YES Problem: active(from(X)) -> mark(cons(X,from(s(X)))) active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) mark(from(X)) -> active(from(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(X)) mark(nil()) -> active(nil()) mark(0()) -> active(0()) mark(length1(X)) -> active(length1(X)) from(mark(X)) -> from(X) from(active(X)) -> from(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) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) length1(mark(X)) -> length1(X) length1(active(X)) -> length1(X) Proof: Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [length1](x0) = [0 0 0]x0 [0 0 0] , [0] [0] = [0] [0], [1 0 0] [length](x0) = [0 0 0]x0 [0 0 0] , [0] [nil] = [0] [0], [1 1 0] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [1 1 0] , [1 1 0] [0] [s](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 1 0] [active](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [1] [from](x0) = [1 1 0]x0 + [1] [0 0 0] [0] orientation: [2 2 0] [2] [2 2 0] [1] 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] [0] [0] active(length(nil())) = [0] >= [0] = mark(0()) [0] [0] [1 1 0] [1 0 0] [1 0 0] active(length(cons(X,Y))) = [0 0 0]X + [0 0 0]Y >= [0 0 0]Y = mark(s(length1(Y))) [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(length1(X)) = [0 0 0]X >= [0 0 0]X = mark(length(X)) [0 0 0] [0 0 0] [2 2 0] [2] [2 2 0] [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 1 0] [1 0 0] [1 1 0] [1 0 0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(cons(mark(X1),X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] mark(s(X)) = [0 0 0]X >= [0 0 0]X = active(s(mark(X))) [0 0 0] [0 0 0] [1 0 0] [1 0 0] mark(length(X)) = [0 0 0]X >= [0 0 0]X = active(length(X)) [0 0 0] [0 0 0] [0] [0] mark(nil()) = [0] >= [0] = active(nil()) [0] [0] [0] [0] mark(0()) = [0] >= [0] = active(0()) [0] [0] [1 0 0] [1 0 0] mark(length1(X)) = [0 0 0]X >= [0 0 0]X = active(length1(X)) [0 0 0] [0 0 0] [1 1 0] [1] [1 1 0] [1] from(mark(X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = from(X) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1] [1 1 0] [1] from(active(X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = from(X) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [1 1 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] [1 0 0] cons(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [1 1 0] [0 0 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] cons(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [1 1 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] [1 0 0] cons(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [1 1 0] [0 0 0] [1 1 0] [1 1 0] [0] [1 1 0] [0] s(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(X) [0 0 0] [1] [0 0 0] [1] [1 1 0] [0] [1 1 0] [0] s(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(X) [0 0 0] [1] [0 0 0] [1] [1 1 0] [1 0 0] length(mark(X)) = [0 0 0]X >= [0 0 0]X = length(X) [0 0 0] [0 0 0] [1 1 0] [1 0 0] length(active(X)) = [0 0 0]X >= [0 0 0]X = length(X) [0 0 0] [0 0 0] [1 1 0] [1 0 0] length1(mark(X)) = [0 0 0]X >= [0 0 0]X = length1(X) [0 0 0] [0 0 0] [1 1 0] [1 0 0] length1(active(X)) = [0 0 0]X >= [0 0 0]X = length1(X) [0 0 0] [0 0 0] problem: active(length(nil())) -> mark(0()) active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) mark(from(X)) -> active(from(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(X)) mark(nil()) -> active(nil()) mark(0()) -> active(0()) mark(length1(X)) -> active(length1(X)) from(mark(X)) -> from(X) from(active(X)) -> from(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) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) length1(mark(X)) -> length1(X) length1(active(X)) -> length1(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [length1](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [0] [0] = [0] [0], [1 0 0] [0] [length](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [0] [nil] = [0] [0], [1 1 0] [0] [mark](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 1 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 