NO Problem: nats() -> adx(zeros()) zeros() -> cons(n__0(),n__zeros()) incr(cons(X,Y)) -> cons(n__s(activate(X)),n__incr(activate(Y))) adx(cons(X,Y)) -> incr(cons(activate(X),n__adx(activate(Y)))) hd(cons(X,Y)) -> activate(X) tl(cons(X,Y)) -> activate(Y) 0() -> n__0() zeros() -> n__zeros() s(X) -> n__s(X) incr(X) -> n__incr(X) adx(X) -> n__adx(X) activate(n__0()) -> 0() activate(n__zeros()) -> zeros() activate(n__s(X)) -> s(X) activate(n__incr(X)) -> incr(activate(X)) activate(n__adx(X)) -> adx(activate(X)) activate(X) -> X Proof: Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [s](x0) = [0 1 0]x0 [0 1 0] , [0] [0] = [0] [0], [1 0 0] [1] [tl](x0) = [0 0 1]x0 + [0] [0 0 1] [1], [1 0 0] [hd](x0) = [0 0 1]x0 [1 1 0] , [1 0 0] [0] [n__adx](x0) = [1 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [0] [n__incr](x0) = [1 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [n__s](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [activate](x0) = [0 1 0]x0 [0 1 1] , [1 0 0] [0] [incr](x0) = [1 0 0]x0 + [0] [1 0 0] [1], [1 1 0] [1 0 0] [cons](x0, x1) = [0 0 1]x0 + [0 0 0]x1 [0 1 0] [0 1 1] , [0] [n__zeros] = [1] [0], [0] [n__0] = [0] [0], [1 0 0] [0] [adx](x0) = [1 0 0]x0 + [0] [1 0 0] [1], [0] [zeros] = [1] [1], [0] [nats] = [1] [1] orientation: [0] [0] nats() = [1] >= [0] = adx(zeros()) [1] [1] [0] [0] zeros() = [1] >= [0] = cons(n__0(),n__zeros()) [1] [1] [1 1 0] [1 0 0] [0] [1 1 0] [1 0 0] [0] incr(cons(X,Y)) = [1 1 0]X + [1 0 0]Y + [0] >= [0 0 0]X + [0 0 0]Y + [0] = cons(n__s(activate(X)),n__incr(activate(Y))) [1 1 0] [1 0 0] [1] [0 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [0] [1 1 0] [1 0 0] [0] adx(cons(X,Y)) = [1 1 0]X + [1 0 0]Y + [0] >= [1 1 0]X + [1 0 0]Y + [0] = incr(cons(activate(X),n__adx(activate(Y)))) [1 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [1 0 0] hd(cons(X,Y)) = [0 1 0]X + [0 1 1]Y >= [0 1 0]X = activate(X) [1 1 1] [1 0 0] [0 1 1] [1 1 0] [1 0 0] [1] [1 0 0] tl(cons(X,Y)) = [0 1 0]X + [0 1 1]Y + [0] >= [0 1 0]Y = activate(Y) [0 1 0] [0 1 1] [1] [0 1 1] [0] [0] 0() = [0] >= [0] = n__0() [0] [0] [0] [0] zeros() = [1] >= [1] = n__zeros() [1] [0] [1 0 0] [1 0 0] s(X) = [0 1 0]X >= [0 1 0]X = n__s(X) [0 1 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] incr(X) = [1 0 0]X + [0] >= [1 0 0]X + [0] = n__incr(X) [1 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] adx(X) = [1 0 0]X + [0] >= [1 0 0]X + [0] = n__adx(X) [1 0 0] [1] [0 0 0] [1] [0] [0] activate(n__0()) = [0] >= [0] = 0() [0] [0] [0] [0] activate(n__zeros()) = [1] >= [1] = zeros() [1] [1] [1 0 0] [1 0 0] activate(n__s(X)) = [0 1 0]X >= [0 1 0]X = s(X) [0 1 0] [0 1 0] [1 0 0] [0] [1 0 0] [0] activate(n__incr(X)) = [1 0 0]X + [0] >= [1 0 0]X + [0] = incr(activate(X)) [1 0 0] [1] [1 0 0] [1] [1 0 0] [0] [1 0 0] [0] activate(n__adx(X)) = [1 0 0]X + [0] >= [1 0 0]X + [0] = adx(activate(X)) [1 0 0] [1] [1 0 0] [1] [1 0 0] activate(X) = [0 1 0]X >= X = X [0 1 1] problem: nats() -> adx(zeros()) zeros() -> cons(n__0(),n__zeros()) incr(cons(X,Y)) -> cons(n__s(activate(X)),n__incr(activate(Y))) adx(cons(X,Y)) -> incr(cons(activate(X),n__adx(activate(Y)))) hd(cons(X,Y)) -> activate(X) 0() -> n__0() zeros() -> n__zeros() s(X) -> n__s(X) incr(X) -> n__incr(X) adx(X) -> n__adx(X) activate(n__0()) -> 0() activate(n__zeros()) -> zeros() activate(n__s(X)) -> s(X) activate(n__incr(X)) -> incr(activate(X)) activate(n__adx(X)) -> adx(activate(X)) activate(X) -> X Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [0] [0] = [0] [0], [1 0 0] [1] [hd](x0) = [1 0 1]x0 + [0] [1 0 0] [0], [1 0 0] [n__adx](x0) = [0 0 1]x0 [0 0 0] , [1 0 0] [n__incr](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [n__s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [activate](x0) = [0 1 1]x0 [0 0 1] , [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 1 0]x1 [0 1 