YES Problem: active(2nd(cons(X,cons(Y,Z)))) -> mark(Y) active(from(X)) -> mark(cons(X,from(s(X)))) mark(2nd(X)) -> active(2nd(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(from(X)) -> active(from(mark(X))) mark(s(X)) -> active(s(mark(X))) 2nd(mark(X)) -> 2nd(X) 2nd(active(X)) -> 2nd(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) from(mark(X)) -> from(X) from(active(X)) -> from(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) Proof: Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [1] [from](x0) = [0 0 0]x0 + [0] [1 0 1] [1], [1 0 1] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [active](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [2nd](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [1 0 0] [1] [cons](x0, x1) = [0 0 0]x0 + [1 0 0]x1 + [0] [0 0 0] [0 0 0] [0] orientation: [1 0 1] [1 0 1] [1 0 0] [2] [1 0 1] active(2nd(cons(X,cons(Y,Z)))) = [0 0 0]X + [0 0 0]Y + [0 0 0]Z + [0] >= [0 0 0]Y = mark(Y) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [2 0 2] [2] [2 0 2] [2] active(from(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(cons(X,from(s(X)))) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1 0 1] mark(2nd(X)) = [0 0 0]X >= [0 0 0]X = active(2nd(mark(X))) [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1] [1 0 1] [1 0 0] [1] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = active(cons(mark(X1),X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [2 0 2] [2] [2 0 2] [2] mark(from(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(from(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1 0 1] mark(s(X)) = [0 0 0]X >= [0 0 0]X = active(s(mark(X))) [0 0 0] [0 0 0] [1 0 1] [1 0 1] 2nd(mark(X)) = [0 0 0]X >= [0 0 0]X = 2nd(X) [0 0 0] [0 0 0] [1 0 1] [1 0 1] 2nd(active(X)) = [0 0 0]X >= [0 0 0]X = 2nd(X) [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1] [1 0 1] [1 0 0] [1] cons(mark(X1),X2) = [0 0 0]X1 + [1 0 0]X2 + [0] >= [0 0 0]X1 + [1 0 0]X2 + [0] = cons(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 0] [1] cons(X1,mark(X2)) = [0 0 0]X1 + [1 0 1]X2 + [0] >= [0 0 0]X1 + [1 0 0]X2 + [0] = cons(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 1] [1 0 0] [1] [1 0 1] [1 0 0] [1] cons(active(X1),X2) = [0 0 0]X1 + [1 0 0]X2 + [0] >= [0 0 0]X1 + [1 0 0]X2 + [0] = cons(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 0] [1] cons(X1,active(X2)) = [0 0 0]X1 + [1 0 1]X2 + [0] >= [0 0 0]X1 + [1 0 0]X2 + [0] = cons(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 1] [1] [1 0 1] [1] from(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = from(X) [1 0 1] [1] [1 0 1] [1] [1 0 1] [1] [1 0 1] [1] from(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = from(X) [1 0 1] [1] [1 0 1] [1] [1 0 1] [1 0 1] s(mark(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 0 1] [1 0 1] s(active(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] problem: active(from(X)) -> mark(cons(X,from(s(X)))) mark(2nd(X)) -> active(2nd(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(from(X)) -> active(from(mark(X))) mark(s(X)) -> active(s(mark(X))) 2nd(mark(X)) -> 2nd(X) 2nd(active(X)) -> 2nd(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) from(mark(X)) -> from(X) from(active(X)) -> from(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dim=2 interpretation: [1 1] [s](x0) = [0 0]x0, [2 2] [2] [from](x0) = [1 1]x0 + [2], [1 1] [mark](x0) = [0 0]x0, [1 1] [active](x0) = [0 0]x0, [1 1] [2nd](x0) = [0 0]x0, [1 1] [1 0] [cons](x0, x1) = [0 0]x0 + [0 0]x1 orientation: [3 3] [4] [3 3] [2] active(from(X)) = [0 0]X + [0] >= [0 0]X + [0] = mark(cons(X,from(s(X)))) [1 1] [1 1] mark(2nd(X)) = [0 0]X >= [0 0]X = active(2nd(mark(X))) [1 1] [1 0] [1 1] [1 0] mark(cons(X1,X2)) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = active(cons(mark(X1),X2)) [3 3] [4] [3 3] [4] mark(from(X)) = [0 0]X + [0] >= [0 0]X + [0] = active(from(mark(X))) [1 1] [1 1] mark(s(X)) = [0 0]X >= [0 0]X = active(s(mark(X))) [1 1] [1 1] 2nd(mark(X)) = [0 0]X >= [0 0]X = 2nd(X) [1 1] [1 1] 2nd(active(X)) = [0 0]X >= [0 0]X = 2nd(X) [1 1] [1 0] [1 1] [1 0] cons(mark(X1),X2) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = cons(X1,X2) [1 1] [1 1] [1 1] [1 0] cons(X1,mark(X2)) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = cons(X1,X2) [1 