1 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [active](x0) = [0 0 0]x0 [0 1 1] , [1 1 0] [from](x0) = [0 0 0]x0 [0 0 1] orientation: [1] [0] active(length(nil())) = [0] >= [0] = mark(0()) [1] [1] [1 1 0] [1 0 0] [1] [1 0 0] [1] active(length(cons(X,Y))) = [0 0 0]X + [0 0 0]Y + [0] >= [0 0 0]Y + [0] = mark(s(length1(Y))) [0 0 0] [0 0 0] [1] [0 0 0] [1] [1 0 0] [1] [1 0 0] [1] active(length1(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(length(X)) [0 0 0] [1] [0 0 0] [1] [1 1 0] [0] [1 1 0] [0] mark(from(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(from(mark(X))) [0 0 1] [1] [0 0 1] [1] [1 1 0] [1 0 0] [0] [1 1 0] [1 0 0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = active(cons(mark(X1),X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1 1 0] [0] [1 1 0] mark(s(X)) = [0 0 0]X + [0] >= [0 0 0]X = active(s(mark(X))) [0 0 0] [1] [0 0 0] [1 0 0] [1] [1 0 0] [1] mark(length(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(length(X)) [0 0 0] [1] [0 0 0] [1] [0] [0] mark(nil()) = [0] >= [0] = active(nil()) [1] [0] [0] [0] mark(0()) = [0] >= [0] = active(0()) [1] [0] [1 0 0] [1] [1 0 0] [1] mark(length1(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(length1(X)) [0 0 0] [1] [0 0 0] [1] [1 1 0] [0] [1 1 0] from(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X = from(X) [0 0 1] [1] [0 0 1] [1 1 0] [1 1 0] from(active(X)) = [0 0 0]X >= [0 0 0]X = from(X) [0 1 1] [0 0 1] [1 1 0] [1 0 0] [1 1 0] [1 0 0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 0 0] cons(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] cons(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 0 0] cons(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 1 0] [1 1 0] s(active(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 1 0] [0] [1 0 0] [0] length(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = length(X) [0 0 0] [0] [0 0 0] [0] [1 1 0] [0] [1 0 0] [0] length(active(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = length(X) [0 0 0] [0] [0 0 0] [0] [1 1 0] [0] [1 0 0] [0] length1(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = length1(X) [0 0 0] [0] [0 0 0] [0] [1 1 0] [0] [1 0 0] [0] length1(active(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = length1(X) [0 0 0] [0] [0 0 0] [0] problem: active(length(cons(X,Y))) -> mark(s(length1(Y))) active(length1(X)) -> mark(length(X)) mark(from(X)) -> active(from(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(X)) mark(nil()) -> active(nil()) mark(0()) -> active(0()) mark(length1(X)) -> active(length1(X)) from(mark(X)) -> from(X) from(active(X)) -> from(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) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) length1(mark(X)) -> length1(X) length1(active(X)) -> length1(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [length1](x0) = [0 0 0]x0 [0 0 1] , [0] [0] = [0] [1], [1 0 0] [length](x0) = [0 0 0]x0 [0 0 1] , [0] [nil] = [0] [0], [1 0 0] [mark](x0) = [0 0 0]x0 [0 1 1] , [1 0 0] [1 0 1] [1] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [1 1 1] [0] [s](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 1 0] [active](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [from](x0) = [0 0 0]x0 [0 0 0] orientation: [1 0 0] [1 0 1] [1] [1 0 1] [0] active(length(cons(X,Y))) = [0 0 0]X + [0 0 0]Y + [0] >= [0 0 0]Y + [0] = mark(s(length1(Y))) [0 0 0] [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] active(length1(X)) = [0 0 0]X >= [0 0 0]X = mark(length(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] mark(from(X)) = [0 0 0]X >= [0 0 0]X = active(from(mark(X))) [0 0 0] [0 0 0] [1 0 0] [1 0 1] [1] [1 0 0] [1 0 1] [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] [1] [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 0 0] [1 0 0] mark(length(X)) = [0 0 0]X >= [0 0 0]X = active(length(X)) [0 0 1] [0 0 1] [0] [0] mark(nil()) = [0] >= [0] = active(nil()) [0] [0] [0] [0] mark(0()) = [0] >= [0] = active(0()) [1] [1] [1 0 0] [1 0 