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