt(),X)) = 2X + 5 >= X = mark(X) active(plus(N,0())) = N + 9 >= N = mark(N) active(plus(N,s(M))) = 4M + N + 5 >= 4M + N + 2 = mark(s(plus(N,M))) active(and(X1,X2)) = 5X1 + 2X2 >= 5X1 + 2X2 = and(active(X1),X2) active(plus(X1,X2)) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = plus(X1,active(X2)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) and(mark(X1),X2) = 5X1 + 2X2 >= 5X1 + 2X2 = mark(and(X1,X2)) plus(mark(X1),X2) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = mark(plus(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) proper(and(X1,X2)) = 5X1 + 2X2 >= 5X1 + 2X2 = and(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(plus(X1,X2)) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = plus(proper(X1),proper(X2)) proper(0()) = 2 >= 2 = ok(0()) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) and(ok(X1),ok(X2)) = 5X1 + 2X2 >= 5X1 + 2X2 = ok(and(X1,X2)) plus(ok(X1),ok(X2)) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = ok(plus(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) top(mark(X)) = X >= X = top(proper(X)) top(ok(X)) = X >= X = 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)) 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: [1 0 1] [top](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [ok](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [proper](x0) = [0 0 1]x0 [0 0 0] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [1 0 0] [plus](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [0 0 1] , [0] [0] = [0] [1], [1 1 0] [1] [mark](x0) = [0 0 0]x0 + [0] [0 0 1] [0], [1 0 0] [active](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [1 0 1] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [0 0 0] , [0] [tt] = [0] [1] orientation: [1 0 0] [1 0 1] [1 0 0] [1 0 1] active(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = and(active(X1),X2) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(plus(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = plus(active(X1),X2) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(plus(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = plus(X1,active(X2)) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1 0 0] [1 0 0] active(s(X)) = [0 0 0]X >= [0 0 0]X = s(active(X)) [0 0 1] [0 0 1] [1 1 0] [1 0 1] [1] [1 0 0] [1 0 1] [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 1] [0 0 0] [0] [0 0 1] [0 0 0] [0] [1 1 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 1] [0 0 1] [0] [0 0 1] [0 0 1] [0] [1 0 0] [1 1 0] [1] [1 0 0] [1 0 0] [1] plus(X1,mark(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 1] [0 0 1] [0] [0 0 1] [0 0 1] [0] [1 1 0] [1] [1 0 0] [1] s(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(s(X)) [0 0 1] [0] [0 0 1] [0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] proper(and(X1,X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = and(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1] [1] proper(tt()) = [1] >= [0] = ok(tt()) [0] [0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] proper(plus(X1,X2)) = [0 0 1]X1 + [0 0 1]X2 >= [0 0 0]X1 + [0 0 0]X2 = plus(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1] [1] proper(0()) = [1] >= [0] = ok(0()) [0] [0] [1 0 1] [1 0 1] proper(s(X)) = [0 0 1]X >= [0 0 0]X = s(proper(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] and(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(and(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] plus(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(plus(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] 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 0 1] top(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X = top(proper(X)) [0 0 0] [0] [0 0 0] [1 0 1] [1 0 1] 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: [top](x0) = x0, [ok](x0) = 5x0 + 4, [proper](x0) = 5x0 + 4, [s](x0) = 3x0 + 2, [plus](x0, x1) = 2x0 + x1 + 2, [0] = 0, [mark](x0) = x0, [active](x0) = x0, [and](x0, x1) = x0 + 2x1 + 2, [tt] = 2 orientation: active(and(X1,X2)) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = and(active(X1),X2) active(plus(X1,X2)) = 2X1 + X2 + 2 >= 2X1 + X2 + 2 = plus(active(X1),X2) active(plus(X1,X2)) = 2X1 + X2 + 2 >= 2X1 + X2 + 2 = plus(X1,active(X2)) active(s(X)) = 3X + 2 >= 3X + 2 = s(active(X)) and(mark(X1),X2) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = mark(and(X1,X2)) plus(mark(X1),X2) = 2X1 + X2 + 2 >= 2X1 + X2 + 2 = mark(plus(X1,X2)) plus(X1,mark(X2)) = 2X1 + X2 + 2 >= 2X1 + X2 + 2 = mark(plus(X1,X2)) s(mark(X)) = 3X + 2 >= 3X + 2 = mark(s(X)) proper(and(X1,X2)) = 5X1 + 10X2 + 14 >= 5X1 + 10X2 + 14 = and(proper(X1),proper(X2)) proper(tt()) = 14 >= 14 = ok(tt()) proper(plus(X1,X2)) = 10X1 + 5X2 + 14 >= 10X1 + 5X2 + 14 = plus(proper(X1),proper(X2)) proper(0()) = 4 >= 4 = ok(0()) proper(s(X)) = 15X + 14 >= 15X + 14 = s(proper(X)) and(ok(X1),ok(X2)) = 5X1 + 10X2 + 14 >= 5X1 + 10X2 + 14 = ok(and(X1,X2)) plus(ok(X1),ok(X2)) = 10X1 + 5X2 + 14 >= 10X1 + 5X2 + 14 = ok(plus(X1,X2)) s(ok(X)) = 15X + 14 >= 15X + 14 = ok(s(X)) top(ok(X)) = 5X + 4 >= X = 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)) 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)) Matrix Interpretation Processor: dim=1 interpretation: [ok](x0) = 4x0, [proper](x0) = 4x0, [s](x0) = 2x0, [plus](x0, x1) = 4x0 + 4x1, [0] = 0, [mark](x0) = x0 + 1, [active](x0) = x0, [and](x0, x1) = x0 + 4x1, [tt] = 6 orientation: active(and(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = and(active(X1),X2) active(plus(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = plus(active(X1),X2) active(plus(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = plus(X1,active(X2)) active(s(X)) = 2X >= 2X = s(active(X)) and(mark(X1),X2) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = mark(and(X1,X2)) plus(mark(X1),X2) = 4X1 + 4X2 + 4 >= 4X1 + 4X2 + 1 = mark(plus(X1,X2)) plus(X1,mark(X2)) = 4X1 + 4X2 + 4 >= 4X1 + 4X2 + 1 = mark(plus(X1,X2)) s(mark(X)) = 2X + 2 >= 2X + 1 = mark(s(X)) proper(and(X1,X2)) = 4X1 + 16X2 >= 4X1 + 16X2 = and(proper(X1),proper(X2)) proper(tt()) = 24 >= 24 = ok(tt()) proper(plus(X1,X2)) = 16X1 + 16X2 >= 16X1 + 16X2 = plus(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(s(X)) = 8X >= 8X = s(proper(X)) and(ok(X1),ok(X2)) = 4X1 + 16X2 >= 4X1 + 16X2 = ok(and(X1,X2)) plus(ok(X1),ok(X2)) = 16X1 + 16X2 >= 16X1 + 16X2 = ok(plus(X1,X2)) s(ok(X)) = 8X >= 8X = ok(s(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)) 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)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [ok](x0) = [0 0 0]x0 [0 0 0] , [1 1 1] [proper](x0) = [0 0 0]x0 [1 0 1] , [1 0 0] [0] [s](x0) = [0 0 0]x0 + [0] [0 1 1] [1], [1 0 0] [1 0 0] [plus](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 1 1] [0 1 1] , [1] [0] = [0] [0], [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [active](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [1 0 0] [0] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 1 1] [0 1 1] [1], [1] [tt] = [0] [0] orientation: [1 1 1] [1 1 1] [1] [1 0 1] [1 0 0] [0] active(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = and(active(X1),X2) [0 1 1] [0 1 1] [1] [0 0 1] [0 1 1] [1] [1 1 1] [1 1 1] [1 0 1] [1 0 0] active(plus(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = plus(active(X1),X2) [0 1 1] [0 1 1] [0 0 1] [0 1 1] [1 1 1] [1 1 1] [1 0 0] [1 0 1] active(plus(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = plus(X1,active(X2)) [0 1 1] [0 1 1] [0 1 1] [0 0 1] [1 1 1] [1] [1 0 1] [0] active(s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(active(X)) [0 