YES Problem: active(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = 2x0, [ok](x0) = x0, [proper](x0) = x0, [0] = 0, [s](x0) = x0 + 3, [plus](x0, x1) = 4x0 + 5x1, [mark](x0) = x0, [U12](x0, x1, x2) = 4x0 + 5x1 + 4x2, [active](x0) = x0, [U11](x0, x1, x2) = 7x0 + 5x1 + 4x2 + 5, [tt] = 1 orientation: active(U11(tt(),M,N)) = 5M + 4N + 12 >= 5M + 4N + 4 = mark(U12(tt(),M,N)) active(U12(tt(),M,N)) = 5M + 4N + 4 >= 5M + 4N + 3 = mark(s(plus(N,M))) active(plus(N,0())) = 4N >= N = mark(N) active(plus(N,s(M))) = 5M + 4N + 15 >= 5M + 4N + 12 = mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) = 7X1 + 5X2 + 4X3 + 5 >= 7X1 + 5X2 + 4X3 + 5 = U11(active(X1),X2,X3) active(U12(X1,X2,X3)) = 4X1 + 5X2 + 4X3 >= 4X1 + 5X2 + 4X3 = U12(active(X1),X2,X3) active(s(X)) = X + 3 >= X + 3 = s(active(X)) active(plus(X1,X2)) = 4X1 + 5X2 >= 4X1 + 5X2 = plus(active(X1),X2) active(plus(X1,X2)) = 4X1 + 5X2 >= 4X1 + 5X2 = plus(X1,active(X2)) U11(mark(X1),X2,X3) = 7X1 + 5X2 + 4X3 + 5 >= 7X1 + 5X2 + 4X3 + 5 = mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) = 4X1 + 5X2 + 4X3 >= 4X1 + 5X2 + 4X3 = mark(U12(X1,X2,X3)) s(mark(X)) = X + 3 >= X + 3 = mark(s(X)) plus(mark(X1),X2) = 4X1 + 5X2 >= 4X1 + 5X2 = mark(plus(X1,X2)) plus(X1,mark(X2)) = 4X1 + 5X2 >= 4X1 + 5X2 = mark(plus(X1,X2)) proper(U11(X1,X2,X3)) = 7X1 + 5X2 + 4X3 + 5 >= 7X1 + 5X2 + 4X3 + 5 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X1,X2,X3)) = 4X1 + 5X2 + 4X3 >= 4X1 + 5X2 + 4X3 = U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X + 3 >= X + 3 = s(proper(X)) proper(plus(X1,X2)) = 4X1 + 5X2 >= 4X1 + 5X2 = plus(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) U11(ok(X1),ok(X2),ok(X3)) = 7X1 + 5X2 + 4X3 + 5 >= 7X1 + 5X2 + 4X3 + 5 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = 4X1 + 5X2 + 4X3 >= 4X1 + 5X2 + 4X3 = ok(U12(X1,X2,X3)) s(ok(X)) = X + 3 >= X + 3 = ok(s(X)) plus(ok(X1),ok(X2)) = 4X1 + 5X2 >= 4X1 + 5X2 = ok(plus(X1,X2)) top(mark(X)) = 2X >= 2X = top(proper(X)) top(ok(X)) = 2X >= 2X = top(active(X)) problem: active(plus(N,0())) -> mark(N) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) 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 1] [ok](x0) = [1 0 0]x0 [0 0 0] , [1 0 1] [0] [proper](x0) = [1 0 0]x0 + [1] [0 0 0] [0], [0] [0] = [0] [1], [s](x0) = x0 , [1 0 1] [1 0 1] [plus](x0, x1) = [1 0 0]x0 + [1 0 0]x1 [0 0 0] [0 0 0] , [1 0 1] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [1 0 1] [U12](x0, x1, x2) = x0 + [1 0 0]x1 + [1 0 1]x2 [0 0 0] [0 0 0] , [1 0 1] [active](x0) = [1 0 0]x0 [0 0 0] , [1 0 1] [1 0 1] [U11](x0, x1, x2) = x0 + [1 0 0]x1 + [1 0 1]x2 [0 0 0] [0 0 0] , [0] [tt] = [0] [0] orientation: [1 0 1] [1] [1 0 1] active(plus(N,0())) = [1 0 1]N + [1] >= [0 0 0]N = mark(N) [0 0 0] [0] [0 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] [1 0 1] [1 0 1] active(U11(X1,X2,X3)) = [1 0 0]X1 + [1 0 1]X2 + [1 0 1]X3 >= [1 0 0]X1 + [1 0 0]X2 + [1 0 1]X3 = U11(active(X1),X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] [1 0 1] [1 0 1] active(U12(X1,X2,X3)) = [1 0 0]X1 + [1 0 1]X2 + [1 0 1]X3 >= [1 0 0]X1 + [1 0 0]X2 + [1 0 1]X3 = U12(active(X1),X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] active(s(X)) = [1 0 0]X >= [1 0 0]X = s(active(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] active(plus(X1,X2)) = [1 0 1]X1 + [1 0 1]X2 >= [1 0 1]X1 + [1 0 0]X2 = plus(active(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] active(plus(X1,X2)) = [1 0 1]X1 + [1 0 1]X2 >= [1 0 0]X1 + [1 0 1]X2 = plus(X1,active(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] [1 0 1] [1 0 1] U11(mark(X1),X2,X3) = [0 0 0]X1 + [1 0 0]X2 + [1 0 1]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = mark(U11(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] [1 0 1] [1 0 1] U12(mark(X1),X2,X3) = [0 0 0]X1 + [1 0 0]X2 + [1 0 1]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = mark(U12(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] s(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(s(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] plus(mark(X1),X2) = [1 