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=3 interpretation: [1 0 1] [0] [length](x0) = [0 1 0]x0 + [1] [0 0 1] [0], [1 0 1] [1 0 1] [0] [U12](x0, x1) = [0 1 0]x0 + [0 1 0]x1 + [1] [0 0 0] [0 0 0] [0], [1 0 0] [1 0 1] [0] [U11](x0, x1) = [0 1 0]x0 + [0 1 0]x1 + [1] [0 0 0] [0 0 1] [0], [0] [nil] = [0] [0], [1 0 0] [s](x0) = [0 1 0]x0 [0 0 0] , [1 0 1] [0] [a__length](x0) = [0 1 0]x0 + [1] [0 0 1] [0], [1 1 0] [0] [mark](x0) = [0 1 0]x0 + [0] [0 0 1] [1], [1 0 1] [1 1 1] [0] [a__U12](x0, x1) = [0 1 0]x0 + [0 1 0]x1 + [1] [0 0 0] [0 0 0] [0], [1 0 0] [1 1 1] [0] [a__U11](x0, x1) = [0 1 0]x0 + [0 1 0]x1 + [1] [0 0 0] [0 0 1] [0], [1] [tt] = [0] [0], [1 0 0] [1] [cons](x0, x1) = x0 + [0 1 0]x1 + [0] [0 1 1] [0], [0] [zeros] = [1] [0], [0] [0] = [0] [0], [1] [a__zeros] = [1] [1] orientation: [1] [1] a__zeros() = [1] >= [1] = cons(0(),zeros()) [1] [1] [1 1 1] [1] [1 1 1] [1] a__U11(tt(),L) = [0 1 0]L + [1] >= [0 1 0]L + [1] = a__U12(tt(),L) [0 0 1] [0] [0 0 0] [0] [1 1 1] [1] [1 1 1] [1] a__U12(tt(),L) = [0 1 0]L + [1] >= [0 1 0]L + [1] = s(a__length(mark(L))) [0 0 0] [0] [0 0 0] [0] [0] [0] a__length(nil()) = [1] >= [0] = 0() [0] [0] [1 1 1] [1 0 1] [1] [1 1 1] [1] a__length(cons(N,L)) = [0 1 0]L + [0 1 0]N + [1] >= [0 1 0]L + [1] = a__U11(tt(),L) [0 1 1] [0 0 1] [0] [0 0 1] [0] [1] [1] mark(zeros()) = [1] >= [1] = a__zeros() [1] [1] [1 1 0] [1 1 1] [1] [1 1 0] [1 1 1] [0] mark(U11(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = a__U11(mark(X1),X2) [0 0 0] [0 0 1] [1] [0 0 0] [0 0 1] [0] [1 1 1] [1 1 1] [1] [1 1 1] [1 1 1] [1] mark(U12(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = a__U12(mark(X1),X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0] [1 1 1] [1] [1 1 1] [1] mark(length(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = a__length(mark(X)) [0 0 1] [1] [0 0 1] [1] [1 1 0] [1 1 0] [1] [1 1 0] [1 0 0] [1] mark(cons(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0] = cons(mark(X1),X2) [0 0 1] [0 1 1] [1] [0 0 1] [0 1 1] [1] [0] [0] mark(0()) = [0] >= [0] = 0() [1] [0] [1] [1] mark(tt()) = [0] >= [0] = tt() [1] [0] [1 1 0] [0] [1 1 0] mark(s(X)) = [0 1 0]X + [0] >= [0 1 0]X = s(mark(X)) [0 0 0] [1] [0 0 0] [0] [0] mark(nil()) = [0] >= [0] = nil() [1] [0] [1] [0] a__zeros() = [1] >= [1] = zeros() [1] [0] [1 0 0] [1 1 1] [0] [1 0 0] [1 0 1] [0] a__U11(X1,X2) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = U11(X1,X2) [0 0 0] [0 0 1] [0] [0 0 0] [0 0 1] [0] [1 0 1] [1 1 1] [0] [1 0 1] [1 0 1] [0] a__U12(X1,X2) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = U12(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 1] [0] [1 0 1] [0] a__length(X) = [0 1 0]X + [1] >= [0 1 0]X + [1] = length(X) [0 0 1] [0] [0 0 1] [0] 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(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__U11(X1,X2) -> U11(X1,X2) a__U12(X1,X2) -> U12(X1,X2) a__length(X) -> length(X) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [length](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [1 1 0] [U12](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1] [nil] = [0] [0], [1 0 0] [s](x0) = [0 1 0]x0 [0 0 0] , [1 1 0] [a__length](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 1 1] [1 1 0] [a__U12](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 1 0] [a__U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [tt] = [0] [0], [1 0 0] [1 1 0] [cons](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [zeros] = [0] [0], [0] [0] = [0] [0], [0] [a__zeros] = [0] [0] orientation: [0] [0] a__zeros() = [0] >= [0] = cons(0(),zeros()) [0] [0] [1 1 0] [1 1 0] a__U11(tt(),L) = [0 0 0]L >= [0 0 0]L = a__U12(tt(),L) [0 0 0] [0 0 0] [1 1 0] [1 1 0] a__U12(tt(),L) = [0 0 0]L >= [0 0 0]L = s(a__length(mark(L))) [0 0 0] [0 0 0] [1] [0] a__length(nil()) = [0] >= [0] = 0() [0] [0] [1 1 0] [1 1 0] [1 1 0] a__length(cons(N,L)) = [0 0 0]L + [0 0 0]N >= [0 0 0]L = a__U11(tt(),L) [0 0 0] [0 0 0] [0 0 0] [0] [0] mark(zeros()) = [0] >= [0] = a__zeros() [0] [0] [1 1 1] [1 1 0] [1 1 0] [1 1 0] mark(U12(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = a__U12(mark(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] mark(length(X)) = [0 0 0]X >= [0 0 0]X = a__length(mark(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(mark(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [0] [0] mark(tt()) = [0] >= [0] = tt() [0] [0] [1 1 0] [1 1 0] mark(s(X)) = [0 0 0]X >= [0 0 0]X = s(mark(X)) [0 0 0] [0 0 0] [1] [1] mark(nil()) = [0] >= [0] = nil() [0] [0] [1 0 0] [1 1 0] [1 0 0] [1 0 0] a__U11(X1,X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U11(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 1 0] [1 0 1] [1 1 0] a__U12(X1,X2) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = U12(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] a__length(X) = [0 0 0]X >= [0 0 0]X = length(X) [0 0 0] [0 0 0] 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(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__U11(X1,X2) -> U11(X1,X2) a__U12(X1,X2) -> U12(X1,X2) a__length(X) -> length(X) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [0] [length](x0) = [0 0 1]x0 + [1] [0 0 0] [0], [1 1 0] [1 1 0] [0] [U12](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [0], [1 0 0] [1 0 0] [U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [nil] = [0] [0], [1 0 0] [s](x0) = [0 1 0]x0 [0 0 0] , [1 1 0] [0] [a__length](x0) = [0 0 1]x0 + [1] [0 0 0] [0], [1 1 0] [0] [mark](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 1 0] [1 1 0] [0] [a__U12](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [0], [1 0 0] [1 1 0] [1] [a__U11](x0, x1) = [0 0 1]x0 + [0 0 0]x1 + [1] [0 0 1] [0 0 0] [0], [0] [tt] = [1] [0], [1 1 0] [1 1 1] [0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [0], [0] [zeros] = [1] [0], [0] [0] = [0] [0], [1] [a__zeros] = [1] [0] orientation: [1] [1] a__zeros() = [1] >= [1] = cons(0(),zeros()) [0] [0] [1 1 0] [1] [1 1 0] [1] a__U11(tt(),L) = [0 0 0]L + [1] >= [0 0 0]L + [1] = a__U12(tt(),L) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1] [1 1 0] [1] a__U12(tt(),L) = [0 0 0]L + [1] >= [0 0 0]L + [1] = s(a__length(mark(L))) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1 1 0] [1] [1 1 0] [1] a__length(cons(N,L)) = [0 0 0]L + [0 0 0]N + [1] >= [0 0 0]L + [1] = a__U11(tt(),L) [0 0 0] [0 0 0] [0] [0 0 0] [0] [1] [1] mark(zeros()) = [1] >= [1] = a__zeros() [0] [0] [1 1 0] [1 1 0] [1] [1 1 0] [1 1 0] [1] mark(U12(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = a__U12(mark(X1),X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 1] [1] [1 1 0] [1] mark(length(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = a__length(mark(X)) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 1 1] [1] [1 1 0] [1 1 1] [1] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = cons(mark(X1),X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] [0] mark(0()) = [1] >= [0] = 0() [0] [0] [1] [0] mark(tt()) = [1] >= [1] = tt() [0] [0] [1 1 0] [0] [1 1 0] [0] mark(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(mark(X)) [0 