YES Problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Proof: Matrix Interpretation Processor: dim=1 interpretation: [nil] = 2, [isNatIList](x0) = 2x0, [isNatList](x0) = 2x0, [isNat](x0) = 2x0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [U11](x0, x1) = 2x0 + x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = 4x0 + 5x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L >= L = mark(s(length(L))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 2V1 >= 2V1 = mark(isNatList(V1)) active(isNat(s(V1))) = 2V1 >= 2V1 = mark(isNat(V1)) active(isNatIList(V)) = 2V >= 2V = mark(isNatList(V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 8V1 + 10V2 >= 2V1 + 2V2 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 4 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 8V1 + 10V2 >= 2V1 + 2V2 = mark(and(isNat(V1),isNatList(V2))) active(length(nil())) = 2 >= 0 = mark(0()) active(length(cons(N,L))) = 5L + 4N >= 5L + 4N = mark(U11(and(isNatList(L),isNat(N)),L)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 4X1 + 5X2 >= 4X1 + 5X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = X >= X = active(length(mark(X))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNat(X)) = 2X >= 2X = active(isNat(X)) mark(isNatList(X)) = 2X >= 2X = active(isNatList(X)) mark(isNatIList(X)) = 2X >= 2X = active(isNatIList(X)) mark(nil()) = 2 >= 2 = active(nil()) cons(mark(X1),X2) = 4X1 + 5X2 >= 4X1 + 5X2 = cons(X1,X2) cons(X1,mark(X2)) = 4X1 + 5X2 >= 4X1 + 5X2 = cons(X1,X2) cons(active(X1),X2) = 4X1 + 5X2 >= 4X1 + 5X2 = cons(X1,X2) cons(X1,active(X2)) = 4X1 + 5X2 >= 4X1 + 5X2 = cons(X1,X2) U11(mark(X1),X2) = 2X1 + X2 >= 2X1 + X2 = U11(X1,X2) U11(X1,mark(X2)) = 2X1 + X2 >= 2X1 + X2 = U11(X1,X2) U11(active(X1),X2) = 2X1 + X2 >= 2X1 + X2 = U11(X1,X2) U11(X1,active(X2)) = 2X1 + X2 >= 2X1 + X2 = U11(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X >= X = length(X) length(active(X)) = X >= X = length(X) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNat(mark(X)) = 2X >= 2X = isNat(X) isNat(active(X)) = 2X >= 2X = isNat(X) isNatList(mark(X)) = 2X >= 2X = isNatList(X) isNatList(active(X)) = 2X >= 2X = isNatList(X) isNatIList(mark(X)) = 2X >= 2X = isNatIList(X) isNatIList(active(X)) = 2X >= 2X = isNatIList(X) problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Matrix Interpretation Processor: dim=1 interpretation: [nil] = 0, [isNatIList](x0) = x0, [isNatList](x0) = x0, [isNat](x0) = x0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0 + 3, [U11](x0, x1) = 2x0 + x1 + 3, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = 2x0 + 3x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = L + 3 >= L + 3 = mark(s(length(L))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 + 3 >= V1 = mark(isNatList(V1)) active(isNat(s(V1))) = V1 >= V1 = mark(isNat(V1)) active(isNatIList(V)) = V >= V = mark(isNatList(V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2V1 + 3V2 >= V1 + V2 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(cons(V1,V2))) = 2V1 + 3V2 >= V1 + V2 = mark(and(isNat(V1),isNatList(V2))) active(length(cons(N,L))) = 3L + 2N + 3 >= 3L + 2N + 3 = mark(U11(and(isNatList(L),isNat(N)),L)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 