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 1] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [length1](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 1] [active](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [1 0 0] [0] [cons](x0, x1) = [0 0 0]x0 + [1 0 0]x1 + [1] [0 0 0] [0 0 0] [0], [0] [nil] = [0] [0], [1 0 1] [from](x0) = [0 0 0]x0 [1 0 1] , [0] [0] = [0] [0], [1 0 0] [0] [length](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 1] [s](x0) = [0 0 0]x0 [0 0 0] orientation: [2 0 2] [2 0 2] active(from(X)) = [0 0 0]X >= [0 0 0]X = mark(cons(X,from(s(X)))) [0 0 0] [0 0 0] [1] [0] active(length(nil())) = [0] >= [0] = mark(0()) [0] [0] [1 0 1] [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] [0] [0 0 0] [0] [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] [0] [0 0 0] [0] [2 0 2] [2 0 2] mark(from(X)) = [0 0 0]X >= [0 0 0]X = active(from(mark(X))) [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1 0 1] [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 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 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] [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] [1 0 0] [1] mark(length1(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(length1(X)) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1 0 1] from(mark(X)) = [0 0 0]X >= [0 0 0]X = from(X) [1 0 1] [1 0 1] [1 0 1] [1 0 1] from(active(X)) = [0 0 0]X >= [0 0 0]X = from(X) [1 0 1] [1 0 1] [1 0 1] [1 0 0] [0] [1 0 1] [1 0 0] [0] cons(mark(X1),X2) = [0 0 0]X1 + [1 0 0]X2 + [1] >= [0 0 0]X1 + [1 0 0]X2 + [1] = cons(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 0] [0] cons(X1,mark(X2)) = [0 0 0]X1 + [1 0 1]X2 + [1] >= [0 0 0]X1 + [1 0 0]X2 + [1] = cons(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 1] [1 0 0] [0] cons(active(X1),X2) = [0 0 0]X1 + [1 0 0]X2 + [1] >= [0 0 0]X1 + [1 0 0]X2 + [1] = cons(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 0] [0] cons(X1,active(X2)) = [0 0 0]X1 + [1 0 1]X2 + [1] >= [0 0 0]X1 + [1 0 0]X2 + [1] = cons(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [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] [1 0 1] [0] [1 0 0] [0] length(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length(X) [0 0 0] [1] [0 0 0] [1] [1 0 1] [0] [1 0 0] [0] length(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length(X) [0 0 0] [1] [0 0 0] [1] [1 0 1] [0] [1 0 0] [0] length1(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length1(X) [0 0 0] [1] [0 0 0] [1] [1 0 1] [0] [1 0 0] [0] length1(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length1(X) [0 0 0] [1] [0 0 0] [1] problem: active(from(X)) -> mark(cons(X,from(s(X)))) 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=2 interpretation: [1 2] [mark](x0) = [0 0]x0, [1 0] [2] [length1](x0) = [0 0]x0 + [1], [1 2] [active](x0) = [0 0]x0, [1 2] [2 0] [2] [cons](x0, x1) = [0 0]x0 + [0 0]x1 + [0], [0] [nil] = [2], [1 2] [0] [from](x0) = [1 2]x0 + [2], [0] [0] = [0], [1 0] [0] [length](x0) = [0 0]x0 + [1], [1 2] [s](x0) = [0 0]x0 orientation: [3 6] [4] [3 6] [2] active(from(X)) = [0 0]X + [0] >= [0 0]X + [0] = mark(cons(X,from(s(X)))) [1 2] [2 0] [4] [1 0] [4] active(length(cons(X,Y))) = [0 0]X + [0 0]Y + [0] >= [0 0]Y + [0] = mark(s(length1(Y))) [1 0] [4] [1 0] [2] active(length1(X)) = [0 0]X + [0] >= [0 0]X + [0] = mark(length(X)) [3 6] [4] [3 6] [4] mark(from(X)) = [0 0]X + [0] >= [0 0]X + [0] = active(from(mark(X))) [1 2] [2 0] [2] [1 2] [2 0] [2] mark(cons(X1,X2)) = [0 0]X1 + [0 0]X2 + [0] >= [0 0]X1 + [0 0]X2 + [0] = active(cons(mark(X1),X2)) [1 2] [1 2] mark(s(X)) = [0 0]X >= [0 0]X = active(s(mark(X))) [1 0] [2] [1 0] [2] mark(length(X)) = [0 0]X + [0] >= [0 0]X + [0] = active(length(X)) [4] [4] mark(nil()) = [0] >= [0] = active(nil()) [0] [0] mark(0()) = [0] >= [0] = active(0()) [1 0] [4] [1 0] [4] mark(length1(X)) = [0 0]X + [0] >= [0 0]X + [0] = active(length1(X)) [1 