Problem: zeros() -> cons(0(),n__zeros()) and(tt(),X) -> activate(X) length(nil()) -> 0() length(cons(N,L)) -> s(length(activate(L))) zeros() -> n__zeros() activate(n__zeros()) -> zeros() activate(X) -> X Proof: Matrix Interpretation Processor: dim=1 interpretation: [s](x0) = x0, [length](x0) = x0, [nil] = 6, [activate](x0) = x0, [and](x0, x1) = 3x0 + x1 + 1, [tt] = 7, [cons](x0, x1) = 2x0 + x1, [n__zeros] = 0, [0] = 0, [zeros] = 0 orientation: zeros() = 0 >= 0 = cons(0(),n__zeros()) and(tt(),X) = X + 22 >= X = activate(X) length(nil()) = 6 >= 0 = 0() length(cons(N,L)) = L + 2N >= L = s(length(activate(L))) zeros() = 0 >= 0 = n__zeros() activate(n__zeros()) = 0 >= 0 = zeros() activate(X) = X >= X = X problem: zeros() -> cons(0(),n__zeros()) length(cons(N,L)) -> s(length(activate(L))) zeros() -> n__zeros() activate(n__zeros()) -> zeros() activate(X) -> X Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [s](x0) = [0 0 0]x0 [1 0 0] , [1 0 0] [0] [length](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 1] [1] [activate](x0) = [0 1 0]x0 + [1] [0 0 1] [0], [1 0 0] [1 0 1] [1] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [1 0 1] [0], [0] [n__zeros] = [0] [0], [0] [0] = [0] [0], [1] [zeros] = [1] [0] orientation: [1] [1] zeros() = [1] >= [0] = cons(0(),n__zeros()) [0] [0] [1 0 1] [1 0 0] [1] [1 0 1] [1] length(cons(N,L)) = [0 0 0]L + [0 0 0]N + [0] >= [0 0 0]L + [0] = s(length(activate(L))) [1 0 1] [0 0 0] [1] [1 0 1] [1] [1] [0] zeros() = [1] >= [0] = n__zeros() [0] [0] [1] [1] activate(n__zeros()) = [1] >= [1] = zeros() [0] [0] [1 0 1] [1] activate(X) = [0 1 0]X + [1] >= X = X [0 0 1] [0] problem: zeros() -> cons(0(),n__zeros()) length(cons(N,L)) -> s(length(activate(L))) activate(n__zeros()) -> zeros() Unfolding Processor: loop length: 3 terms: length(cons(N,n__zeros())) s(length(activate(n__zeros()))) s(length(zeros())) context: s([]) substitution: N -> 0() Qed