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