2X1 + 3X2 >= 2X1 + 3X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 2X1 + X2 + 3 >= 2X1 + X2 + 3 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = X + 3 >= X + 3 = active(length(mark(X))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(isNatList(X)) = X >= X = active(isNatList(X)) mark(isNatIList(X)) = X >= X = active(isNatIList(X)) mark(nil()) = 0 >= 0 = active(nil()) cons(mark(X1),X2) = 2X1 + 3X2 >= 2X1 + 3X2 = cons(X1,X2) cons(X1,mark(X2)) = 2X1 + 3X2 >= 2X1 + 3X2 = cons(X1,X2) cons(active(X1),X2) = 2X1 + 3X2 >= 2X1 + 3X2 = cons(X1,X2) cons(X1,active(X2)) = 2X1 + 3X2 >= 2X1 + 3X2 = cons(X1,X2) U11(mark(X1),X2) = 2X1 + X2 + 3 >= 2X1 + X2 + 3 = U11(X1,X2) U11(X1,mark(X2)) = 2X1 + X2 + 3 >= 2X1 + X2 + 3 = U11(X1,X2) U11(active(X1),X2) = 2X1 + X2 + 3 >= 2X1 + X2 + 3 = U11(X1,X2) U11(X1,active(X2)) = 2X1 + X2 + 3 >= 2X1 + X2 + 3 = U11(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X + 3 >= X + 3 = length(X) length(active(X)) = X + 3 >= X + 3 = length(X) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) isNatList(mark(X)) = X >= X = isNatList(X) isNatList(active(X)) = X >= X = isNatList(X) isNatIList(mark(X)) = X >= X = isNatIList(X) isNatIList(active(X)) = X >= X = isNatIList(X) problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Matrix Interpretation Processor: dim=1 interpretation: [nil] = 0, [isNatIList](x0) = 2x0 + 1, [isNatList](x0) = x0, [isNat](x0) = x0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = 2x0 + 4, [U11](x0, x1) = 2x0 + 4x1 + 4, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + 4x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = 4L + 4 >= 2L + 4 = mark(s(length(L))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(V1))) = V1 >= V1 = mark(isNat(V1)) active(isNatIList(V)) = 2V + 1 >= V = mark(isNatList(V)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2V1 + 8V2 + 1 >= V1 + 2V2 + 1 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(cons(V1,V2))) = V1 + 4V2 >= V1 + V2 = mark(and(isNat(V1),isNatList(V2))) active(length(cons(N,L))) = 8L + 2N + 4 >= 6L + 2N + 4 = mark(U11(and(isNatList(L),isNat(N)),L)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 2X1 + 4X2 + 4 >= 2X1 + 4X2 + 4 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = 2X + 4 >= 2X + 4 = active(length(mark(X))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(isNatList(X)) = X >= X = active(isNatList(X)) mark(isNatIList(X)) = 2X + 1 >= 2X + 1 = active(isNatIList(X)) mark(nil()) = 0 >= 0 = active(nil()) cons(mark(X1),X2) = X1 + 4X2 >= X1 + 4X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + 4X2 >= X1 + 4X2 = cons(X1,X2) cons(active(X1),X2) = X1 + 4X2 >= X1 + 4X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + 4X2 >= X1 + 4X2 = cons(X1,X2) U11(mark(X1),X2) = 2X1 + 4X2 + 4 >= 2X1 + 4X2 + 4 = U11(X1,X2) U11(X1,mark(X2)) = 2X1 + 4X2 + 4 >= 2X1 + 4X2 + 4 = U11(X1,X2) U11(active(X1),X2) = 2X1 + 4X2 + 4 >= 2X1 + 4X2 + 4 = U11(X1,X2) U11(X1,active(X2)) = 2X1 + 4X2 + 4 >= 2X1 + 4X2 + 4 = U11(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = 2X + 4 >= 2X + 4 = length(X) length(active(X)) = 2X + 4 >= 2X + 4 = length(X) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) isNatList(mark(X)) = X >= X = isNatList(X) isNatList(active(X)) = X >= X = isNatList(X) isNatIList(mark(X)) = 2X + 1 >= 2X + 1 = isNatIList(X) isNatIList(active(X)) = 2X + 1 >= 2X + 1 = isNatIList(X) problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Matrix Interpretation Processor: dim=2 interpretation: [2] [nil] = [0], [2 0] [isNatIList](x0) = [0 0]x0, [1 0] [isNatList](x0) = [1 0]x0, [2 0] [isNat](x0) = [2 0]x0, [1 0] [1 0] [and](x0, x1) = [0 0]x0 + [1 0]x1, [1 0] [s](x0) = [0 0]x0, [2 0] [length](x0) = [0 0]x0, [1 0] [2 0] [U11](x0, x1) = [0 0]x0 + [2 0]x1, [0] [tt] = [0], [2 0] [mark](x0) = [0 0]x0, [2 0] [3 0] [cons](x0, x1) = [0 0]x0 + [0 0]x1, [0] [0] = [0], [1 1] [active](x0) = [0 0]x0, [0] [zeros] = [0] orientation: [0] [0] active(zeros()) = [0] >= [0] = mark(cons(0(),zeros())) [4 0] [4 0] active(U11(tt(),L)) = [0 0]L >= [0 0]L = mark(s(length(L))) [2 0] [2 0] active(and(tt(),X)) = [0 0]X >= [0 0]X = mark(X) [0] [0] active(isNat(0())) = [0] >= [0] = mark(tt()) [4 0] [4 0] active(isNat(s(V1))) = [0 0]V1 >= [0 0]V1 = mark(isNat(V1)) [4 0] [6 0] [4 0] [4 0] active(isNatIList(cons(V1,V2))) = [0 0]V1 + [0 0]V2 >= [0 0]V1 + [0 0]V2 = mark(and(isNat(V1),isNatIList(V2))) [4 0] [6 0] [4 0] [2 0] active(isNatList(cons(V1,V2))) = [0 0]V1 + [0 0]V2 >= [0 0]V1 + [0 0]V2 = mark(and(isNat(V1),isNatList(V2))) [6 0] [4 0] [6 0] [4 0] active(length(cons(N,L))) = [0 0]L + [0 0]N >= [0 0]L + [0 0]N = mark(U11(and(isNatList(L),isNat(N)),L)) [0] [0] mark(zeros()) = [0] >= [0] = active(zeros()) [4 0] [6 0] [4 0] [3 0] mark(cons(X1,X2)) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = active(cons(mark(X1),X2)) [0] [0] mark(0()) = [0] >= [0] = active(0()) [2 0] [4 0] [2 0] [4 0] mark(U11(X1,X2)) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = active(U11(mark(X1),X2)) [0] [0] mark(tt()) = [0] >= [0] = active(tt()) [2 0] [2 0] mark(s(X)) = [0 0]X >= [0 0]X = active(s(mark(X))) [4 0] [4 0] mark(length(X)) = [0 0]X >= [0 0]X = active(length(mark(X))) [2 0] [2 0] [2 0] [2 0] mark(and(X1,X2)) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = active(and(mark(X1),X2)) [4 0] [4 0] mark(isNat(X)) = [0 0]X >= [0 0]X = active(isNat(X)) [2 0] [2 0] mark(isNatList(X)) = [0 0]X >= [0 0]X = active(isNatList(X)) [4 0] [2 0] mark(isNatIList(X)) = [0 0]X >= [0 0]X = active(isNatIList(X)) [4] [2] mark(nil()) = [0] >= [0] = active(nil()) [4 0] [3 0] [2 0] [3 0] cons(mark(X1),X2) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = cons(X1,X2) [2 0] [6 0] [2 0] [3 0] cons(X1,mark(X2)) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = cons(X1,X2) [2 2] [3 0] [2 0] [3 0] cons(active(X1),X2) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = cons(X1,X2) [2 0] [3 3] [2 0] [3 0] cons(X1,active(X2)) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = cons(X1,X2) [2 0] [2 0] [1 0] [2 0] U11(mark(X1),X2) = [0 0]X1 + [2 0]X2 >= [0 0]X1 + [2 0]X2 = U11(X1,X2) [1 0] [4 0] [1 0] [2 0] U11(X1,mark(X2)) = [0 0]X1 + [4 0]X2 >= [0 0]X1 + [2 0]X2 = U11(X1,X2) [1 1] [2 0] [1 0] [2 0] U11(active(X1),X2) = [0 0]X1 + [2 0]X2 >= [0 0]X1 + [2 0]X2 = U11(X1,X2) [1 0] [2 2] [1 0] [2 0] U11(X1,active(X2)) = [0 0]X1 + [2 2]X2 >= [0 0]X1 + [2 0]X2 = U11(X1,X2) [2 0] [1 0] s(mark(X)) = [0 0]X >= [0 