1 1] [1] [0 0 1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] and(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = mark(and(X1,X2)) [0 0 0] [0 1 1] [1] [0 0 0] [0 0 0] [1 1 1] [1 1 1] [1] [1 1 1] [1 1 1] [0] proper(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = and(proper(X1),proper(X2)) [1 1 1] [1 1 1] [1] [1 0 1] [1 0 1] [1] [1] [1] proper(tt()) = [0] >= [0] = ok(tt()) [1] [0] [1 1 1] [1 1 1] [1 1 1] [1 1 1] proper(plus(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = plus(proper(X1),proper(X2)) [1 1 1] [1 1 1] [1 0 1] [1 0 1] [1] [1] proper(0()) = [0] >= [0] = ok(0()) [1] [0] [1 1 1] [1] [1 1 1] [0] proper(s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(proper(X)) [1 1 1] [1] [1 0 1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] and(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = ok(and(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] plus(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(plus(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] s(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X = ok(s(X)) [0 0 0] [1] [0 0 0] problem: active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(tt()) -> ok(tt()) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) s(ok(X)) -> ok(s(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [ok](x0) = [0 1 0]x0 [0 0 0] , [1 1 0] [proper](x0) = [0 1 0]x0 [0 0 0] , [1 0 1] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 1] [0] [plus](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 0] , [1 1 0] [0] [active](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [1 0 0] [1 1 0] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [tt] = [0] [0] orientation: [1 1 0] [1 1 1] [1] [1 1 0] [1 0 1] [0] active(plus(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = plus(active(X1),X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0] [1 1 0] [1 1 1] [1] [1 0 0] [1 1 0] [1] active(plus(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = plus(X1,active(X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0] [1 0 0] [1 1 0] [1 0 0] [1 1 0] and(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(and(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] proper(tt()) = [0] >= [0] = ok(tt()) [0] [0] [1 1 0] [1 1 1] [1] [1 1 0] [1 1 0] [0] proper(plus(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = plus(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 0 1] [1 1 1] [1 0 0] [1 1 0] and(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(and(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [0] [1 0 0] [1 0 1] [0] plus(ok(X1),ok(X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = ok(plus(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 1] [1 0 1] s(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(s(X)) [0 0 0] [0 0 0] problem: active(plus(X1,X2)) -> plus(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(tt()) -> ok(tt()) proper(0()) -> ok(0()) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) s(ok(X)) -> ok(s(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [ok](x0) = [1 0 0]x0 + [1] [0 0 0] [1], [1 0 1] [0] [proper](x0) = [0 0 0]x0 + [1] [0 1 0] [0], [1 0 0] [0] [s](x0) = [1 0 0]x0 + [1] [0 1 0] [0], [1 0 0] [1 0 0] [plus](x0, x1) = [0 1 0]x0 + [1 0 0]x1 [0 0 0] [0 1 0] , [0] [0] = [1] [0], [1 0 0] [1] [mark](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [0] [active](x0) = x0 + [0] [1], [1 0 0] [and](x0, x1) = [1 0 0]x0 + x1 [0 0 0] , [0] [tt] = [1] [1] orientation: [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] active(plus(X1,X2)) = [0 1 0]X1 + [1 0 0]X2 + [0] >= [0 1 0]X1 + [1 0 0]X2 = plus(X1,active(X2)) [0 0 0] [0 1 0] [1] [0 0 0] [0 1 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] and(mark(X1),X2) = [1 0 0]X1 + X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0] = mark(and(X1,X2)) [0 0 0] [0] [0 0 0] [0 0 0] [0] [1] [0] proper(tt()) = [1] >= [1] = ok(tt()) [1] [1] [0] [0] proper(0()) = [1] >= [1] = ok(0()) [1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] and(ok(X1),ok(X2)) = [1 0 0]X1 + [1 0 0]X2 + [1] >= [1 0 0]X1 + [1 0 0]X2 + [1] = 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)) = [1 0 0]X1 + [1 0 0]X2 + [1] >= [1 0 0]X1 + [1 0 0]X2 + [1] = ok(plus(X1,X2)) [0 0 0] [1 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] s(ok(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = ok(s(X)) [1 0 0] [1] [0 0 0] [1] problem: active(plus(X1,X2)) -> plus(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(0()) -> ok(0()) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) s(ok(X)) -> ok(s(X)) Matrix Interpretation Processor: dim=1 interpretation: [ok](x0) = 4x0 + 6, [proper](x0) = 5x0, [s](x0) = 4x0 + 3, [plus](x0, x1) = x0 + x1 + 2, [0] = 6, [mark](x0) = x0 + 6, [active](x0) = x0 + 6, [and](x0, x1) = x0 + x1 + 2 orientation: active(plus(X1,X2)) = X1 + X2 + 8 >= X1 + X2 + 8 = plus(X1,active(X2)) and(mark(X1),X2) = X1 + X2 + 8 >= X1 + X2 + 8 = mark(and(X1,X2)) proper(0()) = 30 >= 30 = ok(0()) and(ok(X1),ok(X2)) = 4X1 + 4X2 + 14 >= 4X1 + 4X2 + 14 = ok(and(X1,X2)) plus(ok(X1),ok(X2)) = 4X1 + 4X2 + 14 >= 4X1 + 4X2 + 14 = ok(plus(X1,X2)) s(ok(X)) = 16X + 27 >= 16X + 18 = ok(s(X)) problem: active(plus(X1,X2)) -> plus(X1,active(X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(0()) -> ok(0()) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [ok](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 0] [0] [proper](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 1] [0] [plus](x0, x1) = [0 0 0]x0 + x1 + [1] [0 0 0] [0], [0] [0] = [0] [0], [1 1 0] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [active](x0) = [1 1 1]x0 [0 0 0] , [1 0 0] [1 0 0] [1] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 1] [0] orientation: [1 0 1] [1 1 0] [1] [1 0 1] [1 1 0] [0] active(plus(X1,X2)) = [1 0 1]X1 + [1 1 1]X2 + [1] >= [0 0 0]X1 + [1 1 1]X2 + [1] = plus(X1,active(X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 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 1] [0] [0 0 0] [0 0 0] [0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [1] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] 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 1] [1] [0 0 0] [0 0 1] [1] [1 0 1] [1 0 0] [1] [1 0 1] [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 1] [1] [0 0 0] [0 0 1] [1] problem: and(mark(X1),X2) -> mark(and(X1,X2)) proper(0()) -> ok(0()) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) Matrix Interpretation Processor: dim=3 interpretation: [ok](x0) = x0 , [1 0 0] [proper](x0) = [0 0 0]x0 [0 0 0] , [0] [0] = [0] [0], [1 1 0] [0] [mark](x0) = [0 0 1]x0 + [1] [0 1 0] [1], [1 1 0] [1 0 0] [and](x0, x1) = [0 0 1]x0 + [0 0 0]x1 [0 1 0] [0 0 0] orientation: [1 1 1] [1 0 0] [1] [1 1 1] [1 0 0] [0] and(mark(X1),X2) = [0 1 0]X1 + [0 0 0]X2 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [1] = mark(and(X1,X2)) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] and(ok(X1),ok(X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 1]X1 + [0 0 0]X2 = ok(and(X1,X2)) [0 1 0] [0 0 0] [0 1 0] [0 0 0] problem: proper(0()) -> ok(0()) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [ok](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 0] [1] [proper](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [0] [0] = [0] [0], [1 0 1] [1 0 0] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 1] orientation: [1] [0] proper(0()) = [0] >= [0] = ok(0()) [1] [1] [1 0 1] [1 0 0] [1] [1 0 1] [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 1] [1] [0 0 0] [0 0 1] [1] problem: Qed