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)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Proof: Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [mark](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [1 1 0] [1 1 0] [1] [U11](x0, x1, x2) = [0 0 0]x0 + [0 1 0]x1 + [0 1 0]x2 + [1] [0 0 0] [0 0 0] [0 0 0] [0], [1 0 0] [1 1 0] [1 1 0] [U12](x0, x1, x2) = [0 1 0]x0 + [0 1 0]x1 + [0 1 0]x2 [0 0 0] [0 1 0] [0 0 0] , [1 1 0] [1 1 0] [plus](x0, x1) = [0 1 0]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [0] [tt] = [1] [0], [0] [0] = [1] [0], [1 0 0] [0] [s](x0) = [0 1 0]x0 + [1] [0 1 0] [0], [1 0 0] [active](x0) = [0 1 0]x0 [0 0 0] orientation: [1 1 0] [1 1 0] [1] [1 1 0] [1 1 0] [0] active(U11(tt(),M,N)) = [0 1 0]M + [0 1 0]N + [1] >= [0 1 0]M + [0 1 0]N + [1] = mark(U12(tt(),M,N)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 1 0] [0] [1 1 0] [1 1 0] [0] active(U12(tt(),M,N)) = [0 1 0]M + [0 1 0]N + [1] >= [0 1 0]M + [0 1 0]N + [1] = mark(s(plus(N,M))) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1] [1 0 0] active(plus(N,0())) = [0 1 0]N + [1] >= [0 1 0]N = mark(N) [0 0 0] [0] [0 0 0] [1 1 0] [1 1 0] [1] [1 1 0] [1 1 0] [1] active(plus(N,s(M))) = [0 1 0]M + [0 1 0]N + [1] >= [0 1 0]M + [0 1 0]N + [1] = mark(U11(tt(),M,N)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 1 0] [1 1 0] [1] [1 0 0] [1 1 0] [1 1 0] [1] mark(U11(X1,X2,X3)) = [0 0 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] >= [0 0 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] = active(U11(mark(X1),X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [0] [0] mark(tt()) = [1] >= [1] = active(tt()) [0] [0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] mark(U12(X1,X2,X3)) = [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 >= [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 = active(U12(mark(X1),X2,X3)) [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] mark(s(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = active(s(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] mark(plus(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = active(plus(mark(X1),mark(X2))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] mark(0()) = [1] >= [1] = active(0()) [0] [0] [1 0 0] [1 1 0] [1 1 0] [1] [1 0 0] [1 1 0] [1 1 0] [1] U11(mark(X1),X2,X3) = [0 0 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] >= [0 0 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] = 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 0] [1 1 0] [1 1 0] [1] [1 0 0] [1 1 0] [1 1 0] [1] U11(X1,mark(X2),X3) = [0 0 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] >= [0 0 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] = 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 0] [1 1 0] [1 1 0] [1] [1 0 0] [1 1 0] [1 1 0] [1] U11(X1,X2,mark(X3)) = [0 0 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] >= [0 0 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] = 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 0] [1 1 0] [1 1 0] [1] [1 0 0] [1 1 0] [1 1 0] [1] U11(active(X1),X2,X3) = [0 0 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] >= [0 0 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] = 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 0] [1 1 0] [1 1 0] [1] [1 0 0] [1 1 0] [1 1 0] [1] U11(X1,active(X2),X3) = [0 0 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] >= [0 0 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] = 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 0] [1 1 0] [1 1 0] [1] [1 0 0] [1 1 0] [1 1 0] [1] U11(X1,X2,active(X3)) = [0 0 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] >= [0 0 