0]X = s(X) [1 1] [1 0] s(active(X)) = [0 0]X >= [0 0]X = s(X) [4 0] [2 0] length(mark(X)) = [0 0]X >= [0 0]X = length(X) [2 2] [2 0] length(active(X)) = [0 0]X >= [0 0]X = length(X) [2 0] [1 0] [1 0] [1 0] and(mark(X1),X2) = [0 0]X1 + [1 0]X2 >= [0 0]X1 + [1 0]X2 = and(X1,X2) [1 0] [2 0] [1 0] [1 0] and(X1,mark(X2)) = [0 0]X1 + [2 0]X2 >= [0 0]X1 + [1 0]X2 = and(X1,X2) [1 1] [1 0] [1 0] [1 0] and(active(X1),X2) = [0 0]X1 + [1 0]X2 >= [0 0]X1 + [1 0]X2 = and(X1,X2) [1 0] [1 1] [1 0] [1 0] and(X1,active(X2)) = [0 0]X1 + [1 1]X2 >= [0 0]X1 + [1 0]X2 = and(X1,X2) [4 0] [2 0] isNat(mark(X)) = [4 0]X >= [2 0]X = isNat(X) [2 2] [2 0] isNat(active(X)) = [2 2]X >= [2 0]X = isNat(X) [2 0] [1 0] isNatList(mark(X)) = [2 0]X >= [1 0]X = isNatList(X) [1 1] [1 0] isNatList(active(X)) = [1 1]X >= [1 0]X = isNatList(X) [4 0] [2 0] isNatIList(mark(X)) = [0 0]X >= [0 0]X = isNatIList(X) [2 2] [2 0] isNatIList(active(X)) = [0 0]X >= [0 0]X = isNatIList(X) problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [isNatIList](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [isNatList](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [isNat](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 1 0] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 1] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [length](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 1] [U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [tt] = [0] [0], [1 0 0] [0] [mark](x0) = [1 1 0]x0 + [1] [0 0 1] [0], [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [1 0 1] , [0] [0] = [0] [0], [1 0 0] [0] [active](x0) = [1 1 0]x0 + [1] [0 0 1] [0], [0] [zeros] = [0] [0] orientation: [0] [0] active(zeros()) = [1] >= [1] = mark(cons(0(),zeros())) [0] [0] [1 0 1] [0] [1 0 1] [0] active(U11(tt(),L)) = [1 0 1]L + [1] >= [1 0 1]L + [1] = mark(s(length(L))) [0 0 0] [0] [0 0 0] [0] [1 1 0] [0] [1 0 0] [0] active(and(tt(),X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = mark(X) [0 0 1] [0] [0 0 1] [0] [0] [0] active(isNat(0())) = [1] >= [1] = mark(tt()) [0] [0] [1 0 0] [0] [1 0 0] [0] active(isNat(s(V1))) = [1 0 0]V1 + [1] >= [1 0 0]V1 + [1] = mark(isNat(V1)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] active(isNatIList(cons(V1,V2))) = [1 0 0]V1 + [1 0 0]V2 + [1] >= [1 0 0]V1 + [1 0 0]V2 + [1] = mark(and(isNat(V1),isNatIList(V2))) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] active(isNatList(cons(V1,V2))) = [1 0 0]V1 + [1 0 0]V2 + [1] >= [1 0 0]V1 + [1 0 0]V2 + [1] = mark(and(isNat(V1),isNatList(V2))) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [2 0 1] [1 0 0] [0] [2 0 1] [1 0 0] [0] active(length(cons(N,L))) = [2 0 1]L + [1 0 0]N + [1] >= [2 0 1]L + [1 0 0]N + [1] = mark(U11(and(isNatList(L),isNat(N)),L)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] [0] mark(zeros()) = [1] >= [1] = active(zeros()) [0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] mark(cons(X1,X2)) = [1 0 0]X1 + [1 0 0]X2 + [1] >= [1 0 0]X1 + [1 0 0]X2 + [1] = active(cons(mark(X1),X2)) [0 0 0] [1 0 1] [0] [0 0 0] [1 0 1] [0] [0] [0] mark(0()) = [1] >= [1] = active(0()) [0] [0] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 1] [0] mark(U11(X1,X2)) = [1 0 0]X1 + [1 