0 1]X1 + [1 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(plus(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(X1,mark(X2)) = [1 0 0]X1 + [1 0 1]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(plus(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] [1 0 1] [0] proper(U11(X1,X2,X3)) = [1 0 0]X1 + [1 0 1]X2 + [1 0 1]X3 + [1] >= [1 0 0]X1 + [1 0 1]X2 + [1 0 1]X3 + [1] = U11(proper(X1),proper(X2),proper(X3)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [0] [0] proper(tt()) = [1] >= [0] = ok(tt()) [0] [0] [1 0 1] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] [1 0 1] [0] proper(U12(X1,X2,X3)) = [1 0 0]X1 + [1 0 1]X2 + [1 0 1]X3 + [1] >= [1 0 0]X1 + [1 0 1]X2 + [1 0 1]X3 + [1] = U12(proper(X1),proper(X2),proper(X3)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 0 1] [0] [1 0 1] [0] proper(s(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = s(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] proper(plus(X1,X2)) = [1 0 1]X1 + [1 0 1]X2 + [1] >= [1 0 1]X1 + [1 0 1]X2 = plus(proper(X1),proper(X2)) [0 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] [1 0 1] [1 0 1] [1 0 1] [1 0 1] U11(ok(X1),ok(X2),ok(X3)) = [1 0 0]X1 + [1 0 1]X2 + [1 0 1]X3 >= [1 0 0]X1 + [1 0 1]X2 + [1 0 1]X3 = ok(U11(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] [1 0 1] [1 0 1] U12(ok(X1),ok(X2),ok(X3)) = [1 0 0]X1 + [1 0 1]X2 + [1 0 1]X3 >= [1 0 0]X1 + [1 0 1]X2 + [1 0 1]X3 = ok(U12(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] s(ok(X)) = [1 0 0]X >= [1 0 0]X = ok(s(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] plus(ok(X1),ok(X2)) = [1 0 1]X1 + [1 0 1]X2 >= [1 0 1]X1 + [1 0 1]X2 = ok(plus(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] top(mark(X)) = [0 0 0]X >= [0 0 0]X = top(proper(X)) [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(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) 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 0] [0] [ok](x0) = [1 0 0]x0 + [1] [0 0 1] [0], [1 0 0] [0] [proper](x0) = [1 0 0]x0 + [1] [0 0 0] [0], [0] [0] = [0] [0], [1 0 0] [s](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [1 0 0] [plus](x0, x1) = [1 0 0]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1] [mark](x0) = [0 0 1]x0 + [0] [0 0 1] [0], [1 0 1] [1 0 0] [1 0 0] [U12](x0, x1, x2) = [1 0 1]x0 + [0 1 0]x1 + [1 0 0]x2 [0 0 1] [0 0 0] [0 0 0] , [1 0 0] [active](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [1 0 0] [1 0 0] [U11](x0, x1, x2) = [1 0 0]x0 + [0 1 0]x1 + [1 0 0]x2 [0 0 0] [0 0 0] [0 0 0] , [0] [tt] = [0] [0] orientation: [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(U11(X1,X2,X3)) = [1 0 0]X1 + [0 1 0]X2 + [1 0 0]X3 >= [1 0 0]X1 + [0 1 0]X2 + [1 0 0]X3 = U11(active(X1),X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(U12(X1,X2,X3)) = [1 0 1]X1 + [0 1 0]X2 + [1 0 0]X3 >= [1 0 0]X1 + [0 1 0]X2 + [1 0 0]X3 = U12(active(X1),X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(s(X)) = [0 1 0]X >= [0 1 0]X = s(active(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(plus(X1,X2)) = [1 0 0]X1 + [0 1 0]X2 >= [1 0 0]X1 + [0 1 0]X2 = plus(active(X1),X2) [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(plus(X1,X2)) = [1 0 0]X1 + [0 1 0]X2 >= [1 0 0]X1 + [0 1 0]X2 = plus(X1,active(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1] U11(mark(X1),X2,X3) = [1 0 0]X1 + [0 1 0]X2 + [1 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = mark(U11(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 0 1] [1 0 0] [1 0 0] [1] [1 0 1] [1 0 0] [1 0 0] [1] U12(mark(X1),X2,X3) = [1 0 1]X1 + [0 1 0]X2 + [1 0 0]X3 + [1] >= [0 0 1]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = mark(U12(X1,X2,X3)) [0 0 1] [0 0 0] [0 0 0] [0] [0 0 1] [0 0 0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] s(mark(X)) = [0 0 1]X + [0] >= [0 0 0]X + [0] = mark(s(X)) [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) = [1 0 0]X1 + [0 1 0]X2 + [1] >= [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)) = [1 0 0]X1 + [0 0 1]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 