/export/starexec/sandbox/solver/bin/starexec_run_ttt2 /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- 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=3 interpretation: [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 0] , [proper](x0) = x0 , [1 0 0] [1 0 0] [1 0 0] [U11](x0, x1, x2) = [1 0 0]x0 + [0 0 0]x1 + [1 0 1]x2 [1 0 0] [0 0 0] [1 0 0] , [1 0 0] [1 0 0] [1 0 0] [U12](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [1 0 0]x2 [0 0 0] [0 0 0] [1 0 0] , [1 0 0] [1 0 0] [1] [plus](x0, x1) = [1 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [0], [1 0 0] [top](x0) = [0 0 0]x0 [0 0 0] , [1] [tt] = [0] [0], [0] [0] = [0] [0], [ok](x0) = x0 , [1 0 0] [0] [s](x0) = [0 1 0]x0 + [1] [0 1 0] [1], [1 0 0] [active](x0) = [0 1 0]x0 [0 1 0] orientation: [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] active(U11(tt(),M,N)) = [0 0 0]M + [1 0 1]N + [1] >= [0 0 0]M + [0 0 0]N + [0] = mark(U12(tt(),M,N)) [0 0 0] [1 0 1] [1] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] active(U12(tt(),M,N)) = [0 0 0]M + [1 0 0]N + [0] >= [0 0 0]M + [0 0 0]N + [0] = mark(s(plus(N,M))) [0 0 0] [1 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1] [1 0 0] active(plus(N,0())) = [1 0 0]N + [0] >= [0 0 0]N = mark(N) [1 0 0] [0] [0 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] active(plus(N,s(M))) = [0 0 0]M + [1 0 0]N + [0] >= [0 0 0]M + [0 0 0]N + [0] = mark(U11(tt(),M,N)) [0 0 0] [1 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] active(U11(X1,X2,X3)) = [1 0 0]X1 + [0 0 0]X2 + [1 0 1]X3 >= [1 0 0]X1 + [0 0 0]X2 + [1 0 1]X3 = U11(active(X1),X2,X3) [1 0 0] [0 0 0] [1 0 1] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(U12(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [1 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [1 0 0]X3 = U12(active(X1),X2,X3) [0 0 0] [0 0 0] [1 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [0] active(s(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = s(active(X)) [0 1 0] [1] [0 1 0] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] active(plus(X1,X2)) = [1 0 0]X1 + [0 0 0]X2 + [0] >= [1 0 0]X1 + [0 0 0]X2 + [0] = plus(active(X1),X2) [1 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] active(plus(X1,X2)) = [1 0 0]X1 + [0 0 0]X2 + [0] >= [1 0 0]X1 + [0 0 0]X2 + [0] = plus(X1,active(X2)) [1 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] U11(mark(X1),X2,X3) = [1 0 0]X1 + [0 0 0]X2 + [1 0 1]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = mark(U11(X1,X2,X3)) [1 0 0] [0 0 0] [1 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] U12(mark(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [1 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = mark(U12(X1,X2,X3)) [0 0 0] [0 0 0] [1 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] s(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X = mark(s(X)) [0 0 0] [1] [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 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)) = [1 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 0 0] [1 0 0] [1 0 0] [1 0 0] proper(U11(X1,X2,X3)) = [1 0 0]X1 + [0 0 0]X2 + [1 0 1]X3 >= [1 0 0]X1 + [0 0 0]X2 + [1 0 1]X3 = U11(proper(X1),proper(X2),proper(X3)) [1 0 0] [0 0 0] [1 0 0] [1 0 0] [0 0 0] [1 0 0] [1] [1] proper(tt()) = [0] >= [0] = ok(tt()) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(U12(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [1 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [1 0 0]X3 = U12(proper(X1),proper(X2),proper(X3)) [0 0 0] [0 0 0] [1 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [0] proper(s(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = s(proper(X)) [0 1 