0 1]X2 + [1] >= [1 0 0]X1 + [1 0 1]X2 + [1] = active(U11(mark(X1),X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] [0] mark(tt()) = [1] >= [1] = active(tt()) [0] [0] [1 0 0] [0] [1 0 0] [0] mark(s(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = active(s(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 0 1] [0] [1 0 1] [0] mark(length(X)) = [1 0 1]X + [1] >= [1 0 1]X + [1] = active(length(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 1 0] [0] [1 0 0] [1 1 0] [0] mark(and(X1,X2)) = [1 0 0]X1 + [1 1 0]X2 + [1] >= [1 0 0]X1 + [1 1 0]X2 + [1] = active(and(mark(X1),X2)) [0 0 0] [0 0 1] [0] [0 0 0] [0 0 1] [0] [1 0 0] [0] [1 0 0] [0] mark(isNat(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = active(isNat(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] mark(isNatList(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = active(isNatList(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] mark(isNatIList(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = active(isNatIList(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [1 0 1] [0 0 0] [1 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [1 0 1] [0 0 0] [1 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [1 0 1] [0 0 0] [1 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [1 0 1] [0 0 0] [1 0 1] [1 0 0] [1 0 1] [1 0 0] [1 0 1] U11(mark(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 0 0] [1 0 1] [1 0 0] [1 0 1] U11(X1,mark(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 0 0] [1 0 1] [1 0 0] [1 0 1] U11(active(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 0 0] [1 0 1] [1 0 0] [1 0 1] U11(X1,active(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 0 0] [1 0 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] s(active(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 0 1] [1 0 1] length(mark(X)) = [0 0 0]X >= [0 0 0]X = length(X) [0 0 0] [0 0 0] [1 0 1] [1 0 1] length(active(X)) = [0 0 0]X >= [0 0 0]X = length(X) [0 0 0] [0 0 0] [1 0 0] [1 1 0] [1 0 0] [1 1 0] and(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = and(X1,X2) [0 0 0] [0 0 1] [0 0 0] [0 0 1] [1 0 0] [2 1 0] [1] [1 0 0] [1 1 0] and(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = and(X1,X2) [0 0 0] [0 0 1] [0] [0 0 0] [0 0 1] [1 0 0] [1 1 0] [1 0 0] [1 1 0] and(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = and(X1,X2) [0 0 0] [0 0 1] [0 0 0] [0 0 1] [1 0 0] [2 1 0] [1] [1 0 0] [1 1 0] and(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = and(X1,X2) [0 0 0] [0 0 1] [0] [0 0 0] [0 0 1] [1 0 0] [1 0 0] isNat(mark(X)) = [0 0 0]X >= [0 0 0]X = isNat(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] isNat(active(X)) = [0 0 0]X >= [0 0 0]X = isNat(X) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] isNatList(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = isNatList(X) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] isNatList(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = isNatList(X) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] isNatIList(mark(X)) = [0 0 0]X >= [0 0 0]X = isNatIList(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] isNatIList(active(X)) = [0 0 0]X >= [0 0 0]X = isNatIList(X) [0 