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] proper(U11(X1,X2,X3)) = [1 0 0]X1 + [1 0 0]X2 + [1 0 0]X3 + [1] >= [1 0 0]X1 + [1 0 0]X2 + [1 0 0]X3 + [1] = U11(proper(X1),proper(X2),proper(X3)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [0] [0] proper(tt()) = [1] >= [1] = ok(tt()) [0] [0] [1 0 1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] proper(U12(X1,X2,X3)) = [1 0 1]X1 + [1 0 0]X2 + [1 0 0]X3 + [1] >= [1 0 0]X1 + [1 0 0]X2 + [1 0 0]X3 + [1] = U12(proper(X1),proper(X2),proper(X3)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] proper(s(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = s(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] proper(plus(X1,X2)) = [1 0 0]X1 + [1 0 0]X2 + [1] >= [1 0 0]X1 + [1 0 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()) = [1] >= [1] = ok(0()) [0] [0] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] U11(ok(X1),ok(X2),ok(X3)) = [1 0 0]X1 + [1 0 0]X2 + [1 0 0]X3 + [1] >= [1 0 0]X1 + [1 0 0]X2 + [1 0 0]X3 + [1] = ok(U11(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 0 1] [1 0 0] [1 0 0] [0] [1 0 1] [1 0 0] [1 0 0] [0] U12(ok(X1),ok(X2),ok(X3)) = [1 0 1]X1 + [1 0 0]X2 + [1 0 0]X3 + [1] >= [1 0 1]X1 + [1 0 0]X2 + [1 0 0]X3 + [1] = ok(U12(X1,X2,X3)) [0 0 1] [0 0 0] [0 0 0] [0] [0 0 1] [0 0 0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] s(ok(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = ok(s(X)) [0 0 0] [0] [0 0 0] [0] [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] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 1] [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 1] [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(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1),X2) -> mark(plus(X1,X2)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = 2x0, [ok](x0) = 2x0, [proper](x0) = x0, [0] = 0, [s](x0) = x0, [plus](x0, x1) = 4x0 + x1, [mark](x0) = x0 + 2, [U12](x0, x1, x2) = x0 + 2x1 + 2x2, [active](x0) = x0, [U11](x0, x1, x2) = 2x0 + 4x1 + 4x2, [tt] = 0 orientation: active(U11(X1,X2,X3)) = 2X1 + 4X2 + 4X3 >= 2X1 + 4X2 + 4X3 = U11(active(X1),X2,X3) active(U12(X1,X2,X3)) = X1 + 2X2 + 2X3 >= X1 + 2X2 + 2X3 = U12(active(X1),X2,X3) active(s(X)) = X >= X = s(active(X)) active(plus(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = plus(active(X1),X2) active(plus(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = plus(X1,active(X2)) U11(mark(X1),X2,X3) = 2X1 + 4X2 + 4X3 + 4 >= 2X1 + 4X2 + 4X3 + 2 = mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) = X1 + 2X2 + 2X3 + 2 >= X1 + 2X2 + 2X3 + 2 = mark(U12(X1,X2,X3)) s(mark(X)) = X + 2 >= X + 2 = mark(s(X)) plus(mark(X1),X2) = 4X1 + X2 + 8 >= 4X1 + X2 + 2 = mark(plus(X1,X2)) plus(X1,mark(X2)) = 4X1 + X2 + 2 >= 4X1 + X2 + 2 = mark(plus(X1,X2)) proper(U11(X1,X2,X3)) = 2X1 + 4X2 + 4X3 >= 2X1 + 4X2 + 4X3 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2,X3)) = X1 + 2X2 + 2X3 >= X1 + 2X2 + 2X3 = U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X >= X = s(proper(X)) proper(plus(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = plus(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) U11(ok(X1),ok(X2),ok(X3)) = 4X1 + 8X2 + 8X3 >= 4X1 + 8X2 + 8X3 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = 2X1 + 4X2 + 4X3 >= 2X1 + 4X2 + 4X3 = ok(U12(X1,X2,X3)) s(ok(X)) = 2X >= 2X = ok(s(X)) plus(ok(X1),ok(X2)) = 8X1 + 2X2 >= 8X1 + 2X2 = ok(plus(X1,X2)) top(ok(X)) = 4X >= 2X = top(active(X)) problem: active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) active(plus(X1,X2)) -> plus(X1,active(X2)) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) 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 1] [0] [ok](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 1] [0] [proper](x0) = [0 0 0]x0 + [1] [0 0 1] [0], [0] [0] = [0] [0], [s](x0) = x0 , [1 1 0] [0] [plus](x0, x1) = [0 0 0]x0 + x1 + [0] [0 0 1] [1], [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [U12](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + x2 [0 0 1] [0 0 1] , [1 0 1] [0] [active](x0) = [0 1 