0 0] [0] [0 0 0] [0] [0] [0] mark(nil()) = [1] >= [0] = nil() [0] [0] [1 0 0] [1 1 0] [1] [1 0 0] [1 0 0] a__U11(X1,X2) = [0 0 1]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 = U11(X1,X2) [0 0 1] [0 0 0] [0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [0] [1 1 0] [1 1 0] [0] a__U12(X1,X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = U12(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [0] [1 1 0] [0] a__length(X) = [0 0 1]X + [1] >= [0 0 1]X + [1] = length(X) [0 0 0] [0] [0 0 0] [0] 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(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(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__U12(X1,X2) -> U12(X1,X2) a__length(X) -> length(X) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [0] [length](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 0] [1 1 0] [U12](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [nil] = [1] [0], [1 0 0] [s](x0) = [0 1 0]x0 [0 0 0] , [1 1 0] [0] [a__length](x0) = [0 0 0]x0 + [1] [0 0 1] [0], [1 1 0] [0] [mark](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 1] [1 1 0] [a__U12](x0, x1) = [0 1 1]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 1] [1 1 0] [0] [a__U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [0], [0] [tt] = [0] [1], [1 1 0] [1 1 0] [0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [0], [0] [zeros] = [0] [0], [0] [0] = [0] [0], [0] [a__zeros] = [1] [0] orientation: [0] [0] a__zeros() = [1] >= [1] = cons(0(),zeros()) [0] [0] [1 1 0] [1] [1 1 0] [1] a__U11(tt(),L) = [0 0 0]L + [1] >= [0 0 0]L + [1] = a__U12(tt(),L) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1] [1 1 0] [1] a__U12(tt(),L) = [0 0 0]L + [1] >= [0 0 0]L + [1] = s(a__length(mark(L))) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 1 0] [1] [1 1 0] [1] a__length(cons(N,L)) = [0 0 0]L + [0 0 0]N + [1] >= [0 0 0]L + [1] = a__U11(tt(),L) [0 0 0] [0 0 0] [0] [0 0 0] [0] [0] [0] mark(zeros()) = [1] >= [1] = a__zeros() [0] [0] [1 1 0] [1 1 0] [0] [1 1 0] [1 1 0] [0] mark(U12(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = a__U12(mark(X1),X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1] [1 1 0] [1] mark(length(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = a__length(mark(X)) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 1 0] [1] [1 1 0] [1 1 0] [1] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = cons(mark(X1),X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] [0] mark(0()) = [1] >= [0] = 0() [0] [0] [1 1 0] [0] [1 1 0] [0] mark(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(mark(X)) [0 0 0] [0] [0 0 0] [0] [1] [0] mark(nil()) = [1] >= [1] = nil() [0] [0] [1 0 1] [1 1 0] [1 0 0] [1 1 0] a__U12(X1,X2) = [0 1 1]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = U12(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [0] [1 1 0] [0] a__length(X) = [0 0 0]X + [1] >= [0 0 0]X + [1] = length(X) [0 0 1] [0] [0 0 0] [0] 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(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(s(X)) -> s(mark(X)) a__U12(X1,X2) -> U12(X1,X2) a__length(X) -> length(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [length](x0) = [0 0 0]x0 + [0] [0 1 1] [1], [1 0 0] [1 0 0] [U12](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [0 1 1] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [0] [a__length](x0) = [0 0 0]x0 + [0] [0 1 1] [1], [1 0 1] [mark](x0) = [0 1 0]x0 [0 0 1] , [1 0 0] [1 0 