1] [1 0] [1 1] [1 0] cons(active(X1),X2) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = cons(X1,X2) [1 1] [1 1] [1 1] [1 0] cons(X1,active(X2)) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = cons(X1,X2) [2 2] [2] [2 2] [2] from(mark(X)) = [1 1]X + [2] >= [1 1]X + [2] = from(X) [2 2] [2] [2 2] [2] from(active(X)) = [1 1]X + [2] >= [1 1]X + [2] = from(X) [1 1] [1 1] s(mark(X)) = [0 0]X >= [0 0]X = s(X) [1 1] [1 1] s(active(X)) = [0 0]X >= [0 0]X = s(X) problem: mark(2nd(X)) -> active(2nd(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(from(X)) -> active(from(mark(X))) mark(s(X)) -> active(s(mark(X))) 2nd(mark(X)) -> 2nd(X) 2nd(active(X)) -> 2nd(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) from(mark(X)) -> from(X) from(active(X)) -> from(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dim=3 interpretation: [0] [s](x0) = x0 + [1] [0], [0] [from](x0) = x0 + [1] [0], [1 1 0] [0] [mark](x0) = [0 1 0]x0 + [0] [0 0 1] [1], [active](x0) = x0 , [0] [2nd](x0) = x0 + [1] [0], [1 0 0] [0] [cons](x0, x1) = x0 + [0 0 0]x1 + [1] [0 0 0] [0] orientation: [1 1 0] [1] [1 1 0] [0] mark(2nd(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = active(2nd(mark(X))) [0 0 1] [1] [0 0 1] [1] [1 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [0] mark(cons(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [1] = active(cons(mark(X1),X2)) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 1 0] [1] [1 1 0] [0] mark(from(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = active(from(mark(X))) [0 0 1] [1] [0 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 1] [1] [0 0 1] [1] [1 1 0] [0] [0] 2nd(mark(X)) = [0 1 0]X + [1] >= X + [1] = 2nd(X) [0 0 1] [1] [0] [0] [0] 2nd(active(X)) = X + [1] >= X + [1] = 2nd(X) [0] [0] [1 1 0] [1 0 0] [0] [1 0 0] [0] cons(mark(X1),X2) = [0 1 0]X1 + [0 0 0]X2 + [1] >= X1 + [0 0 0]X2 + [1] = cons(X1,X2) [0 0 1] [0 0 0] [1] [0 0 0] [0] [1 1 0] [0] [1 0 0] [0] cons(X1,mark(X2)) = X1 + [0 0 0]X2 + [1] >= X1 + [0 0 0]X2 + [1] = cons(X1,X2) [0 0 0] [0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] cons(active(X1),X2) = X1 + [0 0 0]X2 + [1] >= X1 + [0 0 0]X2 + [1] = cons(X1,X2) [0 0 0] [0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] cons(X1,active(X2)) = X1 + [0 0 0]X2 + [1] >= X1 + [0 0 0]X2 + [1] = cons(X1,X2) [0 0 0] [0] [0 0 0] [0] [1 1 0] [0] [0] from(mark(X)) = [0 1 0]X + [1] >= X + [1] = from(X) [0 0 1] [1] [0] [0] [0] from(active(X)) = X + [1] >= X + [1] = from(X) [0] [0] [1 1 0] [0] [0] s(mark(X)) = [0 1 0]X + [1] >= X + [1] = s(X) [0 0 1] [1] [0] [0] [0] s(active(X)) = X + [1] >= X + [1] = s(X) [0] [0] problem: 2nd(mark(X)) -> 2nd(X) 2nd(active(X)) -> 2nd(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) from(mark(X)) -> from(X) from(active(X)) -> from(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [from](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [mark](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 0] [0] [active](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 1] [2nd](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [1 0 1] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [1 0 0] orientation: [1 0 1] [1] [1 0 1] 2nd(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X = 2nd(X) [0 0 0] [0] [0 0 0] [1 0 1] [1] [1 0 1] 2nd(active(X)) = [0 0 0]X + [0] >= [0 0 0]X = 2nd(X) [0 0 0] [0] [0 0 0] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [1 0 0] [0] [0 0 0] [1 0 0] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] cons(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [1 0 0] [0] [0 0 0] [1 0 0] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] cons(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [1 0 0] [0] [0 0 0] [1 0 0] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] cons(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [1 0 0] [0] [0 0 0] [1 0 0] [1 0 1] [1] [1 0 1] from(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X = from(X) [0 0 0] [0] [0 0 0] [1 0 1] [1] [1 0 1] from(active(X)) = [0 0 0]X + [0] >= [0 0 0]X = from(X) [0 0 0] [0] [0 0 0] [1 0 1] [1] [1 0 1] s(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X = s(X) [0 0 0] [0] [0 0 0] [1 0 1] [1] [1 0 1] s(active(X)) = [0 0 0]X + [0] >= [0 0 0]X = s(X) [0 0 0] [0] [0 0 0] problem: Qed