0] mark(length1(X)) = [0 0 0]X >= [0 0 0]X = active(length1(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] from(mark(X)) = [0 0 0]X >= [0 0 0]X = from(X) [0 0 0] [0 0 0] [1 1 0] [1 0 0] from(active(X)) = [0 0 0]X >= [0 0 0]X = from(X) [0 0 0] [0 0 0] [1 0 0] [1 0 1] [1] [1 0 0] [1 0 1] [1] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = cons(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 1 1] [1] [1 0 0] [1 0 1] [1] cons(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = cons(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 0] [1 0 1] [1] [1 0 0] [1 0 1] [1] cons(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = cons(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 1 1] [1] [1 0 0] [1 0 1] [1] cons(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = cons(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 1] [0] [1 1 1] [0] s(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(X) [0 0 0] [1] [0 0 0] [1] [1 1 1] [0] [1 1 1] [0] s(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(X) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] length(mark(X)) = [0 0 0]X >= [0 0 0]X = length(X) [0 1 1] [0 0 1] [1 1 0] [1 0 0] length(active(X)) = [0 0 0]X >= [0 0 0]X = length(X) [0 0 1] [0 0 1] [1 0 0] [1 0 0] length1(mark(X)) = [0 0 0]X >= [0 0 0]X = length1(X) [0 1 1] [0 0 1] [1 1 0] [1 0 0] length1(active(X)) = [0 0 0]X >= [0 0 0]X = length1(X) [0 0 1] [0 0 1] problem: active(length1(X)) -> mark(length(X)) mark(from(X)) -> active(from(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(X)) mark(nil()) -> active(nil()) mark(0()) -> active(0()) mark(length1(X)) -> active(length1(X)) from(mark(X)) -> from(X) from(active(X)) -> from(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) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) length1(mark(X)) -> length1(X) length1(active(X)) -> length1(X) Matrix Interpretation Processor: dim=3 interpretation: [1 1 1] [0] [length1](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [0] [0] = [0] [0], [1 1 0] [length](x0) = [0 0 0]x0 [0 0 0] , [0] [nil] = [0] [0], [1 1 0] [0] [mark](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 1 0] [1 0 0] [1] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [0], [1 1 0] [1] [s](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 1 0] [0] [active](x0) = [0 0 1]x0 + [0] [0 0 0] [1], [1 1 0] [from](x0) = [0 0 0]x0 [0 0 0] orientation: [1 1 1] [1] [1 1 0] [0] active(length1(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(length(X)) [0 0 0] [1] [0 0 0] [1] [1 1 0] [0] [1 1 0] [0] mark(from(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(from(mark(X))) [0 0 0] [1] [0 0 0] [1] [1 1 0] [1 0 0] [1] [1 1 0] [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] [1] [0 0 0] [0 0 0] [1] [1 1 0] [1] [1 1 0] [1] 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 0] [0] [1 1 0] [0] mark(length(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(length(X)) [0 0 0] [1] [0 0 0] [1] [0] [0] mark(nil()) = [0] >= [0] = active(nil()) [1] [1] [0] [0] mark(0()) = [0] >= [0] = active(0()) [1] [1] [1 1 1] [1] [1 1 1] [1] mark(length1(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(length1(X)) [0 0 0] [1] [0 0 0] [1] [1 1 0] [1 1 0] from(mark(X)) = [0 0 0]X >= [0 0 0]X = from(X) [0 0 0] [0 0 0] [1 1 1] [1 1 0] from(active(X)) = [0 0 0]X >= [0 0 0]X = from(X) [0 0 0] [0 0 0] [1 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [1] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = cons(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 1 0] [1] [1 1 0] [1 0 0] [1] cons(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = cons(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 1] [1 0 0] [1] [1 1 0] [1 0 0] [1] cons(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = cons(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 1 0] [1] [1 1 0] [1 0 0] [1] cons(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = cons(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1] [1 1 