1] [a__U12](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [0 1 1] , [1 0 0] [1 1 1] [0] [a__U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 1 1] [1], [0] [tt] = [0] [1], [1 0 0] [1 1 1] [cons](x0, x1) = [0 0 0]x0 + [0 1 1]x1 [0 0 1] [0 0 0] , [1] [zeros] = [0] [0], [0] [0] = [1] [0], [1] [a__zeros] = [0] [0] orientation: [1] [1] a__zeros() = [0] >= [0] = cons(0(),zeros()) [0] [0] [1 1 1] [0] [1 0 1] [0] a__U11(tt(),L) = [0 0 0]L + [0] >= [0 0 0]L + [0] = a__U12(tt(),L) [0 1 1] [1] [0 1 1] [1] [1 0 1] [0] [1 0 1] [0] a__U12(tt(),L) = [0 0 0]L + [0] >= [0 0 0]L + [0] = s(a__length(mark(L))) [0 1 1] [1] [0 1 1] [1] [1 1 1] [1 0 0] [0] [1 1 1] [0] a__length(cons(N,L)) = [0 0 0]L + [0 0 0]N + [0] >= [0 0 0]L + [0] = a__U11(tt(),L) [0 1 1] [0 0 1] [1] [0 1 1] [1] [1] [1] mark(zeros()) = [0] >= [0] = a__zeros() [0] [0] [1 0 1] [1 1 1] [1 0 1] [1 0 1] mark(U12(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = a__U12(mark(X1),X2) [0 0 1] [0 1 1] [0 0 1] [0 1 1] [1 1 1] [1] [1 0 1] [0] mark(length(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = a__length(mark(X)) [0 1 1] [1] [0 1 1] [1] [1 0 1] [1 1 1] [1 0 1] [1 1 1] mark(cons(X1,X2)) = [0 0 0]X1 + [0 1 1]X2 >= [0 0 0]X1 + [0 1 1]X2 = cons(mark(X1),X2) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [0] [0] mark(0()) = [1] >= [1] = 0() [0] [0] [1 0 1] [1 0 1] mark(s(X)) = [0 0 0]X >= [0 0 0]X = s(mark(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 1] [1 0 0] [1 0 0] a__U12(X1,X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U12(X1,X2) [0 0 1] [0 1 1] [0 0 1] [0 1 1] [1 0 0] [0] [1 0 0] [0] a__length(X) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length(X) [0 1 1] [1] [0 1 1] [1] 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(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__U12(X1,X2) -> U12(X1,X2) a__length(X) -> length(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [length](x0) = [0 0 0]x0 [0 0 0] , [1 1 1] [1 1 1] [U12](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 1 1] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1] [a__length](x0) = [0 0 1]x0 + [0] [0 0 0] [0], [1 0 1] [mark](x0) = [0 0 0]x0 [0 1 0] , [1 1 1] [1 1 1] [a__U12](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 1 1] [1] [a__U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [0], [0] [tt] = [0] [1], [1 1 1] [cons](x0, x1) = x0 + [0 0 0]x1 [0 0 0] , [0] [zeros] = [0] [1], [0] [0] = [0] [0], [1] [a__zeros] = [0] [0] orientation: [1] [1] a__zeros() = [0] >= [0] = cons(0(),zeros()) [0] [0] [1 1 1] [1] [1 1 1] [1] a__U11(tt(),L) = [0 0 0]L + [0] >= [0 0 0]L + [0] = a__U12(tt(),L) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1] [1 1 1] [1] a__U12(tt(),L) = [0 0 0]L + [0] >= [0 0 0]L + [0] = s(a__length(mark(L))) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1 0 0] [1] [1 1 1] [1] a__length(cons(N,L)) = [0 0 0]L + [0 0 1]N + [0] >= [0 0 0]L + [0] = a__U11(tt(),L) [0 0 0] [0 0 0] [0] [0 0 0] [0] [1] [1] mark(zeros()) = [0] >= [0] = a__zeros() [0] [0] [1 1 1] [1 1 1] [1 1 1] [1 1 1] mark(U12(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = a__U12(mark(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 1 1] [1 0 1] [1 1 1] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(mark(X1),X2) [0 1 0] [0 0 0] [0 1 0] [0 0 0] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 1 1] [1 1 1] mark(s(X)) = [0 0 0]X >= [0 0 0]X = s(mark(X)) [0 0 0] [0 0 0] [1 1 1] [1 1 1] [1 1 1] [1 1 1] a__U12(X1,X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U12(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1] [1 