YES Problem: app(nil(),k) -> k app(l,nil()) -> l app(cons(x,l),k) -> cons(x,app(l,k)) sum(cons(x,nil())) -> cons(x,nil()) sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) plus(0(),y) -> y plus(s(x),y) -> s(plus(x,y)) sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l)))) pred(cons(s(x),nil())) -> cons(x,nil()) Proof: Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [plus](x0, x1) = [0 0 0]x0 + x1 + [1] [1 0 0] [0], [1] [app](x0, x1) = x0 + x1 + [0] [0], [1 0 0] [sum](x0) = [0 0 0]x0 [0 0 0] , [0] [0] = [0] [0], [0] [nil] = [0] [0], [1 0 0] [pred](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] orientation: [1] app(nil(),k) = k + [0] >= k = k [0] [1] app(l,nil()) = l + [0] >= l = l [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1] app(cons(x,l),k) = k + [0 0 0]l + [0 0 0]x + [0] >= [0 0 0]k + [0 0 0]l + [0 0 0]x + [0] = cons(x,app(l,k)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] sum(cons(x,nil())) = [0 0 0]x >= [0 0 0]x = cons(x,nil()) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] sum(cons(x,cons(y,l))) = [0 0 0]l + [0 0 0]x + [0 0 0]y >= [0 0 0]l + [0 0 0]x + [0 0 0]y = sum(cons(plus(x,y),l)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1] sum(app(l,cons(x,cons(y,k)))) = [0 0 0]k + [0 0 0]l + [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]k + [0 0 0]l + [0 0 0]x + [0 0 0]y + [0] = sum(app(l,sum(cons(x,cons(y,k))))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] plus(0(),y) = y + [1] >= y = y [0] [1 0 0] [0] [1 0 0] [1 0 0] plus(s(x),y) = [0 0 0]x + y + [1] >= [0 0 0]x + [0 0 0]y = s(plus(x,y)) [1 0 0] [0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] sum(plus(cons(0(),x),cons(y,l))) = [0 0 0]l + [0 0 0]x + [0 0 0]y >= [0 0 0]l + [0 0 0]x + [0 0 0]y = pred(sum(cons(s(x),cons(y,l)))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] pred(cons(s(x),nil())) = [0 0 0]x >= [0 0 0]x = cons(x,nil()) [0 0 0] [0 0 0] problem: app(cons(x,l),k) -> cons(x,app(l,k)) sum(cons(x,nil())) -> cons(x,nil()) sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) plus(0(),y) -> y plus(s(x),y) -> s(plus(x,y)) sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l)))) pred(cons(s(x),nil())) -> cons(x,nil()) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [plus](x0, x1) = [0 0 0]x0 + [0 1 1]x1 [0 0 0] [0 1 1] , [1 1 1] [1 0 0] [app](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 0 1] [0 0 0] , [1 0 0] [0] [sum](x0) = [1 0 0]x0 + [0] [0 0 0] [1], [0] [0] = [0] [0], [0] [nil] = [0] [0], [1 0 0] [0] [pred](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [s](x0) = [0 0 0]x0 [0 1 0] , [1 0 0] [1 0 0] [0] [cons](x0, x1) = [0 0 0]x0 + [0 1 1]x1 + [0] [0 0 0] [0 0 0] [1] orientation: [1 0 0] [1 1 1] [1 0 0] [1] [1 0 0] [1 1 1] [1 0 0] [0] app(cons(x,l),k) = [0 0 0]k + [0 1 1]l + [0 0 0]x + [0] >= [0 0 0]k + [0 1 1]l + [0 0 0]x + [0] = cons(x,app(l,k)) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] sum(cons(x,nil())) = [1 0 0]x + [0] >= [0 0 0]x + [0] = cons(x,nil()) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] sum(cons(x,cons(y,l))) = [1 0 0]l + [1 0 0]x + [1 0 0]y + [0] >= [1 0 0]l + [1 0 0]x + [1 0 0]y + [0] = sum(cons(plus(x,y),l)) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [1 0 0] [1 1 1] [1 0 0] [1 0 0] [0] [1 0 0] [1 1 1] [1 0 0] [1 0 0] [0] sum(app(l,cons(x,cons(y,k)))) = [1 0 0]k + [1 1 1]l + [1 0 0]x + [1 0 0]y + [0] >= [1 0 0]k + [1 1 1]l + [1 0 0]x + [1 0 0]y + [0] = sum(app(l,sum(cons(x,cons(y,k))))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1] [1 0 0] plus(0(),y) = [0 1 1]y >= y = y [0 1 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] plus(s(x),y) = [0 0 0]x + [0 1 1]y >= [0 0 0]x + [0 0 0]y = s(plus(x,y)) [0 0 0] [0 1 1] [0 0 0] [0 1 1] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] sum(plus(cons(0(),x),cons(y,l))) = [1 