0] [1] s(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(X) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1] [1 1 0] [1] s(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(X) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 1 0] length(mark(X)) = [0 0 0]X >= [0 0 0]X = length(X) [0 0 0] [0 0 0] [1 1 1] [1 1 0] length(active(X)) = [0 0 0]X >= [0 0 0]X = length(X) [0 0 0] [0 0 0] [1 1 1] [1] [1 1 1] [0] length1(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = length1(X) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1] [1 1 1] [0] length1(active(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = length1(X) [0 0 0] [0] [0 0 0] [0] problem: mark(from(X)) -> active(from(mark(X))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(X)) mark(nil()) -> active(nil()) mark(0()) -> active(0()) mark(length1(X)) -> active(length1(X)) from(mark(X)) -> from(X) from(active(X)) -> from(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) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [length1](x0) = [0 0 0]x0 [0 0 0] , [0] [0] = [0] [0], [1 1 0] [length](x0) = [0 0 0]x0 [0 0 0] , [0] [nil] = [0] [0], [1 1 0] [mark](x0) = [0 0 1]x0 [0 1 0] , [1 0 0] [1 0 0] [1] [cons](x0, x1) = [0 1 1]x0 + [0 0 0]x1 + [0] [0 1 1] [0 0 0] [0], [1 0 0] [1] [s](x0) = [0 1 1]x0 + [0] [0 1 1] [0], [active](x0) = x0 , [1 0 0] [0] [from](x0) = [0 1 1]x0 + [1] [0 1 1] [1] orientation: [1 1 1] [1] [1 1 0] [0] mark(from(X)) = [0 1 1]X + [1] >= [0 1 1]X + [1] = active(from(mark(X))) [0 1 1] [1] [0 1 1] [1] [1 1 1] [1 0 0] [1] [1 1 0] [1 0 0] [1] mark(cons(X1,X2)) = [0 1 1]X1 + [0 0 0]X2 + [0] >= [0 1 1]X1 + [0 0 0]X2 + [0] = active(cons(mark(X1),X2)) [0 1 1] [0 0 0] [0] [0 1 1] [0 0 0] [0] [1 1 1] [1] [1 1 0] [1] mark(s(X)) = [0 1 1]X + [0] >= [0 1 1]X + [0] = active(s(mark(X))) [0 1 1] [0] [0 1 1] [0] [1 1 0] [1 1 0] mark(length(X)) = [0 0 0]X >= [0 0 0]X = active(length(X)) [0 0 0] [0 0 0] [0] [0] mark(nil()) = [0] >= [0] = active(nil()) [0] [0] [0] [0] mark(0()) = [0] >= [0] = active(0()) [0] [0] [1 0 0] [1 0 0] mark(length1(X)) = [0 0 0]X >= [0 0 0]X = active(length1(X)) [0 0 0] [0 0 0] [1 1 0] [0] [1 0 0] [0] from(mark(X)) = [0 1 1]X + [1] >= [0 1 1]X + [1] = from(X) [0 1 1] [1] [0 1 1] [1] [1 0 0] [0] [1 0 0] [0] from(active(X)) = [0 1 1]X + [1] >= [0 1 1]X + [1] = from(X) [0 1 1] [1] [0 1 1] [1] [1 1 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] cons(mark(X1),X2) = [0 1 1]X1 + [0 0 0]X2 + [0] >= [0 1 1]X1 + [0 0 0]X2 + [0] = cons(X1,X2) [0 1 1] [0 0 0] [0] [0 1 1] [0 0 0] [0] [1 0 0] [1 1 0] [1] [1 0 0] [1 0 0] [1] cons(X1,mark(X2)) = [0 1 1]X1 + [0 0 0]X2 + [0] >= [0 1 1]X1 + [0 0 0]X2 + [0] = cons(X1,X2) [0 1 1] [0 0 0] [0] [0 1 1] [0 0 0] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] cons(active(X1),X2) = [0 1 1]X1 + [0 0 0]X2 + [0] >= [0 1 1]X1 + [0 0 0]X2 + [0] = cons(X1,X2) [0 1 1] [0 0 0] [0] [0 1 1] [0 0 0] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] cons(X1,active(X2)) = [0 1 1]X1 + [0 0 0]X2 + [0] >= [0 1 1]X1 + [0 0 0]X2 + [0] = cons(X1,X2) [0 1 1] [0 0 0] [0] [0 1 1] [0 0 0] [0] [1 1 0] [1] [1 0 0] [1] s(mark(X)) = [0 1 1]X + [0] >= [0 1 1]X + [0] = s(X) [0 1 1] [0] [0 1 1] [0] [1 0 0] [1] [1 0 0] [1] s(active(X)) = [0 1 1]X + [0] >= [0 1 1]X + [0] = s(X) [0 1 1] [0] [0 1 1] [0] [1 1 1] [1 1 0] length(mark(X)) = [0 0 0]X >= [0 0 0]X = length(X) [0 0 0] [0 0 0] [1 1 0] [1 1 0] length(active(X)) = [0 0 0]X >= [0 0 0]X = length(X) [0 0 0] [0 0 0] problem: mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(X)) mark(nil()) -> active(nil()) mark(0()) -> active(0()) mark(length1(X)) -> active(length1(X)) from(mark(X)) -> from(X) from(active(X)) -> from(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) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [length1](x0) = [0 0 0]x0 [0 0 0] , [0] [0] = [0] [0], [1 0 0] [length](x0) = [0 0 0]x0 [0 0 0] , [0] [nil] = [0] [0], [1 0 1] [0] [mark](x0) = [0 0 0]x0 + [1] [0 0 1] [0], [1 0 0] [1 0 0] [0] [cons](x0, x1) = [1 0 0]x0 + [0 0 0]x1 + [0] [0 0 1] [0 0 0] [1], [1 0 0] [1] [s](x0) = [0 