0] [1] [0 1 0] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] proper(plus(X1,X2)) = [1 0 0]X1 + [0 0 0]X2 + [0] >= [1 0 0]X1 + [0 0 0]X2 + [0] = 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 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 + [0 0 0]X2 + [1 0 1]X3 >= [1 0 0]X1 + [0 0 0]X2 + [1 0 1]X3 = ok(U11(X1,X2,X3)) [1 0 0] [0 0 0] [1 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 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 + [1 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [1 0 0]X3 = ok(U12(X1,X2,X3)) [0 0 0] [0 0 0] [1 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [0] s(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(s(X)) [0 1 0] [1] [0 1 0] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] plus(ok(X1),ok(X2)) = [1 0 0]X1 + [0 0 0]X2 + [0] >= [1 0 0]X1 + [0 0 0]X2 + [0] = ok(plus(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] top(mark(X)) = [0 0 0]X >= [0 0 0]X = top(proper(X)) [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(U11(tt(),M,N)) -> mark(U12(tt(),M,N)) active(U12(tt(),M,N)) -> mark(s(plus(N,M))) 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)) Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = x0, [proper](x0) = x0, [U11](x0, x1, x2) = 2x0 + 2x1 + x2 + 5, [U12](x0, x1, x2) = 2x0 + 2x1 + x2, [plus](x0, x1) = x0 + 2x1 + 3, [top](x0) = 2x0, [tt] = 4, [0] = 4, [ok](x0) = x0, [s](x0) = x0 + 5, [active](x0) = x0 orientation: active(U11(tt(),M,N)) = 2M + N + 13 >= 2M + N + 8 = mark(U12(tt(),M,N)) active(U12(tt(),M,N)) = 2M + N + 8 >= 2M + N + 8 = mark(s(plus(N,M))) active(plus(N,s(M))) = 2M + N + 13 >= 2M + N + 13 = mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) = 2X1 + 2X2 + X3 + 5 >= 2X1 + 2X2 + X3 + 5 = U11(active(X1),X2,X3) active(U12(X1,X2,X3)) = 2X1 + 2X2 + X3 >= 2X1 + 2X2 + X3 = U12(active(X1),X2,X3) active(s(X)) = X + 5 >= X + 5 = s(active(X)) active(plus(X1,X2)) = X1 + 2X2 + 3 >= X1 + 2X2 + 3 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + 2X2 + 3 >= X1 + 2X2 + 3 = plus(X1,active(X2)) U11(mark(X1),X2,X3) = 2X1 + 2X2 + X3 + 5 >= 2X1 + 2X2 + X3 + 5 = mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) = 2X1 + 2X2 + X3 >= 2X1 + 2X2 + X3 = mark(U12(X1,X2,X3)) s(mark(X)) = X + 5 >= X + 5 = mark(s(X)) plus(mark(X1),X2) = X1 + 2X2 + 3 >= X1 + 2X2 + 3 = mark(plus(X1,X2)) plus(X1,mark(X2)) = X1 + 2X2 + 3 >= X1 + 2X2 + 3 = mark(plus(X1,X2)) proper(U11(X1,X2,X3)) = 2X1 + 2X2 + X3 + 5 >= 2X1 + 2X2 + X3 + 5 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 4 >= 4 = ok(tt()) proper(U12(X1,X2,X3)) = 2X1 + 2X2 + X3 >= 2X1 + 2X2 + X3 = U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) = X + 5 >= X + 5 = s(proper(X)) proper(plus(X1,X2)) = X1 + 2X2 + 3 >= X1 + 2X2 + 3 = plus(proper(X1),proper(X2)) proper(0()) = 4 >= 4 = ok(0()) U11(ok(X1),ok(X2),ok(X3)) = 2X1 + 2X2 + X3 + 5 >= 2X1 + 2X2 + X3 + 5 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = 2X1 + 2X2 + X3 >= 2X1 + 2X2 + X3 = ok(U12(X1,X2,X3)) s(ok(X)) = X + 5 >= X + 5 = ok(s(X)) plus(ok(X1),ok(X2)) = X1 + 2X2 + 3 >= X1 + 2X2 + 3 = ok(plus(X1,X2)) top(mark(X)) = 2X >= 2X = top(proper(X)) top(ok(X)) = 2X >= 2X = top(active(X)) problem: active(U12(tt(),M,N)) -> mark(s(plus(N,M))) 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)) Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = x0, [proper](x0) = x0, [U11](x0, x1, x2) = x0 + x1 + x2, [U12](x0, x1, x2) = 2x0 + 3x1 + 4x2 + 4, [plus](x0, x1) = 2x0 + x1, [top](x0) = 4x0, [tt] = 0, [0] = 0, [ok](x0) = x0, [s](x0) = 2x0, [active](x0) = x0 orientation: active(U12(tt(),M,N)) = 3M + 4N + 4 >= 2M + 4N = mark(s(plus(N,M))) active(plus(N,s(M))) = 2M + 2N >= M + N = mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(active(X1),X2,X3) active(U12(X1,X2,X3)) = 2X1 + 3X2 + 4X3 + 4 >= 2X1 + 3X2 + 4X3 + 4 = U12(active(X1),X2,X3) active(s(X)) = 2X >= 2X = s(active(X)) active(plus(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = plus(active(X1),X2) active(plus(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = plus(X1,active(X2)) U11(mark(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) = 2X1 + 3X2 + 4X3 + 4 >= 2X1 + 3X2 + 4X3 + 4 = mark(U12(X1,X2,X3)) s(mark(X)) = 2X >= 2X = mark(s(X)) plus(mark(X1),X2) = 2X1 + X2 >= 2X1 + X2 = mark(plus(X1,X2)) plus(X1,mark(X2)) = 2X1 + X2 >= 2X1 + X2 = mark(plus(X1,X2)) proper(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2,X3)) = 2X1 + 3X2 + 4X3 + 4 >= 2X1 + 3X2 + 4X3 + 4 = U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) = 2X >= 2X = s(proper(X)) proper(plus(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = plus(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) U11(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = 2X1 + 3X2 + 4X3 + 4 >= 2X1 + 3X2 + 4X3 + 4 = ok(U12(X1,X2,X3)) s(ok(X)) = 2X >= 2X = ok(s(X)) plus(ok(X1),ok(X2)) = 2X1 + X2 >= 2X1 + X2 = ok(plus(X1,X2)) top(mark(X)) = 4X >= 4X = top(proper(X)) top(ok(X)) = 4X >= 4X = top(active(X)) problem: 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)) Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = x0 + 6, [proper](x0) = x0, [U11](x0, x1, x2) = x0 + x1 + 4x2 + 2, [U12](x0, x1, x2) = 4x0 + 4x1 + x2, [plus](x0, x1) = 4x0 + 4x1, [top](x0) = x0, [tt] = 0, [0] = 0, [ok](x0) = x0, [s](x0) = 2x0 + 2, [active](x0) = x0 orientation: active(plus(N,s(M))) = 8M + 4N + 8 >= M + 4N + 8 = mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) = X1 + X2 + 4X3 + 2 >= X1 + X2 + 4X3 + 2 = U11(active(X1),X2,X3) active(U12(X1,X2,X3)) = 4X1 + 4X2 + X3 >= 4X1 + 4X2 + X3 = U12(active(X1),X2,X3) active(s(X)) = 2X + 2 >= 2X + 2 = s(active(X)) active(plus(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = plus(active(X1),X2) active(plus(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = plus(X1,active(X2)) U11(mark(X1),X2,X3) = X1 + X2 + 4X3 + 8 >= X1 + X2 + 4X3 + 8 = mark(U11(X1,X2,X3)) U12(mark(X1),X2,X3) = 4X1 + 4X2 + X3 + 24 >= 4X1 + 4X2 + X3 + 6 = mark(U12(X1,X2,X3)) s(mark(X)) = 2X + 14 >= 2X + 8 = mark(s(X)) plus(mark(X1),X2) = 4X1 + 4X2 + 24 >= 4X1 + 4X2 + 6 = mark(plus(X1,X2)) plus(X1,mark(X2)) = 4X1 + 4X2 + 24 >= 4X1 + 4X2 + 6 = mark(plus(X1,X2)) proper(U11(X1,X2,X3)) = X1 + X2 + 4X3 + 2 >= X1 + X2 + 4X3 + 2 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2,X3)) = 4X1 + 4X2 + X3 >= 4X1 + 4X2 + X3 = U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) = 2X + 2 >= 2X + 2 = s(proper(X)) proper(plus(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = plus(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) U11(ok(X1),ok(X2),ok(X3)) = X1 + X2 + 4X3 + 2 >= X1 + X2 + 4X3 + 2 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = 4X1 + 4X2 + X3 >= 4X1 + 4X2 + X3 = ok(U12(X1,X2,X3)) s(ok(X)) = 2X + 2 >= 2X + 2 = ok(s(X)) plus(ok(X1),ok(X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = ok(plus(X1,X2)) top(mark(X)) = X + 6 >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(X)) problem: 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)) 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: [mark](x0) = x0, [proper](x0) = 6x0, [U11](x0, x1, x2) = 3x0 + 4x1 + x2, [U12](x0, x1, x2) = 2x0 + 4x1 + x2, [plus](x0, x1) = x0 + 2x1, [top](x0) = x0 + 2, [tt] = 0, [0] = 3, [ok](x0) = 4x0, [s](x0) = 2x0, [active](x0) = x0 orientation: active(plus(N,s(M))) = 4M + N >= 4M + N = mark(U11(tt(),M,N)) active(U11(X1,X2,X3)) = 3X1 + 4X2 + X3 >= 3X1 + 4X2 + X3 = U11(active(X1),X2,X3) active(U12(X1,X2,X3)) = 2X1 + 4X2 + X3 >= 2X1 + 4X2 + X3 = U12(active(X1),X2,X3) active(s(X)) = 2X >= 2X = s(active(X)) active(plus(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = plus(active(X1),X2) active(plus(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = plus(X1,active(X2)) U11(mark(X1),X2,X3) = 3X1 + 4X2 + X3 >= 3X1 + 4X2 + X3 = mark(U11(X1,X2,X3)) proper(U11(X1,X2,X3)) = 18X1 + 24X2 + 6X3 >= 18X1 + 24X2 + 6X3 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X1,X2,X3)) = 12X1 + 24X2 + 6X3 >= 12X1 + 24X2 + 6X3 = U12(proper(X1),proper(X2),proper(X3)) proper(s(X)) = 12X >= 12X = s(proper(X)) proper(plus(X1,X2)) = 6X1 + 12X2 >= 6X1 + 12X2 = plus(proper(X1),proper(X2)) proper(0()) = 18 >= 12 = ok(0()) U11(ok(X1),ok(X2),ok(X3)) = 12X1 + 16X2 + 4X3 >= 12X1 + 16X2 + 4X3 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = 8X1 + 16X2 + 4X3 >= 8X1 + 16X2 + 4X3 = ok(U12(X1,X2,X3)) s(ok(X)) = 8X >= 8X = ok(s(X)) plus(ok(X1),ok(X2)) = 4X1 + 8X2 >= 4X1 + 8X2 = ok(plus(X1,X2)) top(ok(X)) = 4X + 2 >= X + 2 = top(active(X)) problem: 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)) 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)) 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: [mark](x0) = x0 , [1 0 0] [proper](x0) = [0 0 0]x0 [0 0 1] , [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] [1 1 0] [1 0 1] [0 0 0] [1], [1 0 1] [1 0 1] [1 0 0] [0] [U12](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [0] [0 1 0] [0 0 0] [0 0 0] [1], [1 0 0] [1 0 0] [1] [plus](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [1 0 0] [1], [1 0 0] [top](x0) = [0 0 0]x0 [0 0 0] , [0] [tt] = [0] [0], [ok](x0) = x0 , [1 0 1] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [active](x0) = [0 0 0]x0 [0 0 1] orientation: [1 0 1] [1 0 0] [1] [1 0 0] [1 0 0] [0] active(plus(N,s(M))) = [0 0 0]M + [0 0 0]N + [0] >= [0 0 0]M + [0 0 0]N + [0] = mark(U11(tt(),M,N)) [1 0 1] [0 0 0] [1] [1 0 1] [0 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] active(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(active(X1),X2,X3) [1 1 0] [1 0 1] [0 0 0] [1] [1 0 0] [1 0 1] [0 0 0] [1] [1 0 1] [1 0 1] [1 0 0] [0] [1 0 1] [1 0 1] [1 0 0] [0] active(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] = U12(active(X1),X2,X3) [0 1 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [1 0 1] [1 0 1] active(s(X)) = [0 0 0]X >= [0 0 0]X = s(active(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] active(plus(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = plus(active(X1),X2) [0 0 0] [1 0 0] [1] [0 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] active(plus(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = plus(X1,active(X2)) [0 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] [0] U11(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 + [0] = mark(U11(X1,X2,X3)) [1 1 0] [1 0 1] [0 0 0] [1] [1 1 0] [1 0 1] [0 0 0] [1] [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)) = [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)) [1 1 0] [1 0 1] [0 0 0] [1] [1 0 0] [1 0 1] [0 0 0] [1] [0] [0] proper(tt()) = [0] >= [0] = ok(tt()) [0] [0] [1 0 1] [1 0 1] [1 0 0] [0] [1 0 1] [1 0 1] [1 0 0] [0] proper(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] = U12(proper(X1),proper(X2),proper(X3)) [0 1 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [1 0 1] [1 0 1] proper(s(X)) = [0 0 0]X >= [0 0 0]X = s(proper(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] proper(plus(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = plus(proper(X1),proper(X2)) [0 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] [0] U11(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 + [0] = ok(U11(X1,X2,X3)) [1 1 0] [1 0 1] [0 0 0] [1] [1 1 0] [1 0 1] [0 0 0] [1] [1 0 1] [1 0 1] [1 0 0] [0] [1 0 1] [1 0 1] [1 0 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 + [0] = ok(U12(X1,X2,X3)) [0 1 0] [0 0 0] [0 0 0] [1] [0 1 0] [0 0 0] [0 0 0] [1] [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 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] plus(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = ok(plus(X1,X2)) [0 0 0] [1 0 0] [1] [0 0 0] [1 0 0] [1] [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(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)) 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)) 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] [0] [mark](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 0] [proper](x0) = [0 0 1]x0 [0 0 1] , [1 0 1] [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 0] [0 0 0] , [1 0 1] [1 0 0] [1 0 0] [U12](x0, x1, x2) = [0 1 0]x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 1] [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [plus](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 1] [1] [top](x0) = [0 0 1]x0 + [1] [0 0 0] [0], [0] [tt] = [0] [1], [1 0 0] [ok](x0) = [0 0 0]x0 [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 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(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 = 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)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 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 0 0]X >= [0 0 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)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 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)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = plus(X1,active(X2)) [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] [0] U11(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 + [0] = mark(U11(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 0 0] [1 0 0] [1 0 1] [1 0 0] [1 0 0] proper(U11(X1,X2,X3)) = [0 0 1]X1 + [0 0 0]X2 + [0 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 0] [0 0 0] [0 0 1] [0 0 0] [0 0 0] [0] [0] proper(tt()) = [1] >= [0] = ok(tt()) [1] [1] [1 0 1] [1 0 0] [1 0 0] [1 0 1] [1 0 0] [1 0 0] proper(U12(X1,X2,X3)) = [0 0 1]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 1]X1 + [0 0 0]X2 + [0 0 0]X3 = U12(proper(X1),proper(X2),proper(X3)) [0 0 1] [0 0 0] [0 0 0] [0 0 1] [0 0 0] [0 0 0] [1 0 0] [1 0 0] proper(s(X)) = [0 0 0]X >= [0 0 0]X = s(proper(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] 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)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 0 0] [1 0 0] [1 0 1] [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 1 1] [0 0 0] [0 0 0] [0 0 1] [0 0 0] [0 0 0] [1 1 1] [1 0 0] [1 0 0] [1 0 1] [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]X1 + [0 0 0]X2 + [0 0 0]X3 = ok(U12(X1,X2,X3)) [0 1 1] [0 0 0] [0 0 0] [0 1 1] [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 1 1] [1] [1 0 0] [1] top(ok(X)) = [0 1 1]X + [1] >= [0 0 0]X + [1] = top(active(X)) [0 0 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)) 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)) 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 1] [0] [proper](x0) = [0 0 0]x0 + [1] [0 0 1] [0], [1 0 0] [1 1 0] [0] [U11](x0, x1, x2) = x0 + [0 0 0]x1 + [0 0 0]x2 + [0] [0 0 1] [0 0 1] [1], [1 0 0] [1 0 0] [U12](x0, x1, x2) = x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 1] [0 0 1] , [1 0 0] [plus](x0, x1) = [0 0 0]x0 + x1 [0 0 1] , [1 0 0] [top](x0) = [0 0 0]x0 [0 0 0] , [0] [tt] = [0] [1], [1 0 