YES Problem: active(and(tt(),X)) -> mark(X) active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Proof: Matrix Interpretation Processor: dim=1 interpretation: [0] = 0, [and](x0, x1) = 4x0 + 4x1, [mark](x0) = x0, [plus](x0, x1) = x0 + x1, [tt] = 6, [s](x0) = x0, [active](x0) = x0 orientation: active(and(tt(),X)) = 4X + 24 >= X = mark(X) active(plus(N,0())) = N >= N = mark(N) active(plus(N,s(M))) = M + N >= M + N = mark(s(plus(N,M))) mark(and(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = active(and(mark(X1),X2)) mark(tt()) = 6 >= 6 = active(tt()) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) mark(s(X)) = X >= X = active(s(mark(X))) and(mark(X1),X2) = 4X1 + 4X2 >= 4X1 + 4X2 = and(X1,X2) and(X1,mark(X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = and(X1,X2) and(active(X1),X2) = 4X1 + 4X2 >= 4X1 + 4X2 = and(X1,X2) and(X1,active(X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = and(X1,X2) plus(mark(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,mark(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(active(X1),X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) problem: active(plus(N,0())) -> mark(N) active(plus(N,s(M))) -> mark(s(plus(N,M))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dim=1 interpretation: [0] = 0, [and](x0, x1) = x0 + 2x1, [mark](x0) = x0, [plus](x0, x1) = x0 + 4x1 + 5, [tt] = 0, [s](x0) = x0 + 3, [active](x0) = x0 orientation: active(plus(N,0())) = N + 5 >= N = mark(N) active(plus(N,s(M))) = 4M + N + 17 >= 4M + N + 8 = mark(s(plus(N,M))) mark(and(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = active(and(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(plus(X1,X2)) = X1 + 4X2 + 5 >= X1 + 4X2 + 5 = active(plus(mark(X1),mark(X2))) mark(0()) = 0 >= 0 = active(0()) mark(s(X)) = X + 3 >= X + 3 = active(s(mark(X))) and(mark(X1),X2) = X1 + 2X2 >= X1 + 2X2 = and(X1,X2) and(X1,mark(X2)) = X1 + 2X2 >= X1 + 2X2 = and(X1,X2) and(active(X1),X2) = X1 + 2X2 >= X1 + 2X2 = and(X1,X2) and(X1,active(X2)) = X1 + 2X2 >= X1 + 2X2 = and(X1,X2) plus(mark(X1),X2) = X1 + 4X2 + 5 >= X1 + 4X2 + 5 = plus(X1,X2) plus(X1,mark(X2)) = X1 + 4X2 + 5 >= X1 + 4X2 + 5 = plus(X1,X2) plus(active(X1),X2) = X1 + 4X2 + 5 >= X1 + 4X2 + 5 = plus(X1,X2) plus(X1,active(X2)) = X1 + 4X2 + 5 >= X1 + 4X2 + 5 = plus(X1,X2) s(mark(X)) = X + 3 >= X + 3 = s(X) s(active(X)) = X + 3 >= X + 3 = s(X) problem: mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dim=3 interpretation: [1] [0] = [0] [0], [1 0 0] [1 0 0] [0] [and](x0, x1) = [0 1 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [0], [1 1 0] [mark](x0) = [1 1 0]x0 [0 0 0] , [1 0 0] [1 0 0] [1] [plus](x0, x1) = [0 1 0]x0 + [0 1 0]x1 + [0] [0 0 0] [1 0 0] [0], [1] [tt] = [0] [0], [1 0 0] [s](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [active](x0) = [0 1 0]x0 [0 0 0] orientation: [1 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [0] mark(and(X1,X2)) = [1 1 0]X1 + [1 0 0]X2 + [1] >= [1 1 0]X1 + [0 0 0]X2 + [1] = active(and(mark(X1),X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1] [1] mark(tt()) = [1] >= [0] = active(tt()) [0] [0] [1 1 0] [1 1 0] [1] [1 1 0] [1 1 0] [1] mark(plus(X1,X2)) = [1 1 0]X1 + [1 1 0]X2 + [1] >= [1 1 0]X1 + [1 1 0]X2 + [0] = active(plus(mark(X1),mark(X2))) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1] [1] mark(0()) = [1] >= [0] = active(0()) [0] [0] [1 1 0] [1 1 0] mark(s(X)) = [1 1 0]X >= [1 1 0]X = active(s(mark(X))) [0 0 0] [0 0 0] [1 1 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] and(mark(X1),X2) = [1 1 0]X1 + [0 0 0]X2 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [1] = and(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 1 0] [0] [1 0 0] [1 0 0] [0] and(X1,mark(X2)) = [0 1 0]X1 + [0 0 0]X2 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [1] = and(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] and(active(X1),X2) = [0 1 0]X1 + [0 0 0]X2 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [1] = and(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] and(X1,active(X2)) = [0 1 0]X1 + [0 0 0]X2 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [1] = and(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] plus(mark(X1),X2) = [1 1 0]X1 + [0 1 0]X2 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0] = plus(X1,X2) [0 0 0] [1 0 0] [0] [0 0 0] [1 0 0] [0] [1 0 0] [1 