0]x0 + [1] [0 0 1] [0], [1 0 0] [1 1 0] [0] [U11](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + x2 + [0] [0 0 1] [0 0 1] [1], [0] [tt] = [0] [0] orientation: [1 0 1] [1 1 1] [1 0 1] [1] [1 0 1] [1 1 0] [0] active(U11(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 1 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + X3 + [0] = U11(active(X1),X2,X3) [0 0 1] [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] [1 0 1] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 0] active(U12(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 1 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + X3 = U12(active(X1),X2,X3) [0 0 1] [0 0 1] [0 0 1] [0] [0 0 1] [0 0 1] [1 0 1] [0] [1 0 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 1 1] [1 0 1] [1] [1 1 1] [1] active(plus(X1,X2)) = [0 0 0]X1 + [0 1 0]X2 + [1] >= [0 0 0]X1 + X2 + [0] = plus(active(X1),X2) [0 0 1] [0 0 1] [1] [0 0 1] [1] [1 1 1] [1 0 1] [1] [1 1 0] [1 0 1] [0] active(plus(X1,X2)) = [0 0 0]X1 + [0 1 0]X2 + [1] >= [0 0 0]X1 + [0 1 0]X2 + [1] = plus(X1,active(X2)) [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] U12(mark(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = mark(U12(X1,X2,X3)) [0 0 0] [0 0 1] [0 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 0] [0 0 0] [1 1 0] [1 0 0] [0] [1 1 0] [1 0 0] plus(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = mark(plus(X1,X2)) [0 0 1] [0 0 0] [1] [0 0 0] [0 0 0] [1 0 1] [1 1 1] [1 0 1] [1] [1 0 1] [1 0 1] [1 0 1] [1] proper(U11(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] = U11(proper(X1),proper(X2),proper(X3)) [0 0 1] [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [0 0 1] [1] [0] [0] proper(tt()) = [1] >= [1] = ok(tt()) [0] [0] [1 0 1] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] [1 0 1] [0] proper(U12(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] = U12(proper(X1),proper(X2),proper(X3)) [0 0 1] [0 0 1] [0 0 1] [0] [0 0 1] [0 0 1] [0 0 1] [0] [1 0 1] [0] [1 0 1] [0] proper(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(proper(X)) [0 0 1] [0] [0 0 1] [0] [1 1 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] proper(plus(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = plus(proper(X1),proper(X2)) [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] [0] [0] proper(0()) = [1] >= [1] = ok(0()) [0] [0] [1 1 1] [1 1 1] [1 1 1] [1] [1 0 1] [1 1 1] [1 1 1] [1] U11(ok(X1),ok(X2),ok(X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] = ok(U11(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [0] [1 1 1] [1 1 1] [1 1 1] [0] [1 0 1] [1 0 1] [1 1 1] [0] U12(ok(X1),ok(X2),ok(X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] = ok(U12(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 1 1] [0] [1 1 1] [0] s(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(s(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1 1 1] [1] [1 1 1] [1 1 1] [1] plus(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = ok(plus(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0] [1 1 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(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) proper(0()) -> ok(0()) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1] [top](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [ok](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [proper](x0) = [1 1 0]x0 [0 0 1] , [0] [0] = [1] [1], [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] , [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [1 0 0] [0] [U12](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [0] [0 0 1] [0 0 1] [0 0 1] [1], [1 0 0] [0] [active](x0) = [0 0 0]x0 + [1] [0 0 1] [0], [1 0 0] [1 0 0] [1 0 0] [U11](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 1] [0 0 1] [0 0 1] , [1] [tt] = [0] [0] orientation: [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] active(U12(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = U12(active(X1),X2,X3) [0 0 1] [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [0 0 1] [1] [1 0 0] [0] [1 0 0] active(s(X)) = [0 