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] = 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 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] U12(mark(X1),X2,X3) = [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 >= [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 = U12(X1,X2,X3) [0 0 0] [0 1 0] [0 0 0] [0 0 0] [0 1 0] [0 0 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] U12(X1,mark(X2),X3) = [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 >= [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 = U12(X1,X2,X3) [0 0 0] [0 1 0] [0 0 0] [0 0 0] [0 1 0] [0 0 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] U12(X1,X2,mark(X3)) = [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 >= [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 = U12(X1,X2,X3) [0 0 0] [0 1 0] [0 0 0] [0 0 0] [0 1 0] [0 0 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] U12(active(X1),X2,X3) = [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 >= [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 = U12(X1,X2,X3) [0 0 0] [0 1 0] [0 0 0] [0 0 0] [0 1 0] [0 0 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] U12(X1,active(X2),X3) = [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 >= [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 = U12(X1,X2,X3) [0 0 0] [0 1 0] [0 0 0] [0 0 0] [0 1 0] [0 0 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] U12(X1,X2,active(X3)) = [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 >= [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 = U12(X1,X2,X3) [0 0 0] [0 1 0] [0 0 0] [0 0 0] [0 1 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] = s(X) [0 1 0] [0] [0 1 0] [0] [1 0 0] [0] [1 0 0] [0] s(active(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = s(X) [0 1 0] [0] [0 1 0] [0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] plus(mark(X1),X2) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = plus(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] plus(X1,mark(X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = plus(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] plus(active(X1),X2) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = plus(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] plus(X1,active(X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = plus(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] problem: active(U12(tt(),M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt(),M,N)) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [mark](x0) = [0 0 0]x0 [0 1 0] , [1 1 1] [1 0 0] [1 1 1] [U11](x0, x1, x2) = [1 1 1]x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 0] [0 0 0] [0 0 0] , [1 1 1] [1 1 1] [1 1 1] [0] [U12](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [1] [0 0 0] [0 0 0] [0 0 0] [0], [1 1 1] [1 1 1] [plus](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [tt] = [0] [0], [0] [0] = [0] [0], [1 1 1] [0] [s](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 0] [active](x0) = [0 0 1]x0 [0 1 0] orientation: [1 1 1] [1 1 1] [0] [1 1 1] [1 1 1] [0] active(U12(tt(),M,N)) = [0 0 0]M + [0 0 0]N + [0] >= [0 0 0]M + [0 0 0]N + [0] = mark(s(plus(N,M))) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 1] [1 1 1] [1] [1 0 0] [1 1 1] active(plus(N,s(M))) = [0 0 0]M + [0 0 0]N + [0] >= [0 0 0]M + [0 0 0]N = mark(U11(tt(),M,N)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [1 1 1] [1 0 0] [1 1 1] [1 1 1] [1 0 0] [1 1 1] mark(U11(X1,X2,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 = active(U11(mark(X1),X2,X3)) [1 1 1] [0 0 0] [0 0 0] [1 1 1] [0 0 0] [0 0 0] [0] [0] mark(tt()) = [0] >= [0] = active(tt()) [0] [0] [1 1 1] [1 1 1] [1 1 