0] [0 0 0] , [0] [n__zeros] = [1] [0], [0] [n__0] = [0] [0], [1 0 0] [adx](x0) = [0 0 1]x0 [0 0 0] , [0] [zeros] = [1] [0], [0] [nats] = [0] [1] orientation: [0] [0] nats() = [0] >= [0] = adx(zeros()) [1] [0] [0] [0] zeros() = [1] >= [1] = cons(n__0(),n__zeros()) [0] [0] [1 0 1] [1 0 0] [1 0 0] [1 0 0] incr(cons(X,Y)) = [0 0 0]X + [0 0 0]Y >= [0 0 0]X + [0 0 0]Y = cons(n__s(activate(X)),n__incr(activate(Y))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1 0 1] [1 0 0] adx(cons(X,Y)) = [0 1 0]X + [0 0 0]Y >= [0 0 0]X + [0 0 0]Y = incr(cons(activate(X),n__adx(activate(Y)))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1] [1 0 0] hd(cons(X,Y)) = [1 1 1]X + [1 0 0]Y + [0] >= [0 1 1]X = activate(X) [1 0 1] [1 0 0] [0] [0 0 1] [0] [0] 0() = [0] >= [0] = n__0() [0] [0] [0] [0] zeros() = [1] >= [1] = n__zeros() [0] [0] [1 0 0] [1 0 0] s(X) = [0 0 0]X >= [0 0 0]X = n__s(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] incr(X) = [0 0 0]X >= [0 0 0]X = n__incr(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] adx(X) = [0 0 1]X >= [0 0 1]X = n__adx(X) [0 0 0] [0 0 0] [0] [0] activate(n__0()) = [0] >= [0] = 0() [0] [0] [0] [0] activate(n__zeros()) = [1] >= [1] = zeros() [0] [0] [1 0 0] [1 0 0] activate(n__s(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] activate(n__incr(X)) = [0 0 0]X >= [0 0 0]X = incr(activate(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] activate(n__adx(X)) = [0 0 1]X >= [0 0 1]X = adx(activate(X)) [0 0 0] [0 0 0] [1 0 0] activate(X) = [0 1 1]X >= X = X [0 0 1] problem: nats() -> adx(zeros()) zeros() -> cons(n__0(),n__zeros()) incr(cons(X,Y)) -> cons(n__s(activate(X)),n__incr(activate(Y))) adx(cons(X,Y)) -> incr(cons(activate(X),n__adx(activate(Y)))) 0() -> n__0() zeros() -> n__zeros() s(X) -> n__s(X) incr(X) -> n__incr(X) adx(X) -> n__adx(X) activate(n__0()) -> 0() activate(n__zeros()) -> zeros() activate(n__s(X)) -> s(X) activate(n__incr(X)) -> incr(activate(X)) activate(n__adx(X)) -> adx(activate(X)) activate(X) -> X Matrix Interpretation Processor: dim=1 interpretation: [s](x0) = x0, [0] = 0, [n__adx](x0) = 4x0, [n__incr](x0) = x0, [n__s](x0) = x0, [activate](x0) = x0, [incr](x0) = x0, [cons](x0, x1) = 4x0 + 4x1, [n__zeros] = 0, [n__0] = 0, [adx](x0) = 4x0, [zeros] = 0, [nats] = 4 orientation: nats() = 4 >= 0 = adx(zeros()) zeros() = 0 >= 0 = cons(n__0(),n__zeros()) incr(cons(X,Y)) = 4X + 4Y >= 4X + 4Y = cons(n__s(activate(X)),n__incr(activate(Y))) adx(cons(X,Y)) = 16X + 16Y >= 4X + 16Y = incr(cons(activate(X),n__adx(activate(Y)))) 0() = 0 >= 0 = n__0() zeros() = 0 >= 0 = n__zeros() s(X) = X >= X = n__s(X) incr(X) = X >= X = n__incr(X) adx(X) = 4X >= 4X = n__adx(X) activate(n__0()) = 0 >= 0 = 0() activate(n__zeros()) = 0 >= 0 = zeros() activate(n__s(X)) = X >= X = s(X) activate(n__incr(X)) = X >= X = incr(activate(X)) activate(n__adx(X)) = 4X >= 4X = adx(activate(X)) activate(X) = X >= X = X problem: zeros() -> cons(n__0(),n__zeros()) incr(cons(X,Y)) -> cons(n__s(activate(X)),n__incr(activate(Y))) adx(cons(X,Y)) -> incr(cons(activate(X),n__adx(activate(Y)))) 0() -> n__0() zeros() -> n__zeros() s(X) -> n__s(X) incr(X) -> n__incr(X) adx(X) -> n__adx(X) activate(n__0()) -> 0() activate(n__zeros()) -> zeros() activate(n__s(X)) -> s(X) activate(n__incr(X)) -> incr(activate(X)) activate(n__adx(X)) -> adx(activate(X)) activate(X) -> X Unfolding Processor: loop length: 6 terms: incr(cons(X,n__adx(n__zeros()))) cons(n__s(activate(X)),n__incr(activate(n__adx(n__zeros())))) cons(n__s(activate(X)),n__incr(adx(activate(n__zeros())))) cons(n__s(activate(X)),n__incr(adx(zeros()))) cons(n__s(activate(X)),n__incr(adx(cons(n__0(),n__zeros())))) cons(n__s(activate(X)),n__incr(incr(cons(activate(n__0()),n__adx(activate(n__zeros())))))) context: cons(n__s(activate(X)),n__incr([])) substitution: X -> activate(n__0()) Qed