NO Problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> a__U12(tt(),L) a__U12(tt(),L) -> s(a__length(mark(L))) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U11(tt(),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X1,X2) -> U12(X1,X2) a__length(X) -> length(X) Proof: Matrix Interpretation Processor: dim=1 interpretation: [length](x0) = x0 + 1, [U12](x0, x1) = x0 + x1, [U11](x0, x1) = x0 + x1, [nil] = 4, [s](x0) = x0, [a__length](x0) = x0 + 1, [mark](x0) = x0 + 1, [a__U12](x0, x1) = x0 + x1, [a__U11](x0, x1) = x0 + x1, [tt] = 2, [cons](x0, x1) = x0 + 4x1 + 1, [zeros] = 0, [0] = 0, [a__zeros] = 1 orientation: a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt(),L) = L + 2 >= L + 2 = a__U12(tt(),L) a__U12(tt(),L) = L + 2 >= L + 2 = s(a__length(mark(L))) a__length(nil()) = 5 >= 0 = 0() a__length(cons(N,L)) = 4L + N + 2 >= L + 2 = a__U11(tt(),L) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__U11(mark(X1),X2) mark(U12(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__U12(mark(X1),X2) mark(length(X)) = X + 2 >= X + 2 = a__length(mark(X)) mark(cons(X1,X2)) = X1 + 4X2 + 2 >= X1 + 4X2 + 2 = cons(mark(X1),X2) mark(0()) = 1 >= 0 = 0() mark(tt()) = 3 >= 2 = tt() mark(s(X)) = X + 1 >= X + 1 = s(mark(X)) mark(nil()) = 5 >= 4 = nil() a__zeros() = 1 >= 0 = zeros() a__U11(X1,X2) = X1 + X2 >= X1 + X2 = U11(X1,X2) a__U12(X1,X2) = X1 + X2 >= X1 + X2 = U12(X1,X2) a__length(X) = X + 1 >= X + 1 = length(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> a__U12(tt(),L) a__U12(tt(),L) -> s(a__length(mark(L))) a__length(cons(N,L)) -> a__U11(tt(),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__U11(X1,X2) -> U11(X1,X2) a__U12(X1,X2) -> U12(X1,X2) a__length(X) -> length(X) Matrix Interpretation Processor: dim=1 interpretation: [length](x0) = x0 + 1, [U12](x0, x1) = x0 + 2x1, [U11](x0, x1) = x0 + 2x1, [s](x0) = x0, [a__length](x0) = x0 + 1, [mark](x0) = 4x0, [a__U12](x0, x1) = x0 + 4x1, [a__U11](x0, x1) = x0 + 4x1, [tt] = 1, [cons](x0, x1) = 4x0 + 4x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),L) = 4L + 1 >= 4L + 1 = a__U12(tt(),L) a__U12(tt(),L) = 4L + 1 >= 4L + 1 = s(a__length(mark(L))) a__length(cons(N,L)) = 4L + 4N + 1 >= 4L + 1 = a__U11(tt(),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 4X1 + 8X2 >= 4X1 + 4X2 = a__U11(mark(X1),X2) mark(U12(X1,X2)) = 4X1 + 8X2 >= 4X1 + 4X2 = a__U12(mark(X1),X2) mark(length(X)) = 4X + 4 >= 4X + 1 = a__length(mark(X)) mark(cons(X1,X2)) = 16X1 + 16X2 >= 16X1 + 4X2 = cons(mark(X1),X2) mark(s(X)) = 4X >= 4X = s(mark(X)) a__U11(X1,X2) = X1 + 4X2 >= X1 + 2X2 = U11(X1,X2) a__U12(X1,X2) = X1 + 4X2 >= X1 + 2X2 = U12(X1,X2) a__length(X) = X + 1 >= X + 1 = length(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> a__U12(tt(),L) a__U12(tt(),L) -> s(a__length(mark(L))) a__length(cons(N,L)) -> a__U11(tt(),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__U11(X1,X2) -> U11(X1,X2) a__U12(X1,X2) -> U12(X1,X2) a__length(X) -> length(X) Matrix Interpretation Processor: dim=1 interpretation: [length](x0) = x0, [U12](x0, x1) = x0 + x1, [U11](x0, x1) = x0 + x1, [s](x0) = x0, [a__length](x0) = x0 + 1, [mark](x0) = x0, [a__U12](x0, x1) = x0 + x1, [a__U11](x0, x1) = x0 + x1, [tt] = 1, [cons](x0, x1) = 2x0 + x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),L) = L + 1 >= L + 1 = a__U12(tt(),L) a__U12(tt(),L) = L + 1 >= L + 1 = s(a__length(mark(L))) a__length(cons(N,L)) = L + 2N + 1 >= L + 1 = a__U11(tt(),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = X1 + X2 >= X1 + X2 = a__U11(mark(X1),X2) mark(U12(X1,X2)) = X1 + X2 >= X1 + X2 = a__U12(mark(X1),X2) mark(cons(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = cons(mark(X1),X2) mark(s(X)) = X >= X = s(mark(X)) a__U11(X1,X2) = X1 + X2 >= X1 + X2 = U11(X1,X2) a__U12(X1,X2) = X1 + X2 >= X1 + X2 = U12(X1,X2) a__length(X) = X + 1 >= X = length(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> a__U12(tt(),L) a__U12(tt(),L) -> s(a__length(mark(L))) a__length(cons(N,L)) -> a__U11(tt(),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__U11(X1,X2) -> U11(X1,X2) a__U12(X1,X2) -> U12(X1,X2) Matrix Interpretation Processor: dim=1 interpretation: [U12](x0, x1) = x0 + x1 + 2, [U11](x0, x1) = 3x0 + 4x1, [s](x0) = x0, [a__length](x0) = x0 + 3, [mark](x0) = 4x0, [a__U12](x0, x1) = x0 + 4x1 + 2, [a__U11](x0, x1) = 3x0 + 4x1, [tt] = 1, [cons](x0, x1) = x0 + 4x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),L) = 4L + 3 >= 4L + 3 = a__U12(tt(),L) a__U12(tt(),L) = 4L + 3 >= 4L + 3 = s(a__length(mark(L))) a__length(cons(N,L)) = 4L + N + 3 >= 4L + 3 = a__U11(tt(),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 12X1 + 16X2 >= 12X1 + 4X2 = a__U11(mark(X1),X2) mark(U12(X1,X2)) = 4X1 + 4X2 + 8 >= 4X1 + 4X2 + 2 = a__U12(mark(X1),X2) mark(cons(X1,X2)) = 4X1 + 16X2 >= 4X1 + 4X2 = cons(mark(X1),X2) mark(s(X)) = 4X >= 4X = s(mark(X)) a__U11(X1,X2) = 3X1 + 4X2 >= 3X1 + 4X2 = U11(X1,X2) a__U12(X1,X2) = X1 + 4X2 + 2 >= X1 + X2 + 2 = U12(X1,X2) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> a__U12(tt(),L) a__U12(tt(),L) -> s(a__length(mark(L))) a__length(cons(N,L)) -> a__U11(tt(),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__U11(X1,X2) -> U11(X1,X2) a__U12(X1,X2) -> U12(X1,X2) Matrix Interpretation Processor: dim=1 interpretation: [U12](x0, x1) = 2x0 + 2x1, [U11](x0, x1) = 2x0 + 4x1 + 4, [s](x0) = x0, [a__length](x0) = 2x0 + 4, [mark](x0) = 2x0 + 4, [a__U12](x0, x1) = 2x0 + 4x1 + 4, [a__U11](x0, x1) = 2x0 + 6x1 + 4, [tt] = 4, [cons](x0, x1) = 2x0 + 4x1 + 4, [zeros] = 0, [0] = 0, [a__zeros] = 4 orientation: a__zeros() = 4 >= 4 = cons(0(),zeros()) a__U11(tt(),L) = 6L + 12 >= 4L + 12 = a__U12(tt(),L) a__U12(tt(),L) = 4L + 12 >= 4L + 12 = s(a__length(mark(L))) a__length(cons(N,L)) = 8L + 4N + 12 >= 6L + 12 = a__U11(tt(),L) mark(zeros()) = 4 >= 4 = a__zeros() mark(U11(X1,X2)) = 4X1 + 8X2 + 12 >= 4X1 + 6X2 + 12 = a__U11(mark(X1),X2) mark(cons(X1,X2)) = 4X1 + 8X2 + 12 >= 4X1 + 4X2 + 12 = cons(mark(X1),X2) mark(s(X)) = 2X + 4 >= 2X + 4 = s(mark(X)) a__U11(X1,X2) = 2X1 + 6X2 + 4 >= 2X1 + 4X2 + 4 = U11(X1,X2) a__U12(X1,X2) = 2X1 + 4X2 + 4 >= 2X1 + 2X2 = U12(X1,X2) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> a__U12(tt(),L) a__U12(tt(),L) -> s(a__length(mark(L))) a__length(cons(N,L)) -> a__U11(tt(),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__U11(X1,X2) -> U11(X1,X2) Unfolding Processor: loop length: 5 terms: a__U11(tt(),zeros()) a__U12(tt(),zeros()) s(a__length(mark(zeros()))) s(a__length(a__zeros())) s(a__length(cons(0(),zeros()))) context: s([]) substitution: Qed