0 0]X + [1] >= [0 0 0]X = s(active(X)) [0 0 1] [0] [0 0 1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] active(plus(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 = plus(active(X1),X2) [0 0 1] [0 0 1] [0] [0 0 1] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] U12(mark(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = mark(U12(X1,X2,X3)) [0 0 0] [0 0 1] [0 0 1] [1] [0 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 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] plus(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(plus(X1,X2)) [0 0 1] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] [1 0 1] [1 0 1] proper(U11(X1,X2,X3)) = [1 0 0]X1 + [1 0 0]X2 + [1 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = U11(proper(X1),proper(X2),proper(X3)) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1] [1] proper(tt()) = [1] >= [0] = ok(tt()) [0] [0] [1 0 1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1 0 1] [0] proper(U12(X1,X2,X3)) = [1 0 0]X1 + [1 0 0]X2 + [1 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = U12(proper(X1),proper(X2),proper(X3)) [0 0 1] [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [0 0 1] [1] [1 0 1] [1 0 1] proper(s(X)) = [1 0 0]X >= [0 0 0]X = s(proper(X)) [0 0 1] [0 0 1] [1 0 1] [1 0 1] [1 0 1] [1 0 1] proper(plus(X1,X2)) = [1 0 0]X1 + [1 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = plus(proper(X1),proper(X2)) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1] [0] proper(0()) = [1] >= [0] = ok(0()) [1] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] U11(ok(X1),ok(X2),ok(X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = ok(U11(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] U12(ok(X1),ok(X2),ok(X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = ok(U12(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [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] [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] [1] [1 0 0] [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(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = x0 + 4, [ok](x0) = 4x0 + 3, [proper](x0) = 4x0 + 3, [s](x0) = 3x0 + 2, [plus](x0, x1) = x0 + x1 + 1, [mark](x0) = x0, [U12](x0, x1, x2) = x0 + 5x1 + x2 + 6, [active](x0) = x0, [U11](x0, x1, x2) = 2x0 + 2x1 + 4x2 + 7, [tt] = 0 orientation: active(U12(X1,X2,X3)) = X1 + 5X2 + X3 + 6 >= X1 + 5X2 + X3 + 6 = U12(active(X1),X2,X3) active(s(X)) = 3X + 2 >= 3X + 2 = s(active(X)) active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2) U12(mark(X1),X2,X3) = X1 + 5X2 + X3 + 6 >= X1 + 5X2 + X3 + 6 = mark(U12(X1,X2,X3)) s(mark(X)) = 3X + 2 >= 3X + 2 = mark(s(X)) plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2)) proper(U11(X1,X2,X3)) = 8X1 + 8X2 + 16X3 + 31 >= 8X1 + 8X2 + 16X3 + 31 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 3 >= 3 = ok(tt()) proper(s(X)) = 12X + 11 >= 12X + 11 = s(proper(X)) proper(plus(X1,X2)) = 4X1 + 4X2 + 7 >= 4X1 + 4X2 + 7 = plus(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3)) = 8X1 + 8X2 + 16X3 + 31 >= 8X1 + 8X2 + 16X3 + 31 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = 4X1 + 20X2 + 4X3 + 27 >= 4X1 + 20X2 + 4X3 + 27 = ok(U12(X1,X2,X3)) s(ok(X)) = 12X + 11 >= 12X + 11 = ok(s(X)) plus(ok(X1),ok(X2)) = 4X1 + 4X2 + 7 >= 4X1 + 4X2 + 7 = ok(plus(X1,X2)) top(ok(X)) = 4X + 7 >= X + 4 = top(active(X)) problem: active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) active(plus(X1,X2)) -> plus(active(X1),X2) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [ok](x0) = [1 0 0]x0 [0 0 0] , [1 0 0] [proper](x0) = [0 1 1]x0 [0 0 1] , [1 0 0] [0] [s](x0) = [1 0 0]x0 + [0] [0 0 1] [1], [1 0 0] [1] [plus](x0, x1) = x0 + [1 0 0]x1 + [1] [0 0 0] [0], [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [1 0 0] [0] [U12](x0, x1, x2) = [1 0 0]x0 + [1 0 0]x1 + [1 0 0]x2 + [1] [0 0 1] [0 0 0] [0 0 0] [1], [1 0 1] [active](x0) = [1 0 1]x0 [0 0 1] , [1 0 0] [1 0 0] [U11](x0, x1, x2) = [1 0 1]x0 + x1 + [1 0 0]x2 [0 0 0] [0 0 0] , [0] [tt] = [0] [0] orientation: [1 0 1] [1 0 0] [1 