YES Problem: active(zeros()) -> mark(cons(0(),zeros())) active(tail(cons(X,XS))) -> mark(XS) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(tail(X)) -> active(tail(mark(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) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Proof: Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1] [tail](x0) = [1 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [mark](x0) = [1 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [1 0 0]x1 [0 0 0] [1 0 0] , [0] [0] = [0] [0], [1 0 0] [active](x0) = [1 0 0]x0 [0 0 0] , [1] [zeros] = [0] [0] orientation: [1] [1] active(zeros()) = [1] >= [1] = mark(cons(0(),zeros())) [0] [0] [1 0 0] [1 0 0] [1] [1 0 0] active(tail(cons(X,XS))) = [1 0 0]X + [1 0 0]XS + [1] >= [1 0 0]XS = mark(XS) [0 0 0] [0 0 0] [0] [0 0 0] [1] [1] mark(zeros()) = [1] >= [1] = active(zeros()) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] mark(cons(X1,X2)) = [1 0 0]X1 + [1 0 0]X2 >= [1 0 0]X1 + [1 0 0]X2 = active(cons(mark(X1),X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] mark(0()) = [0] >= [0] = active(0()) [0] [0] [1 0 0] [1] [1 0 0] [1] mark(tail(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = active(tail(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(mark(X1),X2) = [0 0 0]X1 + [1 0 0]X2 >= [0 0 0]X1 + [1 0 0]X2 = cons(X1,X2) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(X1,mark(X2)) = [0 0 0]X1 + [1 0 0]X2 >= [0 0 0]X1 + [1 0 0]X2 = cons(X1,X2) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(active(X1),X2) = [0 0 0]X1 + [1 0 0]X2 >= [0 0 0]X1 + [1 0 0]X2 = cons(X1,X2) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(X1,active(X2)) = [0 0 0]X1 + [1 0 0]X2 >= [0 0 0]X1 + [1 0 0]X2 = cons(X1,X2) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1] tail(mark(X)) = [1 0 0]X + [0] >= [1 0 0]X + [0] = tail(X) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] tail(active(X)) = [1 0 0]X + [0] >= [1 0 0]X + [0] = tail(X) [0 0 0] [0] [0 0 0] [0] problem: active(zeros()) -> mark(cons(0(),zeros())) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(tail(X)) -> active(tail(mark(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) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [1] [tail](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 1] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [1 0 0] [cons](x0, x1) = [1 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [0] = [0] [0], [1 0 1] [active](x0) = [0 0 0]x0 [0 0 0] , [0] [zeros] = [0] [1] orientation: [1] [0] active(zeros()) = [0] >= [0] = mark(cons(0(),zeros())) [0] [0] [1] [1] mark(zeros()) = [0] >= [0] = active(zeros()) [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] [0] [0] mark(0()) = [0] >= [0] = active(0()) [0] [0] [1 0 1] [1] [1 0 1] [1] mark(tail(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(tail(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1 0 0] [1 0 1] [1 0 0] cons(mark(X1),X2) = [1 0 1]X1 + [0 0 0]X2 >= [1 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 0] cons(X1,mark(X2)) = [1 0 0]X1 + [0 0 0]X2 >= [1 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 0] [1 0 1] [1 0 0] cons(active(X1),X2) = [1 0 1]X1 + [0 0 0]X2 >= [1 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 0] cons(X1,active(X2)) = [1 0 0]X1 + [0 0 0]X2 >= [1 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] [1] tail(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = tail(X) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1] [1 0 1] [1] tail(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = tail(X) [0 0 0] [0] [0 0 0] [0] problem: mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(tail(X)) -> active(tail(mark(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) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Matrix Interpretation Processor: dim=1 interpretation: [tail](x0) = 4x0 + 3, [mark](x0) = 2x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 4, [active](x0) = x0, [zeros] = 4 orientation: mark(zeros()) = 9 >= 4 = active(zeros()) mark(cons(X1,X2)) = 2X1 + 2X2 + 1 >= 2X1 + X2 + 1 = active(cons(mark(X1),X2)) mark(0()) = 9 >= 4 = active(0()) mark(tail(X)) = 8X + 7 >= 8X + 7 = active(tail(mark(X))) cons(mark(X1),X2) = 2X1 + X2 + 1 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + 2X2 + 1 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) tail(mark(X)) = 8X + 7 >= 4X + 3 = tail(X) tail(active(X)) = 4X + 3 >= 4X + 3 = tail(X) problem: mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(tail(X)) -> active(tail(mark(X))) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) tail(active(X)) -> tail(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [tail](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 1 0] [0] [mark](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [1 0 0] [1 0 0] [0] [cons](x0, x1) = [0 1 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [0], [1 0 0] [0] [active](x0) = [0 1 0]x0 + [0] [0 0 0] [1] orientation: [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 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 0] [1] [1 1 0] [0] mark(tail(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = active(tail(mark(X))) [0 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) = [0 1 0]X1 + [0 0 0]X2 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [1] = cons(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] cons(X1,active(X2)) = [0 1 0]X1 + [0 0 0]X2 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [1] = cons(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] tail(active(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = tail(X) [0 0 0] [0] [0 0 0] [0] problem: cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) tail(active(X)) -> tail(X) Matrix Interpretation Processor: dim=1 interpretation: [tail](x0) = 4x0 + 3, [cons](x0, x1) = x0 + 2x1 + 4, [active](x0) = x0 + 5 orientation: cons(active(X1),X2) = X1 + 2X2 + 9 >= X1 + 2X2 + 4 = cons(X1,X2) cons(X1,active(X2)) = X1 + 2X2 + 14 >= X1 + 2X2 + 4 = cons(X1,X2) tail(active(X)) = 4X + 23 >= 4X + 3 = tail(X) problem: Qed