0 0] a__length(X) = [0 0 1]X + [0] >= [0 0 0]X = length(X) [0 0 0] [0] [0 0 0] 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(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__U12(X1,X2) -> U12(X1,X2) Matrix Interpretation Processor: dim=3 interpretation: [1 1 1] [1 1 0] [U12](x0, x1) = [0 0 1]x0 + [0 0 0]x1 [0 1 0] [0 0 0] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [0] [a__length](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [1] [mark](x0) = [0 0 1]x0 + [0] [0 1 0] [0], [1 1 1] [1 1 0] [a__U12](x0, x1) = [0 0 1]x0 + [0 0 0]x1 [0 1 0] [0 0 0] , [1 1 0] [1 1 0] [0] [a__U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [0] [tt] = [1] [0], [1 0 0] [1 1 0] [0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [1], [0] [zeros] = [1] [1], [0] [0] = [0] [0], [1] [a__zeros] = [1] [1] orientation: [1] [1] a__zeros() = [1] >= [1] = cons(0(),zeros()) [1] [1] [1 1 0] [1] [1 1 0] [1] a__U11(tt(),L) = [0 0 0]L + [0] >= [0 0 0]L + [0] = a__U12(tt(),L) [0 0 0] [1] [0 0 0] [1] [1 1 0] [1] [1 1 0] [1] a__U12(tt(),L) = [0 0 0]L + [0] >= [0 0 0]L + [0] = s(a__length(mark(L))) [0 0 0] [1] [0 0 0] [0] [1 1 0] [1 0 0] [1] [1 1 0] [1] a__length(cons(N,L)) = [0 0 0]L + [0 0 0]N + [0] >= [0 0 0]L + [0] = a__U11(tt(),L) [0 0 0] [0 0 0] [1] [0 0 0] [1] [1] [1] mark(zeros()) = [1] >= [1] = a__zeros() [1] [1] [1 1 1] [1 1 0] [1] [1 1 1] [1 1 0] [1] mark(U12(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 + [0] >= [0 1 0]X1 + [0 0 0]X2 + [0] = a__U12(mark(X1),X2) [0 0 1] [0 0 0] [0] [0 0 1] [0 0 0] [0] [1 0 0] [1 1 0] [1] [1 0 0] [1 1 0] [1] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = cons(mark(X1),X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 0 0] [1] [1 0 0] [1] mark(s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(mark(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1 1 0] [1 1 1] [1 1 0] a__U12(X1,X2) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 1]X1 + [0 0 0]X2 = U12(X1,X2) [0 1 0] [0 0 0] [0 1 0] [0 0 0] 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(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__U12(X1,X2) -> U12(X1,X2) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [1 1 0] [U12](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [s](x0) = x0 , [1 0 0] [1] [a__length](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 1 0] [mark](x0) = [1 1 1]x0 [0 0 1] , [1 0 1] [1 1 0] [a__U12](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 1] [1 1 0] [a__U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [tt] = [0] [1], [1 1 0] [0] [cons](x0, x1) = x0 + [0 0 0]x1 + [1] [0 0 0] [0], [0] [zeros] = [0] [1], [0] [0] = [0] [0], [0] [a__zeros] = [1] [0] orientation: [0] [0] a__zeros() = [1] >= [1] = cons(0(),zeros()) [0] [0] [1 1 0] [1] [1 1 0] [1] a__U11(tt(),L) = [0 0 0]L + [0] >= [0 0 0]L + [0] = a__U12(tt(),L) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1] [1 1 0] [1] a__U12(tt(),L) = [0 0 0]L + [0] >= [0 0 0]L + [0] = s(a__length(mark(L))) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 0 0] [1] [1 1 0] [1] a__length(cons(N,L)) = [0 0 0]L + [0 0 0]N + [0] >= [0 0 0]L + [0] = a__U11(tt(),L) [0 0 0] [0 0 0] [0] [0 0 0] [0] [0] [0] mark(zeros()) = [1] >= [1] = a__zeros() [1] [0] [1 1 1] [1 1 0] [1 1 1] [1 1 0] mark(U12(X1,X2)) = [1 1 1]X1 + [1 1 0]X2 >= [1 1 1]X1 + [0 0 0]X2 = a__U12(mark(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1] [1 1 0] [1 1 0] [0] mark(cons(X1,X2)) = [1 1 1]X1 + [1 1 0]X2 + [1] >= [1 1 1]X1 + [0 0 0]X2 + [1] = cons(mark(X1),X2) [0 0 1] [0 0 0] [0] [0 0 1] [0 0 0] [0] [1 1 0] [1 1 0] mark(s(X)) = [1 1 1]X >= [1 1 1]X = s(mark(X)) [0 0 1] [0 0 1] [1 0 1] [1 1 0] [1 0 1] [1 1 0] a__U12(X1,X2) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = U12(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] 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(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__U12(X1,X2) -> U12(X1,X2) Matrix Interpretation Processor: dim=3 interpretation: [1 1 1] [0] [U12](x0, x1) = x0 + [0 0 0]x1 + [1] [0 1 1] [1], [s](x0) = x0 , [1 0 1] [0] [a__length](x0) = [0 0 0]x0 + [1] [0 0 1] [1], [1 1 0] [mark](x0) = [0 0 1]x0 [0 0 1] , [1 1 1] [0] [a__U12](x0, x1) = x0 + [0 0 0]x1 + [1] [0 1 1] [1], [1 0 0] [1 1 1] [0] [a__U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 1 1] [1], [0] [tt] = [0] [0], [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 1 1] , [1] [zeros] = [0] [1], [0] [0] = [0] [0], [1] [a__zeros] = [0] [1] orientation: [1] [1] a__zeros() = [0] >= [0] = cons(0(),zeros()) [1] [1] [1 1 1] [0] [1 1 1] [0] a__U11(tt(),L) = [0 0 0]L + [1] >= [0 0 0]L + [1] = a__U12(tt(),L) [0 1 1] [1] [0 1 1] [1] [1 1 1] [0] [1 1 1] [0] a__U12(tt(),L) = [0 0 0]L + [1] >= [0 0 0]L + [1] = s(a__length(mark(L))) [0 1 1] [1] [0 0 1] [1] [1 1 1] [1 0 0] [0] [1 1 1] [0] a__length(cons(N,L)) = [0 0 0]L + [0 0 0]N + [1] >= [0 0 0]L + [1] = a__U11(tt(),L) [0 1 1] [0 0 0] [1] [0 1 1] [1] [1] [1] mark(zeros()) = [1] >= [0] = a__zeros() [1] [1] [1 1 0] [1 1 1] [1] [1 1 0] [1 1 1] [0] mark(U12(X1,X2)) = [0 0 1]X1 + [0 1 1]X2 + [1] >= [0 0 1]X1 + [0 0 0]X2 + [1] = a__U12(mark(X1),X2) [0 0 1] [0 1 1] [1] [0 0 1] [0 1 1] [1] [1 1 0] [1 1 0] mark(s(X)) = [0 0 1]X >= [0 0 1]X = s(mark(X)) [0 0 1] [0 0 1] [1 1 1] [0] [1 1 1] [0] a__U12(X1,X2) = X1 + [0 0 0]X2 + [1] >= X1 + [0 0 0]X2 + [1] = U12(X1,X2) [0 1 1] [1] [0 1 1] [1] 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(s(X)) -> s(mark(X)) a__U12(X1,X2) -> U12(X1,X2) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [U12](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [0] [s](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [1] [a__length](x0) = [0 0 1]x0 + [0] [0 0 0] [1], [1 0 0] [0] [mark](x0) = [0 0 1]x0 + [0] [0 0 0] [1], [1 0 0] [1 0 0] [1] [a__U12](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [1 1 0] [1 1 0] [0] [a__U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [0] [tt] = [1] [0], [1 0 0] [1 1 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [0 0 0] , [0] [zeros] = [0] [0], [0] [0] = [0] [1], [0] [a__zeros] = [0] [1] orientation: [0] [0] a__zeros() = [0] >= [0] = cons(0(),zeros()) [1] [1] [1 1 0] [1] [1 0 0] [1] a__U11(tt(),L) = [0 0 0]L + [0] >= [0 0 0]L + [0] = a__U12(tt(),L) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1] [1 0 0] [1] a__U12(tt(),L) = [0 0 0]L + [0] >= [0 0 0]L + [0] = s(a__length(mark(L))) [0 0 0] [1] [0 0 0] [1] [1 1 0] [1 0 0] [1] [1 1 0] [1] a__length(cons(N,L)) = [0 0 0]L + [0 0 1]N + [0] >= [0 0 0]L + [0] = a__U11(tt(),L) [0 0 0] [0 0 0] [1] [0 0 0] [1] [0] [0] mark(zeros()) = [0] >= [0] = a__zeros() [1] [1] [1 0 0] [0] [1 0 0] [0] mark(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = s(mark(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] a__U12(X1,X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = U12(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] 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(s(X)) -> s(mark(X)) 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