0 0] [1] [1 0 1] [1 0 0] [1 0 0] [0] active(U12(X1,X2,X3)) = [1 0 1]X1 + [1 0 0]X2 + [1 0 0]X3 + [1] >= [1 0 1]X1 + [1 0 0]X2 + [1 0 0]X3 + [1] = U12(active(X1),X2,X3) [0 0 1] [0 0 0] [0 0 0] [1] [0 0 1] [0 0 0] [0 0 0] [1] [1 0 1] [1] [1 0 1] [0] active(s(X)) = [1 0 1]X + [1] >= [1 0 1]X + [0] = s(active(X)) [0 0 1] [1] [0 0 1] [1] [1 0 1] [1 0 0] [1] [1 0 1] [1 0 0] [1] active(plus(X1,X2)) = [1 0 1]X1 + [1 0 0]X2 + [1] >= [1 0 1]X1 + [1 0 0]X2 + [1] = plus(active(X1),X2) [0 0 1] [0 0 0] [0] [0 0 1] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] U12(mark(X1),X2,X3) = [1 0 0]X1 + [1 0 0]X2 + [1 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = mark(U12(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] s(mark(X)) = [1 0 0]X + [0] >= [0 0 0]X = mark(s(X)) [0 0 0] [1] [0 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] plus(X1,mark(X2)) = X1 + [1 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0] = mark(plus(X1,X2)) [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(U11(X1,X2,X3)) = [1 0 1]X1 + [0 1 1]X2 + [1 0 0]X3 >= [1 0 1]X1 + [0 1 1]X2 + [1 0 0]X3 = U11(proper(X1),proper(X2),proper(X3)) [0 0 0] [0 0 1] [0 0 0] [0 0 0] [0 0 1] [0 0 0] [0] [0] proper(tt()) = [0] >= [0] = ok(tt()) [0] [0] [1 0 0] [0] [1 0 0] [0] proper(s(X)) = [1 0 1]X + [1] >= [1 0 0]X + [0] = s(proper(X)) [0 0 1] [1] [0 0 1] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] proper(plus(X1,X2)) = [0 1 1]X1 + [1 0 0]X2 + [1] >= [0 1 1]X1 + [1 0 0]X2 + [1] = plus(proper(X1),proper(X2)) [0 0 1] [0 0 0] [0] [0 0 1] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] U11(ok(X1),ok(X2),ok(X3)) = [1 0 0]X1 + [1 0 0]X2 + [1 0 0]X3 >= [1 0 0]X1 + [1 0 0]X2 + [1 0 0]X3 = ok(U11(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] U12(ok(X1),ok(X2),ok(X3)) = [1 0 0]X1 + [1 0 0]X2 + [1 0 0]X3 + [1] >= [1 0 0]X1 + [1 0 0]X2 + [1 0 0]X3 = ok(U12(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] s(ok(X)) = [1 0 0]X + [0] >= [1 0 0]X = ok(s(X)) [0 0 0] [1] [0 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] 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] [0 0 0] [0] [0 0 0] [0 0 0] [0] problem: active(plus(X1,X2)) -> plus(active(X1),X2) U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [ok](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [proper](x0) = [0 1 0]x0 [0 1 0] , [1 1 0] [s](x0) = [1 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [0] [plus](x0, x1) = [0 1 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [0], [1 0 0] [mark](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [1 0 0] [1 0 0] [1] [U12](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [1] [0 0 1] [0 0 0] [0 0 0] [0], [1 1 0] [active](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [1 0 0] [U11](x0, x1, x2) = x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 0] [0 0 0] , [0] [tt] = [1] [0] orientation: [1 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [0] active(plus(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [1] = plus(active(X1),X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1] U12(mark(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] = mark(U12(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 1 0] [1 1 0] s(mark(X)) = [1 0 0]X >= [1 0 0]X = mark(s(X)) [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 0 0]X2 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [1] = 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 0 0] [1 0 0] [1 0 0] [1 0 0] proper(U11(X1,X2,X3)) = [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 = U11(proper(X1),proper(X2),proper(X3)) [0 1 0] [0 0 0] [0 0 0] [0 1 0] [0 0 0] [0 0 0] [0] [0] proper(tt()) = [1] >= [1] = ok(tt()) [1] [0] [1 1 0] [1 1 0] proper(s(X)) = [1 0 0]X >= [1 0 0]X = s(proper(X)) [1 0 0] [0 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] proper(plus(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [1] = plus(proper(X1),proper(X2)) [0 1 0] [0 0 0] [1] [0 0 0] [0 0 0] [0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] [1 0 0] [1 0 