1] [0] [ok](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [s](x0) = x0 , [1 0 1] [active](x0) = [0 1 0]x0 [0 0 1] orientation: [1 0 1] [1 0 1] [1 1 1] [1] [1 0 1] [1 0 0] [1 1 0] [0] active(U11(X1,X2,X3)) = [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = U11(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 1] [1 0 1] [1 0 1] [1 0 1] [1 0 0] [1 0 0] active(U12(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 = U12(active(X1),X2,X3) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1 0 1] [1 0 1] active(s(X)) = [0 1 0]X >= [0 1 0]X = s(active(X)) [0 0 1] [0 0 1] [1 0 1] [1 0 1] [1 0 1] active(plus(X1,X2)) = [0 0 0]X1 + [0 1 0]X2 >= [0 0 0]X1 + X2 = plus(active(X1),X2) [0 0 1] [0 0 1] [0 0 1] [1 0 1] [1 0 1] [1 0 0] [1 0 1] active(plus(X1,X2)) = [0 0 0]X1 + [0 1 0]X2 >= [0 0 0]X1 + [0 1 0]X2 = plus(X1,active(X2)) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1 0 1] [1 0 1] [1 1 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] [1] proper(tt()) = [1] >= [1] = ok(tt()) [1] [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 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 0 1] [1 0 1] [1 1 1] [1] [1 0 1] [1 0 1] [1 1 1] [1] U11(ok(X1),ok(X2),ok(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 + [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 0 1] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] [1 0 1] [0] U12(ok(X1),ok(X2),ok(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 + [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 1] [0] [1 0 1] [0] s(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(s(X)) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] [0] plus(ok(X1),ok(X2)) = [0 0 0]X1 + [0 1 0]X2 + [1] >= [0 0 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] 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) active(plus(X1,X2)) -> plus(X1,active(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)) 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 1 0] [proper](x0) = [1 0 1]x0 [0 0 1] , [1] [U11](x0, x1, x2) = x0 + x1 + x2 + [0] [0], [1 0 1] [0] [U12](x0, x1, x2) = [0 1 0]x0 + x1 + x2 + [1] [0 0 0] [1], [1 0 1] [0] [plus](x0, x1) = x0 + [0 1 0]x1 + [0] [0 0 0] [1], [1 0 0] [1] [top](x0) = [0 0 0]x0 + [0] [0 0 1] [0], [0] [tt] = [0] [1], [1 0 0] [ok](x0) = [0 0 0]x0 [0 0 1] , [s](x0) = x0 , [1 0 0] [active](x0) = [1 1 0]x0 [0 0 1] orientation: [1 0 1] [1 0 0] [1 0 0] [0] [1 0 1] [0] active(U12(X1,X2,X3)) = [1 1 1]X1 + [1 1 0]X2 + [1 1 0]X3 + [1] >= [1 1 0]X1 + X2 + X3 + [1] = U12(active(X1),X2,X3) [0 0 0] [0 0 1] [0 0 1] [1] [0 0 0] [1] [1 0 0] [1 0 0] active(s(X)) = [1 1 0]X >= [1 1 0]X = s(active(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 1] [0] active(plus(X1,X2)) = [1 1 0]X1 + [1 1 1]X2 + [0] >= [1 1 0]X1 + [0 1 0]X2 + [0] = plus(active(X1),X2) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 0 0] [1 0 1] [0] [1 0 1] [0] active(plus(X1,X2)) = [1 1 0]X1 + [1 1 1]X2 + [0] >= X1 + [1 1 0]X2 + [0] = plus(X1,active(X2)) [0 0 1] [0 0 0] [1] [0 0 0] [1] [1 1 0] [1 1 0] [1 1 0] [1] [1 1 0] [1 1 0] [1 1 0] [1] proper(U11(X1,X2,X3)) = [1 0 1]X1 + [1 0 1]X2 + [1 0 1]X3 + [1] >= [1 0 1]X1 + [1 0 1]X2 + [1 0 1]X3 + [0] = U11(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] [0] [0] proper(tt()) = [1] >= [0] = ok(tt()) [1] [1] [1 1 1] [1 1 0] [1 1 0] [1] [1 1 1] [1 1 0] [1 1 0] [0] proper(U12(X1,X2,X3)) = [1 0 1]X1 + [1 0 1]X2 + [1 0 1]X3 + [1] >= [1 0 1]X1 + [1 0 1]X2 + [1 0 1]X3 + [1] = U12(proper(X1),proper(X2),proper(X3)) [0 0 0] [0 0 1] [0 0 1] [1] [0 0 0] [0 0 1] [0 0 1] [1] [1 1 0] [1 1 0] proper(s(X)) = [1 0 1]X >= [1 0 1]X = s(proper(X)) [0 0 1] [0 0 1] [1 1 0] [1 1 1] [0] [1 1 0] [1 1 1] [0] proper(plus(X1,X2)) = [1 0 1]X1 + [1 0 1]X2 + [1] >= [1 0 1]X1 + [1 0 1]X2 + [0] = plus(proper(X1),proper(X2)) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1] U11(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 + [0] = ok(U11(X1,X2,X3)) [0 0 1] [0 0 1] [0 0 1] [0] [0 0 1] [0 0 1] [0 0 1] [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)) = [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] = ok(U12(X1,X2,X3)) [0 0 0] [0 0 1] [0 0 1] [1] [0 0 0] [0 0 1] [0 0 1] [1] [1 0 0] [1 0 0] s(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(s(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 1] [0] plus(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = ok(plus(X1,X2)) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [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 1] [0] [0 0 1] [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) active(plus(X1,X2)) -> plus(X1,active(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: [proper](x0) = 5x0, [U11](x0, x1, x2) = 6x0 + 4x1 + 4x2 + 2, [U12](x0, x1, x2) = 2x0 + x1 + x2, [plus](x0, x1) = 2x0 + x1 + 1, [top](x0) = x0 + 4, [tt] = 2, [ok](x0) = 4x0 + 2, [s](x0) = 2x0, [active](x0) = 4x0 orientation: active(U12(X1,X2,X3)) = 8X1 + 4X2 + 4X3 >= 8X1 + X2 + X3 = U12(active(X1),X2,X3) active(s(X)) = 8X >= 8X = s(active(X)) active(plus(X1,X2)) = 8X1 + 4X2 + 4 >= 8X1 + X2 + 1 = plus(active(X1),X2) active(plus(X1,X2)) = 8X1 + 4X2 + 4 >= 2X1 + 4X2 + 1 = plus(X1,active(X2)) proper(U11(X1,X2,X3)) = 30X1 + 20X2 + 20X3 + 10 >= 30X1 + 20X2 + 20X3 + 2 = U11(proper(X1),proper(X2),proper(X3)) proper(tt()) = 10 >= 10 = ok(tt()) proper(s(X)) = 10X >= 10X = s(proper(X)) proper(plus(X1,X2)) = 10X1 + 5X2 + 5 >= 10X1 + 5X2 + 1 = plus(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3)) = 24X1 + 16X2 + 16X3 + 30 >= 24X1 + 16X2 + 16X3 + 10 = ok(U11(X1,X2,X3)) U12(ok(X1),ok(X2),ok(X3)) = 8X1 + 4X2 + 4X3 + 8 >= 8X1 + 4X2 + 4X3 + 2 = ok(U12(X1,X2,X3)) s(ok(X)) = 8X + 4 >= 8X + 2 = ok(s(X)) plus(ok(X1),ok(X2)) = 8X1 + 4X2 + 7 >= 8X1 + 4X2 + 6 = ok(plus(X1,X2)) top(ok(X)) = 4X + 6 >= 4X + 4 = top(active(X)) problem: active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) Matrix Interpretation Processor: dim=1 interpretation: [proper](x0) = 4x0 + 1, [U12](x0, x1, x2) = x0 + 4x1 + x2 + 1, [tt] = 4, [ok](x0) = 4x0 + 1, [s](x0) = x0 + 7, [active](x0) = x0 orientation: active(U12(X1,X2,X3)) = X1 + 4X2 + X3 + 1 >= X1 + 4X2 + X3 + 1 = U12(active(X1),X2,X3) active(s(X)) = X + 7 >= X + 7 = s(active(X)) proper(tt()) = 17 >= 17 = ok(tt()) proper(s(X)) = 4X + 29 >= 4X + 8 = s(proper(X)) problem: active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) active(s(X)) -> s(active(X)) proper(tt()) -> ok(tt()) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [proper](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [1 0 0] [0] [U12](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], [0] [tt] = [0] [0], [1 0 0] [ok](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [s](x0) = [0 0 1]x0 + [0] [0 1 0] [1], [1 1 1] [0] [active](x0) = [0 0 1]x0 + [0] [0 1 0] [1] orientation: [1 1 1] [1 0 0] [1 0 0] [1] [1 1 1] [1 0 0] [1 0 0] [0] active(U12(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 + [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 1 1] [1] [1 1 1] [0] active(s(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = s(active(X)) [0 0 1] [1] [0 0 1] [1] [0] [0] proper(tt()) = [0] >= [0] = ok(tt()) [0] [0] problem: proper(tt()) -> ok(tt()) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [proper](x0) = [0 0 0]x0 [0 1 0] , [0] [tt] = [1] [0], [1 0 0] [0] [ok](x0) = [0 0 0]x0 + [0] [0 0 0] [1] orientation: [1] [0] proper(tt()) = [0] >= [0] = ok(tt()) [1] [1] problem: Qed