0 0] [0 0 0] problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) and(mark(X1),X2) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Matrix Interpretation Processor: dim=3 interpretation: [1 1 1] [0] [isNatIList](x0) = [0 0 0]x0 + [0] [1 0 0] [1], [1 0 0] [isNatList](x0) = [0 0 0]x0 [1 0 1] , [1 0 0] [isNat](x0) = [0 0 0]x0 [1 0 0] , [1 0 0] [1 1 0] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 1] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [length](x0) = [0 0 0]x0 [1 0 0] , [1 0 0] [1 0 1] [U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 1] , [0] [tt] = [0] [0], [1 0 0] [0] [mark](x0) = [1 1 0]x0 + [1] [0 0 1] [0], [1 0 0] [1 0 1] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [1 1 1] , [0] [0] = [0] [0], [1 0 0] [0] [active](x0) = [1 0 0]x0 + [1] [0 1 1] [0], [0] [zeros] = [0] [0] orientation: [0] [0] active(zeros()) = [1] >= [1] = mark(cons(0(),zeros())) [0] [0] [1 0 1] [0] [1 0 1] [0] active(U11(tt(),L)) = [1 0 1]L + [1] >= [1 0 1]L + [1] = mark(s(length(L))) [0 0 1] [0] [0 0 0] [0] [1 1 0] [0] [1 0 0] [0] active(and(tt(),X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = mark(X) [0 0 1] [0] [0 0 1] [0] [0] [0] active(isNat(0())) = [1] >= [1] = mark(tt()) [0] [0] [1 0 0] [0] [1 0 0] [0] active(isNat(s(V1))) = [1 0 0]V1 + [1] >= [1 0 0]V1 + [1] = mark(isNat(V1)) [1 0 0] [0] [1 0 0] [0] [1 0 0] [2 1 2] [0] [1 0 0] [1 1 1] [0] active(isNatIList(cons(V1,V2))) = [1 0 0]V1 + [2 1 2]V2 + [1] >= [1 0 0]V1 + [1 1 1]V2 + [1] = mark(and(isNat(V1),isNatIList(V2))) [1 0 0] [1 0 1] [1] [0 0 0] [1 0 0] [1] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 0] [0] active(isNatList(cons(V1,V2))) = [1 0 0]V1 + [1 0 1]V2 + [1] >= [1 0 0]V1 + [1 0 0]V2 + [1] = mark(and(isNat(V1),isNatList(V2))) [1 0 0] [2 1 2] [0] [0 0 0] [1 0 1] [0] [2 1 2] [1 0 0] [0] [2 0 1] [1 0 0] [0] active(length(cons(N,L))) = [2 1 2]L + [1 0 0]N + [1] >= [2 0 1]L + [1 0 0]N + [1] = mark(U11(and(isNatList(L),isNat(N)),L)) [1 0 1] [1 0 0] [0] [0 0 1] [0 0 0] [0] [0] [0] mark(zeros()) = [1] >= [1] = active(zeros()) [0] [0] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 1] [0] mark(cons(X1,X2)) = [1 0 0]X1 + [1 0 1]X2 + [1] >= [1 0 0]X1 + [1 0 1]X2 + [1] = active(cons(mark(X1),X2)) [0 0 0] [1 1 1] [0] [0 0 0] [1 1 1] [0] [0] [0] mark(0()) = [1] >= [1] = active(0()) [0] [0] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 1] [0] mark(U11(X1,X2)) = [1 0 0]X1 + [1 0 1]X2 + [1] >= [1 0 0]X1 + [1 0 1]X2 + [1] = active(U11(mark(X1),X2)) [0 0 0] [0 0 1] [0] [0 0 0] [0 0 1] [0] [0] [0] mark(tt()) = [1] >= [1] = active(tt()) [0] [0] [1 0 0] [0] [1 0 0] [0] mark(s(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = active(s(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 0 1] [0] [1 0 1] [0] mark(length(X)) = [1 0 1]X + [1] >= [1 0 1]X + [1] = active(length(mark(X))) [1 0 0] [0] [1 0 0] [0] [1 0 0] [1 1 0] [0] [1 0 0] [1 1 0] [0] mark(and(X1,X2)) = [1 0 0]X1 + [1 1 0]X2 + [1] >= [1 0 0]X1 + [1 1 0]X2 + [1] = active(and(mark(X1),X2)) [0 0 0] [0 0 1] [0] [0 0 0] [0 0 1] [0] [1 0 0] [0] [1 0 0] [0] mark(isNat(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = active(isNat(X)) [1 0 0] [0] [1 0 0] [0] [1 0 0] [0] [1 0 