0] U11(ok(X1),ok(X2),ok(X3)) = [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 = ok(U11(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 0] [1 0 0] [1] U12(ok(X1),ok(X2),ok(X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] = ok(U12(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 1 1] [1 1 0] s(ok(X)) = [1 0 1]X >= [1 0 0]X = ok(s(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] [0] [1 0 0] [1 0 0] [0] plus(ok(X1),ok(X2)) = [0 1 0]X1 + [0 0 0]X2 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [1] = ok(plus(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] problem: U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) s(mark(X)) -> mark(s(X)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) Matrix Interpretation Processor: dim=1 interpretation: [ok](x0) = 4x0, [proper](x0) = 2x0, [s](x0) = x0, [plus](x0, x1) = 4x0 + x1, [mark](x0) = x0 + 4, [U12](x0, x1, x2) = 5x0 + x1 + x2, [U11](x0, x1, x2) = 4x0 + 2x1 + 4x2, [tt] = 0 orientation: U12(mark(X1),X2,X3) = 5X1 + X2 + X3 + 20 >= 5X1 + X2 + X3 + 4 = mark(U12(X1,X2,X3)) s(mark(X)) = X + 4 >= X + 4 = mark(s(X)) plus(X1,mark(X2)) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = mark(plus(X1,X2)) proper(U11(X1,X2,X3)) = 8X1 + 4X2 + 8X3 >= 8X1 + 4X2 + 8X3 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = 2X >= 2X = s(proper(X)) proper(plus(X1,X2)) = 8X1 + 2X2 >= 8X1 + 2X2 = plus(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3)) = 16X1 + 8X2 + 16X3 >= 16X1 + 8X2 + 16X3 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = 20X1 + 4X2 + 4X3 >= 20X1 + 4X2 + 4X3 = ok(U12(X1,X2,X3)) s(ok(X)) = 4X >= 4X = ok(s(X)) plus(ok(X1),ok(X2)) = 16X1 + 4X2 >= 16X1 + 4X2 = ok(plus(X1,X2)) problem: s(mark(X)) -> mark(s(X)) plus(X1,mark(X2)) -> mark(plus(X1,X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) Matrix Interpretation Processor: dim=1 interpretation: [ok](x0) = 4x0 + 3, [proper](x0) = 4x0 + 3, [s](x0) = 6x0 + 5, [plus](x0, x1) = x0 + 2x1 + 2, [mark](x0) = x0 + 1, [U12](x0, x1, x2) = 2x0 + 2x1 + x2 + 4, [U11](x0, x1, x2) = x0 + 4x1 + x2 + 5, [tt] = 2 orientation: s(mark(X)) = 6X + 11 >= 6X + 6 = mark(s(X)) plus(X1,mark(X2)) = X1 + 2X2 + 4 >= X1 + 2X2 + 3 = mark(plus(X1,X2)) proper(U11(X1,X2,X3)) = 4X1 + 16X2 + 4X3 + 23 >= 4X1 + 16X2 + 4X3 + 23 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 11 >= 11 = ok(tt()) proper(s(X)) = 24X + 23 >= 24X + 23 = s(proper(X)) proper(plus(X1,X2)) = 4X1 + 8X2 + 11 >= 4X1 + 8X2 + 11 = plus(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3)) = 4X1 + 16X2 + 4X3 + 23 >= 4X1 + 16X2 + 4X3 + 23 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = 8X1 + 8X2 + 4X3 + 19 >= 8X1 + 8X2 + 4X3 + 19 = ok(U12(X1,X2,X3)) s(ok(X)) = 24X + 23 >= 24X + 23 = ok(s(X)) plus(ok(X1),ok(X2)) = 4X1 + 8X2 + 11 >= 4X1 + 8X2 + 11 = ok(plus(X1,X2)) problem: proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [0] [ok](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 1] [0] [proper](x0) = [0 0 0]x0 + [1] [0 0 1] [0], [s](x0) = x0 , [1 0 0] [plus](x0, x1) = [0 0 0]x0 + x1 [0 0 1] , [1 0 0] [1 0 0] [1 0 0] [1] [U12](x0, x1, x2) = [0 0 0]x0 + [0 1 0]x1 + [0 0 0]x2 + [0] [0 0 0] [0 0 0] [0 0 0] [0], [1 1 0] [1 0 0] [0] [U11](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + x2 + [0] [0 0 1] [0 0 1] [1], [0] [tt] = [0] [1] orientation: [1 1 1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1 0 1] [1] proper(U11(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] = U11(proper(X1),proper(X2),proper(X3)) [0 0 1] [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [0 0 1] [1] [1] [0] proper(tt()) = [1] >= [1] = ok(tt()) [1] [0] [1 0 1] [0] [1 0 1] [0] proper(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(proper(X)) [0 0 1] [0] [0 0 1] [0] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] [0] proper(plus(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = plus(proper(X1),proper(X2)) [0 0 1] [0 0 1] [0] [0 0 1] [0 0 1] [0] [1 1 0] [1 1 0] [1 1 0] [1] [1 1 0] [1 0 0] [1 1 0] [0] U11(ok(X1),ok(X2),ok(X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] = ok(U11(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [0] [1 