1] [0] [1 1 1] [1 1 1] [1 1 1] [0] mark(U12(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] = active(U12(mark(X1),X2,X3)) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [1 1 1] [0] [1 1 1] [0] mark(s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(s(mark(X))) [0 0 0] [1] [0 0 0] [1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] mark(plus(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(plus(mark(X1),mark(X2))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] mark(0()) = [0] >= [0] = active(0()) [0] [0] [1 1 1] [1 0 0] [1 1 1] [1 1 1] [1 0 0] [1 1 1] U11(mark(X1),X2,X3) = [1 1 1]X1 + [0 0 0]X2 + [0 0 0]X3 >= [1 1 1]X1 + [0 0 0]X2 + [0 0 0]X3 = U11(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 0 1] [1 1 1] [1 1 1] [1 0 0] [1 1 1] U11(X1,mark(X2),X3) = [1 1 1]X1 + [0 0 0]X2 + [0 0 0]X3 >= [1 1 1]X1 + [0 0 0]X2 + [0 0 0]X3 = U11(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 0 0] [1 1 1] [1 1 1] [1 0 0] [1 1 1] U11(X1,X2,mark(X3)) = [1 1 1]X1 + [0 0 0]X2 + [0 0 0]X3 >= [1 1 1]X1 + [0 0 0]X2 + [0 0 0]X3 = U11(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 0 0] [1 1 1] [1 1 1] [1 0 0] [1 1 1] U11(active(X1),X2,X3) = [1 1 1]X1 + [0 0 0]X2 + [0 0 0]X3 >= [1 1 1]X1 + [0 0 0]X2 + [0 0 0]X3 = U11(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 0 0] [1 1 1] [1 1 1] [1 0 0] [1 1 1] U11(X1,active(X2),X3) = [1 1 1]X1 + [0 0 0]X2 + [0 0 0]X3 >= [1 1 1]X1 + [0 0 0]X2 + [0 0 0]X3 = U11(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 0 0] [1 1 1] [1 1 1] [1 0 0] [1 1 1] U11(X1,X2,active(X3)) = [1 1 1]X1 + [0 0 0]X2 + [0 0 0]X3 >= [1 1 1]X1 + [0 0 0]X2 + [0 0 0]X3 = U11(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 1 1] [1 1 1] [0] [1 1 1] [1 1 1] [1 1 1] [0] 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] = 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 1] [1 1 1] [0] [1 1 1] [1 1 1] [1 1 1] [0] U12(X1,mark(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(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 1] [1 1 1] [0] [1 1 1] [1 1 1] [1 1 1] [0] U12(X1,X2,mark(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(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 1] [1 1 1] [0] [1 1 1] [1 1 1] [1 1 1] [0] U12(active(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(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 1] [1 1 1] [0] [1 1 1] [1 1 1] [1 1 1] [0] U12(X1,active(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(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 1] [1 1 1] [0] [1 1 1] [1 1 1] [1 1 1] [0] U12(X1,X2,active(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(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(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(X) [0 0 0] [0] [0 0 0] [0] [1 1 1] [0] [1 1 1] [0] s(active(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(X) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1 1 1] [1 1 1] [1 1 1] plus(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = plus(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 1 1] [1 1 1] [1 1 1] plus(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = plus(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 1 1] [1 1 1] [1 1 1] plus(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = plus(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 1 1] [1 1 1] [1 1 1] plus(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = plus(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] problem: active(U12(tt(),M,N)) -> mark(s(plus(N,M))) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = 2x0, [U11](x0, x1, x2) = 4x0 + 4x1 + x2 + 1, [U12](x0, x1, x2) = 6x0 + 2x1 + 2x2 + 2, [plus](x0, x1) = x0 + x1 + 1, [tt] = 1, [0] = 1, [s](x0) = x0 + 1, [active](x0) = x0 + 1 orientation: active(U12(tt(),M,N)) = 2M + 2N + 9 >= 2M + 2N + 4 = mark(s(plus(N,M))) mark(U11(X1,X2,X3)) = 8X1 + 8X2 + 2X3 + 2 >= 8X1 + 4X2 + X3 + 2 = active(U11(mark(X1),X2,X3)) mark(tt()) = 2 >= 2 = active(tt()) mark(U12(X1,X2,X3)) = 12X1 + 4X2 + 4X3 + 4 >= 12X1 + 2X2 + 2X3 + 3 = active(U12(mark(X1),X2,X3)) mark(s(X)) = 2X + 2 >= 2X + 2 = active(s(mark(X))) mark(plus(X1,X2)) = 2X1 + 2X2 + 2 >= 2X1 + 2X2 + 2 = active(plus(mark(X1),mark(X2))) mark(0()) = 2 >= 2 = active(0()) U11(mark(X1),X2,X3) = 8X1 + 4X2 + X3 + 1 >= 4X1 + 4X2 + X3 + 1 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 4X1 + 8X2 + X3 + 1 >= 4X1 + 4X2 + X3 + 1 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 4X1 + 4X2 + 2X3 + 1 >= 4X1 + 4X2 + X3 + 1 = U11(X1,X2,X3) U11(active(X1),X2,X3) = 4X1 + 4X2 + X3 + 5 >= 4X1 + 4X2 + X3 + 1 = U11(X1,X2,X3) U11(X1,active(X2),X3) = 4X1 + 4X2 + X3 + 5 >= 4X1 + 4X2 + X3 + 1 = U11(X1,X2,X3) U11(X1,X2,active(X3)) = 4X1 + 4X2 + X3 + 2 >= 4X1 + 4X2 + X3 + 1 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 12X1 + 2X2 + 2X3 + 2 >= 6X1 + 2X2 + 2X3 + 2 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 6X1 + 4X2 + 2X3 + 2 >= 6X1 + 2X2 + 2X3 + 2 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 6X1 + 2X2 + 4X3 + 2 >= 6X1 + 2X2 + 2X3 + 2 = U12(X1,X2,X3) U12(active(X1),X2,X3) = 6X1 + 2X2 + 2X3 + 8 >= 6X1 + 2X2 + 2X3 + 2 = U12(X1,X2,X3) U12(X1,active(X2),X3) = 6X1 + 2X2 + 2X3 + 4 >= 6X1 + 2X2 + 2X3 + 2 = U12(X1,X2,X3) U12(X1,X2,active(X3)) = 6X1 + 2X2 + 2X3 + 4 >= 6X1 + 2X2 + 2X3 + 2 = U12(X1,X2,X3) s(mark(X)) = 2X + 1 >= X + 1 = s(X) s(active(X)) = X + 2 >= X + 1 = s(X) plus(mark(X1),X2) = 2X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,mark(X2)) = X1 + 2X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 1 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 + 2 >= X1 + X2 + 1 = plus(X1,X2) problem: mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) s(mark(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = 4x0, [U11](x0, x1, x2) = 4x0 + 2x1 + 4x2, [U12](x0, x1, x2) = x0 + x1 + 4x2, [plus](x0, x1) = x0 + 4x1, [tt] = 4, [0] = 2, [s](x0) = 2x0 + 2, [active](x0) = x0 orientation: mark(U11(X1,X2,X3)) = 16X1 + 8X2 + 16X3 >= 16X1 + 2X2 + 4X3 = active(U11(mark(X1),X2,X3)) mark(tt()) = 16 >= 4 = active(tt()) mark(s(X)) = 8X + 8 >= 8X + 2 = active(s(mark(X))) mark(plus(X1,X2)) = 4X1 + 16X2 >= 4X1 + 16X2 = active(plus(mark(X1),mark(X2))) mark(0()) = 8 >= 2 = active(0()) U11(mark(X1),X2,X3) = 16X1 + 2X2 + 4X3 >= 4X1 + 2X2 + 4X3 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = 4X1 + 8X2 + 4X3 >= 4X1 + 2X2 + 4X3 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = 4X1 + 2X2 + 16X3 >= 4X1 + 2X2 + 4X3 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 4X1 + X2 + 4X3 >= X1 + X2 + 4X3 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = X1 + 4X2 + 4X3 >= X1 + X2 + 4X3 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = X1 + X2 + 16X3 >= X1 + X2 + 4X3 = U12(X1,X2,X3) s(mark(X)) = 8X + 2 >= 2X + 2 = s(X) plus(mark(X1),X2) = 4X1 + 4X2 >= X1 + 4X2 = plus(X1,X2) plus(X1,mark(X2)) = X1 + 16X2 >= X1 + 4X2 = plus(X1,X2) problem: mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) s(mark(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [mark](x0) = [0 0 1]x0 [0 0 1] , [1 0 0] [1 0 0] [1 0 0] [U11](x0, x1, x2) = [1 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 1] [0 0 1] [0 0 0] , [1 0 0] [1 0 0] [1 0 0] [1] [U12](x0, x1, x2) = [0 0 0]x0 + [1 0 0]x1 + [0 0 0]x2 + [0] [0 0 1] [0 0 0] [0 0 0] [0], [1 0 0] [1 0 0] [0] [plus](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 1] [0 0 