0 0]x0 + [0] [0 0 1] [0], [1 0 0] [active](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [from](x0) = [0 0 0]x0 [0 0 0] orientation: [1 0 1] [1 0 0] [1] [1 0 1] [1 0 0] [0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0] = active(cons(mark(X1),X2)) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 0 1] [1] [1 0 1] [1] mark(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = active(s(mark(X))) [0 0 1] [0] [0 0 1] [0] [1 0 0] [0] [1 0 0] mark(length(X)) = [0 0 0]X + [1] >= [0 0 0]X = active(length(X)) [0 0 0] [0] [0 0 0] [0] [0] mark(nil()) = [1] >= [0] = active(nil()) [0] [0] [0] [0] mark(0()) = [1] >= [0] = active(0()) [0] [0] [1 0 0] [0] [1 0 0] mark(length1(X)) = [0 0 0]X + [1] >= [0 0 0]X = active(length1(X)) [0 0 0] [0] [0 0 0] [1 0 1] [1 0 0] from(mark(X)) = [0 0 0]X >= [0 0 0]X = from(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] from(active(X)) = [0 0 0]X >= [0 0 0]X = from(X) [0 0 0] [0 0 0] [1 0 1] [1 0 0] [0] [1 0 0] [1 0 0] [0] cons(mark(X1),X2) = [1 0 1]X1 + [0 0 0]X2 + [0] >= [1 0 0]X1 + [0 0 0]X2 + [0] = cons(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 0] [1 0 0] [0] cons(X1,mark(X2)) = [1 0 0]X1 + [0 0 0]X2 + [0] >= [1 0 0]X1 + [0 0 0]X2 + [0] = cons(X1,X2) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] cons(active(X1),X2) = [1 0 0]X1 + [0 0 0]X2 + [0] >= [1 0 0]X1 + [0 0 0]X2 + [0] = cons(X1,X2) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] cons(X1,active(X2)) = [1 0 0]X1 + [0 0 0]X2 + [0] >= [1 0 0]X1 + [0 0 0]X2 + [0] = cons(X1,X2) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 0 1] [1] [1 0 0] [1] s(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(X) [0 0 1] [0] [0 0 1] [0] [1 0 0] [1] [1 0 0] [1] s(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(X) [0 0 1] [0] [0 0 1] [0] [1 0 1] [1 0 0] length(mark(X)) = [0 0 0]X >= [0 0 0]X = length(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] length(active(X)) = [0 0 0]X >= [0 0 0]X = length(X) [0 0 0] [0 0 0] problem: mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(X)) mark(nil()) -> active(nil()) mark(0()) -> active(0()) mark(length1(X)) -> active(length1(X)) from(mark(X)) -> from(X) from(active(X)) -> from(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) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [length1](x0) = [0 0 0]x0 [0 0 0] , [0] [0] = [0] [0], [1 0 0] [1] [length](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [0] [nil] = [0] [0], [1 1 1] [0] [mark](x0) = [0 0 0]x0 + [1] [0 1 1] [0], [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [0] [s](x0) = [0 1 1]x0 + [0] [0 0 0] [1], [1 0 0] [active](x0) = [0 0 1]x0 [0 1 0] , [1 0 0] [from](x0) = [0 0 0]x0 [0 0 0] orientation: [1 1 1] [1] [1 1 1] [0] mark(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(s(mark(X))) [0 1 1] [1] [0 1 1] [1] [1 0 0] [1] [1 0 0] [1] mark(length(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = active(length(X)) [0 0 0] [0] [0 0 0] [0] [0] [0] mark(nil()) = [1] >= [0] = active(nil()) [0] [0] [0] [0] mark(0()) = [1] >= [0] = active(0()) [0] [0] [1 0 0] [0] [1 0 0] mark(length1(X)) = [0 0 0]X + [1] >= [0 0 0]X = active(length1(X)) [0 0 0] [0] [0 0 0] [1 1 1] [1 0 0] from(mark(X)) = [0 0 0]X >= [0 0 0]X = from(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] from(active(X)) = [0 0 0]X >= [0 0 0]X = from(X) [0 0 0] [0 0 0] [1 1 1] [1 0 0] [1 0 0] [1 0 0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 1 1] [1 0 0] [1 0 0] cons(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(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] cons(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(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] cons(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [0] [1 0 0] [0] s(mark(X)) = [0 1 1]X + [1] >= [0 1 1]X + [0] = s(X) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] s(active(X)) = [0 1 1]X + [0] >= [0 1 1]X + [0] = s(X) [0 0 0] [1] [0 0 0] [1] [1 1 1] [1] [1 0 