1 0] [1] [1 0 0] [1 0 0] [1] plus(X1,mark(X2)) = [0 1 0]X1 + [1 1 0]X2 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0] = plus(X1,X2) [0 0 0] [1 1 0] [0] [0 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] plus(active(X1),X2) = [0 1 0]X1 + [0 1 0]X2 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0] = plus(X1,X2) [0 0 0] [1 0 0] [0] [0 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] plus(X1,active(X2)) = [0 1 0]X1 + [0 1 0]X2 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0] = plus(X1,X2) [0 0 0] [1 0 0] [0] [0 0 0] [1 0 0] [0] [1 1 0] [1 0 0] s(mark(X)) = [1 1 0]X >= [0 1 0]X = s(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] s(active(X)) = [0 1 0]X >= [0 1 0]X = s(X) [0 0 0] [0 0 0] problem: mark(tt()) -> active(tt()) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dim=3 interpretation: [0] [0] = [1] [0], [1 0 1] [1 0 0] [0] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [1 1 0] [0] [mark](x0) = [0 1 0]x0 + [0] [0 0 1] [1], [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], [0] [tt] = [1] [0], [1 0 0] [0] [s](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 1] [0] [active](x0) = [0 1 0]x0 + [0] [0 0 0] [1] orientation: [1] [0] mark(tt()) = [1] >= [1] = active(tt()) [1] [1] [1 1 0] [1 1 0] [1] [1 1 0] [1 1 0] [0] mark(plus(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = active(plus(mark(X1),mark(X2))) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1] [0] mark(0()) = [1] >= [1] = active(0()) [1] [1] [1 1 0] [1] [1 1 0] [0] mark(s(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = active(s(mark(X))) [0 0 0] [1] [0 0 0] [1] [1 1 1] [1 0 0] [1] [1 0 1] [1 0 0] [0] and(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = and(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 1] [1 1 0] [0] [1 0 1] [1 0 0] [0] and(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = and(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 1] [1 0 0] [1] [1 0 1] [1 0 0] [0] and(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = and(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 0] [0] and(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = and(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] plus(mark(X1),X2) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = plus(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 1 0] [0] [1 0 0] [1 0 0] [0] plus(X1,mark(X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = plus(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 1] [1 0 0] [0] [1 0 0] [1 0 0] [0] plus(active(X1),X2) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = plus(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 0] [0] plus(X1,active(X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = plus(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [0] [1 0 0] [0] s(mark(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = s(X) [0 0 0] [0] [0 0 0] [0] [1 0 1] [0] [1 0 0] [0] s(active(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = s(X) [0 0 0] [0] [0 0 0] [0] problem: and(X1,mark(X2)) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) 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) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dim=1 interpretation: [and](x0, x1) = 4x0 + 4x1, [mark](x0) = x0 + 4, [plus](x0, x1) = 4x0 + 2x1, [s](x0) = 4x0, [active](x0) = 4x0 orientation: and(X1,mark(X2)) = 4X1 + 4X2 + 16 >= 4X1 + 4X2 = and(X1,X2) and(X1,active(X2)) = 4X1 + 16X2 >= 4X1 + 4X2 = and(X1,X2) plus(mark(X1),X2) = 4X1 + 2X2 + 16 >= 4X1 + 2X2 = plus(X1,X2) plus(X1,mark(X2)) = 4X1 + 2X2 + 8 >= 4X1 + 2X2 = plus(X1,X2) plus(active(X1),X2) = 16X1 + 2X2 >= 4X1 + 2X2 = plus(X1,X2) plus(X1,active(X2)) = 4X1 + 8X2 >= 4X1 + 2X2 = plus(X1,X2) s(mark(X)) = 4X + 16 >= 4X = s(X) s(active(X)) = 16X >= 4X = s(X) problem: and(X1,active(X2)) -> and(X1,X2) plus(active(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) s(active(X)) -> s(X) Matrix Interpretation Processor: dim=1 interpretation: [and](x0, x1) = 4x0 + 6x1 + 3, [plus](x0, x1) = x0 + x1, [s](x0) = x0, [active](x0) = x0 + 1 orientation: and(X1,active(X2)) = 4X1 + 6X2 + 9 >= 4X1 + 6X2 + 3 = and(X1,X2) plus(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 = plus(X1,X2) plus(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 = plus(X1,X2) s(active(X)) = X + 1 >= X = s(X) problem: Qed