2] [0] [1 2] [0] from(mark(X)) = [1 2]X + [2] >= [1 2]X + [2] = from(X) [1 2] [0] [1 2] [0] from(active(X)) = [1 2]X + [2] >= [1 2]X + [2] = from(X) [1 2] [2 0] [2] [1 2] [2 0] [2] cons(mark(X1),X2) = [0 0]X1 + [0 0]X2 + [0] >= [0 0]X1 + [0 0]X2 + [0] = cons(X1,X2) [1 2] [2 4] [2] [1 2] [2 0] [2] cons(X1,mark(X2)) = [0 0]X1 + [0 0]X2 + [0] >= [0 0]X1 + [0 0]X2 + [0] = cons(X1,X2) [1 2] [2 0] [2] [1 2] [2 0] [2] cons(active(X1),X2) = [0 0]X1 + [0 0]X2 + [0] >= [0 0]X1 + [0 0]X2 + [0] = cons(X1,X2) [1 2] [2 4] [2] [1 2] [2 0] [2] cons(X1,active(X2)) = [0 0]X1 + [0 0]X2 + [0] >= [0 0]X1 + [0 0]X2 + [0] = cons(X1,X2) [1 2] [1 2] s(mark(X)) = [0 0]X >= [0 0]X = s(X) [1 2] [1 2] s(active(X)) = [0 0]X >= [0 0]X = s(X) [1 2] [0] [1 0] [0] length(mark(X)) = [0 0]X + [1] >= [0 0]X + [1] = length(X) [1 2] [0] [1 0] [0] length(active(X)) = [0 0]X + [1] >= [0 0]X + [1] = length(X) [1 2] [2] [1 0] [2] length1(mark(X)) = [0 0]X + [1] >= [0 0]X + [1] = length1(X) [1 2] [2] [1 0] [2] length1(active(X)) = [0 0]X + [1] >= [0 0]X + [1] = length1(X) problem: active(length(cons(X,Y))) -> mark(s(length1(Y))) 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=1 interpretation: [mark](x0) = 2x0 + 1, [length1](x0) = 4x0 + 4, [active](x0) = x0 + 3, [cons](x0, x1) = x0 + 4x1 + 3, [nil] = 4, [from](x0) = x0 + 4, [0] = 4, [length](x0) = 6x0 + 4, [s](x0) = 2x0 + 4 orientation: active(length(cons(X,Y))) = 6X + 24Y + 25 >= 16Y + 25 = mark(s(length1(Y))) mark(from(X)) = 2X + 9 >= 2X + 8 = active(from(mark(X))) mark(cons(X1,X2)) = 2X1 + 8X2 + 7 >= 2X1 + 4X2 + 7 = active(cons(mark(X1),X2)) mark(s(X)) = 4X + 9 >= 4X + 9 = active(s(mark(X))) mark(length(X)) = 12X + 9 >= 6X + 7 = active(length(X)) mark(nil()) = 9 >= 7 = active(nil()) mark(0()) = 9 >= 7 = active(0()) mark(length1(X)) = 8X + 9 >= 4X + 7 = active(length1(X)) from(mark(X)) = 2X + 5 >= X + 4 = from(X) from(active(X)) = X + 7 >= X + 4 = from(X) cons(mark(X1),X2) = 2X1 + 4X2 + 4 >= X1 + 4X2 + 3 = cons(X1,X2) cons(X1,mark(X2)) = X1 + 8X2 + 7 >= X1 + 4X2 + 3 = cons(X1,X2) cons(active(X1),X2) = X1 + 4X2 + 6 >= X1 + 4X2 + 3 = cons(X1,X2) cons(X1,active(X2)) = X1 + 4X2 + 15 >= X1 + 4X2 + 3 = cons(X1,X2) s(mark(X)) = 4X + 6 >= 2X + 4 = s(X) s(active(X)) = 2X + 10 >= 2X + 4 = s(X) length(mark(X)) = 12X + 10 >= 6X + 4 = length(X) length(active(X)) = 6X + 22 >= 6X + 4 = length(X) length1(mark(X)) = 8X + 8 >= 4X + 4 = length1(X) length1(active(X)) = 4X + 16 >= 4X + 4 = length1(X) problem: active(length(cons(X,Y))) -> mark(s(length1(Y))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [0] [mark](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 0] [length1](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [active](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 1] [1 0 0] [0] [cons](x0, x1) = [0 1 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [1 0 0] [1] [length](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 1] [0] [s](x0) = [0 0 0]x0 + [1] [0 0 0] [1] orientation: [1 0 1] [1 0 0] [1] [1 0 0] [1] active(length(cons(X,Y))) = [0 0 0]X + [0 0 0]Y + [1] >= [0 0 0]Y + [1] = mark(s(length1(Y))) [0 0 0] [0 0 0] [0] [0 0 0] [0] [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 + [1] = active(cons(mark(X1),X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 1] [1] [1 0 1] [0] mark(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(s(mark(X))) [0 0 0] [0] [0 0 0] [0] problem: active(length(cons(X,Y))) -> mark(s(length1(Y))) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [length1](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1] [active](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 1]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [length](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] orientation: [1 0 0] [1 0 0] [1] [1 0 0] active(length(cons(X,Y))) = [0 0 0]X + [0 0 0]Y + [0] >= [0 0 0]Y = mark(s(length1(Y))) [0 0 0] [0 0 0] [0] [0 0 0] problem: Qed