1] [1], [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [active](x0) = [0 0 0]x0 [0 0 0] orientation: [1 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 0] [1 0 0] mark(U11(X1,X2,X3)) = [0 0 1]X1 + [0 0 1]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = active(U11(mark(X1),X2,X3)) [0 0 1] [0 0 1] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] mark(plus(X1,X2)) = [0 0 1]X1 + [0 0 1]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 = active(plus(mark(X1),mark(X2))) [0 0 1] [0 0 1] [1] [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] U11(mark(X1),X2,X3) = [1 0 1]X1 + [0 0 0]X2 + [0 0 0]X3 >= [1 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = U11(X1,X2,X3) [0 0 1] [0 0 1] [0 0 0] [0 0 1] [0 0 1] [0 0 0] [1 0 0] [1 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] U11(X1,mark(X2),X3) = [1 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [1 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = U11(X1,X2,X3) [0 0 1] [0 0 1] [0 0 0] [0 0 1] [0 0 1] [0 0 0] [1 0 0] [1 0 0] [1 0 1] [1 0 0] [1 0 0] [1 0 0] U11(X1,X2,mark(X3)) = [1 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [1 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = U11(X1,X2,X3) [0 0 1] [0 0 1] [0 0 0] [0 0 1] [0 0 1] [0 0 0] [1 0 1] [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 + [1 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [0] = 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 0 1] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1] U12(X1,mark(X2),X3) = [0 0 0]X1 + [1 0 1]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [0] = 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 0 0] [1 0 1] [1] [1 0 0] [1 0 0] [1 0 0] [1] U12(X1,X2,mark(X3)) = [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [0] = 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 1] [1 0 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 0 1] [1 0 0] [0] [1 0 0] [1 0 0] [0] plus(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = plus(X1,X2) [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 0] [0] plus(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = plus(X1,X2) [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] problem: mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) s(mark(X)) -> s(X) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = x0 + 3, [U11](x0, x1, x2) = x0 + x1 + 6x2, [U12](x0, x1, x2) = 4x0 + 4x1 + 4x2, [plus](x0, x1) = 4x0 + 4x1, [s](x0) = 4x0, [active](x0) = x0 orientation: mark(U11(X1,X2,X3)) = X1 + X2 + 6X3 + 3 >= X1 + X2 + 6X3 + 3 = active(U11(mark(X1),X2,X3)) U11(mark(X1),X2,X3) = X1 + X2 + 6X3 + 3 >= X1 + X2 + 6X3 = U11(X1,X2,X3) U11(X1,mark(X2),X3) = X1 + X2 + 6X3 + 3 >= X1 + X2 + 6X3 = U11(X1,X2,X3) U11(X1,X2,mark(X3)) = X1 + X2 + 6X3 + 18 >= X1 + X2 + 6X3 = U11(X1,X2,X3) U12(mark(X1),X2,X3) = 4X1 + 4X2 + 4X3 + 12 >= 4X1 + 4X2 + 4X3 = U12(X1,X2,X3) U12(X1,mark(X2),X3) = 4X1 + 4X2 + 4X3 + 12 >= 4X1 + 4X2 + 4X3 = U12(X1,X2,X3) U12(X1,X2,mark(X3)) = 4X1 + 4X2 + 4X3 + 12 >= 4X1 + 4X2 + 4X3 = U12(X1,X2,X3) s(mark(X)) = 4X + 12 >= 4X = s(X) plus(mark(X1),X2) = 4X1 + 4X2 + 12 >= 4X1 + 4X2 = plus(X1,X2) plus(X1,mark(X2)) = 4X1 + 4X2 + 12 >= 4X1 + 4X2 = plus(X1,X2) problem: mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [mark](x0) = [0 0 1]x0 [0 1 0] , [1 0 1] [1 1 0] [1 1 1] [0] [U11](x0, x1, x2) = [0 0 1]x0 + [0 0 0]x1 + [0 0 0]x2 + [0] [0 1 0] [0 0 0] [0 0 0] [1], [1 0 0] [active](x0) = [0 1 0]x0 [0 0 0] orientation: [1 1 1] [1 1 0] [1 1 1] [1] [1 1 1] [1 1 0] [1 1 1] mark(U11(X1,X2,X3)) = [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 = active(U11(mark(X1),X2,X3)) [0 0 1] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] problem: Qed