0 0]l + [1 0 0]x + [1 0 0]y + [0] >= [0 0 0]l + [0 0 0]x + [0 0 0]y + [0] = pred(sum(cons(s(x),cons(y,l)))) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] pred(cons(s(x),nil())) = [0 0 0]x + [0] >= [0 0 0]x + [0] = cons(x,nil()) [0 0 0] [1] [0 0 0] [1] problem: sum(cons(x,nil())) -> cons(x,nil()) sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) plus(0(),y) -> y plus(s(x),y) -> s(plus(x,y)) sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l)))) pred(cons(s(x),nil())) -> cons(x,nil()) Matrix Interpretation Processor: dim=1 interpretation: [plus](x0, x1) = x0 + x1, [app](x0, x1) = 2x0 + x1 + 2, [sum](x0) = x0, [0] = 2, [nil] = 7, [pred](x0) = x0 + 2, [s](x0) = x0, [cons](x0, x1) = x0 + x1 + 1 orientation: sum(cons(x,nil())) = x + 8 >= x + 8 = cons(x,nil()) sum(cons(x,cons(y,l))) = l + x + y + 2 >= l + x + y + 1 = sum(cons(plus(x,y),l)) sum(app(l,cons(x,cons(y,k)))) = k + 2l + x + y + 4 >= k + 2l + x + y + 4 = sum(app(l,sum(cons(x,cons(y,k))))) plus(0(),y) = y + 2 >= y = y plus(s(x),y) = x + y >= x + y = s(plus(x,y)) sum(plus(cons(0(),x),cons(y,l))) = l + x + y + 4 >= l + x + y + 4 = pred(sum(cons(s(x),cons(y,l)))) pred(cons(s(x),nil())) = x + 10 >= x + 8 = cons(x,nil()) problem: sum(cons(x,nil())) -> cons(x,nil()) sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) plus(s(x),y) -> s(plus(x,y)) sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l)))) Matrix Interpretation Processor: dim=1 interpretation: [plus](x0, x1) = 2x0 + x1 + 1, [app](x0, x1) = 4x0 + 2x1, [sum](x0) = x0, [0] = 6, [nil] = 0, [pred](x0) = x0 + 3, [s](x0) = x0, [cons](x0, x1) = x0 + x1 orientation: sum(cons(x,nil())) = x >= x = cons(x,nil()) sum(app(l,cons(x,cons(y,k)))) = 2k + 4l + 2x + 2y >= 2k + 4l + 2x + 2y = sum(app(l,sum(cons(x,cons(y,k))))) plus(s(x),y) = 2x + y + 1 >= 2x + y + 1 = s(plus(x,y)) sum(plus(cons(0(),x),cons(y,l))) = l + 2x + y + 13 >= l + x + y + 3 = pred(sum(cons(s(x),cons(y,l)))) problem: sum(cons(x,nil())) -> cons(x,nil()) sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) plus(s(x),y) -> s(plus(x,y)) Matrix Interpretation Processor: dim=1 interpretation: [plus](x0, x1) = 3x0 + x1 + 1, [app](x0, x1) = x0 + x1 + 2, [sum](x0) = x0, [nil] = 0, [s](x0) = x0 + 7, [cons](x0, x1) = x0 + x1 orientation: sum(cons(x,nil())) = x >= x = cons(x,nil()) sum(app(l,cons(x,cons(y,k)))) = k + l + x + y + 2 >= k + l + x + y + 2 = sum(app(l,sum(cons(x,cons(y,k))))) plus(s(x),y) = 3x + y + 22 >= 3x + y + 8 = s(plus(x,y)) problem: sum(cons(x,nil())) -> cons(x,nil()) sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [1 0 0] [1] [app](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [1 1 0] [sum](x0) = [0 1 0]x0 [0 1 0] , [0] [nil] = [0] [1], [1 1 0] [1 1 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 1]x1 [0 0 0] [0 0 0] orientation: [1 1 0] [1] [1 1 0] [0] sum(cons(x,nil())) = [0 0 0]x + [1] >= [0 0 0]x + [1] = cons(x,nil()) [0 0 0] [1] [0 0 0] [0] [1 1 1] [1 0 1] [1 1 0] [1 1 0] [1] [1 1 1] [1 0 1] [1 1 0] [1 1 0] [1] sum(app(l,cons(x,cons(y,k)))) = [0 0 0]k + [0 0 0]l + [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]k + [0 0 0]l + [0 0 0]x + [0 0 0]y + [0] = sum(app(l,sum(cons(x,cons(y,k))))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] problem: sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 1] [app](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [sum](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [1] orientation: [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] sum(app(l,cons(x,cons(y,k)))) = [0 0 0]k + [0 0 0]l + [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]k + [0 0 0]l + [0 0 0]x + [0 0 0]y = sum(app(l,sum(cons(x,cons(y,k))))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] problem: Qed