1 0] [1 1 0] [1 1 0] [1] [1 0 0] [1 1 0] [1 0 0] [1] U12(ok(X1),ok(X2),ok(X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] = ok(U12(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 1 0] [0] [1 1 0] [0] s(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(s(X)) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 1 0] [0] [1 0 0] [1 1 0] [0] plus(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = ok(plus(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] problem: proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) 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 + [1] [0 0 0] [0], [1 0 1] [proper](x0) = [0 1 0]x0 [0 0 1] , [1 0 0] [0] [s](x0) = [0 0 0]x0 + [1] [0 1 1] [0], [1 0 0] [plus](x0, x1) = [0 0 0]x0 + x1 [0 0 1] , [1 0 0] [1 0 0] [1 0 0] [U12](x0, x1, x2) = [1 0 0]x0 + [0 1 0]x1 + [0 0 0]x2 [0 0 0] [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [1 0 0] [0] [U11](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [0] [0 1 1] [0 0 1] [0 0 1] [1] orientation: [1 1 1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1 0 1] [0] proper(U11(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = U11(proper(X1),proper(X2),proper(X3)) [0 1 1] [0 0 1] [0 0 1] [1] [0 1 1] [0 0 1] [0 0 1] [1] [1 1 1] [0] [1 0 1] [0] proper(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(proper(X)) [0 1 1] [0] [0 1 1] [0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] proper(plus(X1,X2)) = [0 0 0]X1 + [0 1 0]X2 >= [0 0 0]X1 + [0 1 0]X2 = plus(proper(X1),proper(X2)) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] U12(ok(X1),ok(X2),ok(X3)) = [1 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] = ok(U12(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] s(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(s(X)) [0 0 0] [1] [0 0 0] [0] [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 + [1] = ok(plus(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] problem: proper(s(X)) -> s(proper(X)) proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [ok](x0) = [0 0 0]x0 [0 0 1] , [1 1 0] [proper](x0) = [0 1 0]x0 [0 1 0] , [1 0 0] [0] [s](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 0] [1 0 0] [0] [plus](x0, x1) = [0 1 0]x0 + [0 1 0]x1 + [1] [0 0 0] [0 0 0] [0], [1 0 0] [1 0 0] [1 0 1] [0] [U12](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [1 0 0]x2 + [1] [0 0 0] [0 0 0] [0 0 0] [0] orientation: [1 1 0] [1] [1 1 0] [0] proper(s(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = s(proper(X)) [0 1 0] [1] [0 0 0] [0] [1 1 0] [1 1 0] [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 1 0] [0 1 0] [1] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 0] [1 0 1] U12(ok(X1),ok(X2),ok(X3)) = [0 0 0]X1 + [0 0 0]X2 + [1 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = ok(U12(X1,X2,X3)) [0 0 0] [0 0 0] [0 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 + [1] >= [0 0 0]X = ok(s(X)) [0 0 0] [0] [0 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] plus(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 = ok(plus(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] problem: U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [ok](x0) = [0 0 1]x0 + [0] [0 1 0] [1], [1 1 1] [s](x0) = [0 1 0]x0 [0 0 1] , [1 1 1] [1 0 0] [0] [plus](x0, x1) = [0 0 0]x0 + [0 0 1]x1 + [0] [0 0 0] [0 1 0] [1], [1 1 1] [1 0 0] [U12](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + x2 [0 0 0] [0 0 0] orientation: [1 1 1] [1 0 0] [1 0 0] [1] [1 1 1] [1 0 0] [1 0 0] [0] U12(ok(X1),ok(X2),ok(X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 1]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 1]X3 + [0] = ok(U12(X1,X2,X3)) [0 0 0] [0 0 0] [0 1 0] [1] [0 0 0] [0 0 0] [0 1 0] [1] [1 1 1] [1] [1 1 1] [0] s(ok(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = ok(s(X)) [0 1 0] [1] [0 1 0] [1] [1 1 1] [1] [1 1 1] [0] plus(ok(X1),ok(X2)) = [0 0 0]X1 + X2 + [1] >= [0 0 0]X1 + X2 + [1] = ok(plus(X1,X2)) [0 0 0] [1] [0 0 0] [1] problem: Qed