0] [1] length(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length(X) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] length(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length(X) [0 0 0] [0] [0 0 0] [0] problem: mark(length(X)) -> active(length(X)) mark(nil()) -> active(nil()) mark(0()) -> active(0()) mark(length1(X)) -> active(length1(X)) from(mark(X)) -> from(X) from(active(X)) -> from(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) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1] [length1](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [0] [0] = [0] [0], [1 0 1] [0] [length](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [0] [nil] = [0] [1], [1 0 1] [0] [mark](x0) = [0 1 0]x0 + [1] [0 0 0] [1], [1 0 1] [1 0 1] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 1] [s](x0) = [0 0 0]x0 [0 0 0] , [0] [active](x0) = x0 + [1] [0], [1 1 0] [from](x0) = [0 0 0]x0 [0 0 0] orientation: [1 0 1] [1] [1 0 1] [0] mark(length(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = active(length(X)) [0 0 0] [1] [0 0 0] [1] [1] [0] mark(nil()) = [1] >= [1] = active(nil()) [1] [1] [0] [0] mark(0()) = [1] >= [1] = active(0()) [1] [0] [1 0 0] [1] [1 0 0] [1] mark(length1(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(length1(X)) [0 0 0] [1] [0 0 0] [0] [1 1 1] [1] [1 1 0] from(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X = from(X) [0 0 0] [0] [0 0 0] [1 1 0] [1] [1 1 0] 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 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] [0 0 0] [0] [0 0 0] [0 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] [0 0 0] [0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] cons(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(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] cons(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 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 0 1] s(active(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 0 1] [1] [1 0 1] [0] length(mark(X)) = [0 1 0]X + [1] >= [0 1 0]X + [0] = length(X) [0 0 0] [1] [0 0 0] [1] [1 0 1] [0] [1 0 1] [0] length(active(X)) = [0 1 0]X + [1] >= [0 1 0]X + [0] = length(X) [0 0 0] [1] [0 0 0] [1] problem: mark(0()) -> active(0()) mark(length1(X)) -> active(length1(X)) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) s(active(X)) -> s(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [length1](x0) = [0 0 0]x0 + [1] [0 1 0] [0], [0] [0] = [0] [0], [1 0 1] [length](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [0] [mark](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 1] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 1]x1 [0 0 0] [0 0 0] , [1 0 0] [0] [s](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 0] [0] [active](x0) = [0 0 0]x0 + [0] [0 0 1] [1] orientation: [0] [0] mark(0()) = [0] >= [0] = active(0()) [1] [1] [1 0 0] [1] [1 0 0] [0] mark(length1(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(length1(X)) [0 1 0] [1] [0 1 0] [1] [1 0 1] [1 0 0] [1] [1 0 1] [1 0 0] cons(active(X1),X2) = [0 0 0]X1 + [0 0 1]X2 + [0] >= [0 0 0]X1 + [0 0 1]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [1 0 1] [1 0 0] [0] [1 0 1] [1 0 0] cons(X1,active(X2)) = [0 0 0]X1 + [0 0 1]X2 + [1] >= [0 0 0]X1 + [0 0 1]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [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 0 1] [1] [1 0 1] length(active(X)) = [0 0 0]X + [0] >= [0 0 0]X = length(X) [0 0 0] [0] [0 0 0] problem: mark(0()) -> active(0()) cons(X1,active(X2)) -> cons(X1,X2) s(active(X)) -> s(X) Matrix Interpretation Processor: dim=3 interpretation: [0] [0] = [1] [0], [1 0 0] [1] [mark](x0) = [0 0 0]x0 + [0] [0 1 0] [0], [1 0 0] [1 0 1] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 1] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [active](x0) = [0 0 0]x0 + [0] [0 0 1] [1] orientation: [1] [0] mark(0()) = [0] >= [0] = active(0()) [1] [1] [1 0 0] [1 0 1] [1] [1 0 0] [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] [0 0 0] [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