0] [0] mark(isNatList(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = active(isNatList(X)) [1 0 1] [0] [1 0 1] [0] [1 1 1] [0] [1 1 1] [0] mark(isNatIList(X)) = [1 1 1]X + [1] >= [1 1 1]X + [1] = active(isNatIList(X)) [1 0 0] [1] [1 0 0] [1] [1 0 0] [1 0 1] [1 0 0] [1 0 1] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [1 1 1] [0 0 0] [1 1 1] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 1] cons(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [2 1 1] [1] [0 0 0] [1 1 1] [1 0 0] [1 0 1] [1 0 0] [1 0 1] cons(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [1 1 1] [0 0 0] [1 1 1] [1 0 0] [1 1 1] [0] [1 0 0] [1 0 1] cons(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [2 1 1] [1] [0 0 0] [1 1 1] [1 0 0] [1 0 1] [1 0 0] [1 0 1] U11(mark(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 1] [0 0 0] [0 0 1] [1 0 0] [1 0 1] [1 0 0] [1 0 1] U11(X1,mark(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 1] [0 0 0] [0 0 1] [1 0 0] [1 0 1] [1 0 0] [1 0 1] U11(active(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 1] [0 0 0] [0 0 1] [1 0 0] [1 1 1] [1 0 0] [1 0 1] U11(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U11(X1,X2) [0 0 0] [0 1 1] [0 0 0] [0 0 1] [1 0 0] [1 0 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] s(active(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 0 1] [1 0 1] length(mark(X)) = [0 0 0]X >= [0 0 0]X = length(X) [1 0 0] [1 0 0] [1 1 1] [1 0 1] length(active(X)) = [0 0 0]X >= [0 0 0]X = length(X) [1 0 0] [1 0 0] [1 0 0] [1 1 0] [1 0 0] [1 1 0] and(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = and(X1,X2) [0 0 0] [0 0 1] [0 0 0] [0 0 1] [1 0 0] [1 1 0] [1 0 0] [1 1 0] and(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = and(X1,X2) [0 0 0] [0 0 1] [0 0 0] [0 0 1] [1 0 0] [1 0 0] isNat(mark(X)) = [0 0 0]X >= [0 0 0]X = isNat(X) [1 0 0] [1 0 0] [1 0 0] [1 0 0] isNat(active(X)) = [0 0 0]X >= [0 0 0]X = isNat(X) [1 0 0] [1 0 0] [1 0 0] [1 0 0] isNatList(mark(X)) = [0 0 0]X >= [0 0 0]X = isNatList(X) [1 0 1] [1 0 1] [1 0 0] [1 0 0] isNatList(active(X)) = [0 0 0]X >= [0 0 0]X = isNatList(X) [1 1 1] [1 0 1] [2 1 1] [1] [1 1 1] [0] isNatIList(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = isNatIList(X) [1 0 0] [1] [1 0 0] [1] [2 1 1] [1] [1 1 1] [0] isNatIList(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = isNatIList(X) [1 0 0] [1] [1 0 0] [1] problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) and(mark(X1),X2) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) Matrix Interpretation Processor: dim=2 interpretation: [2 0] [isNatIList](x0) = [0 0]x0, [1 2] [isNatList](x0) = [0 0]x0, [1 0] [0] [isNat](x0) = [0 0]x0 + [2], [1 0] [and](x0, x1) = [0 0]x0 + x1, [1 0] [s](x0) = [0 0]x0, [2 0] [2] [length](x0) = [0 0]x0 + [0], [2 0] [2 0] [2] [U11](x0, x1) = [0 0]x0 + [0 0]x1 + [0], [0] [tt] = [0], [0] [mark](x0) = x0 + [1], [1 0] [2 2] [cons](x0, x1) = [0 0]x0 + [0 0]x1, [0] [0] = [0], [0] [active](x0) = x0 + [1], [0] [zeros] = [0] orientation: [0] [0] active(zeros()) = [1] >= [1] = mark(cons(0(),zeros())) [2 0] [2] [2 0] [2] active(U11(tt(),L)) = [0 0]L + [1] >= [0 0]L + [1] = mark(s(length(L))) [0] [0] active(and(tt(),X)) = X + [1] >= X + [1] = mark(X) [0] [0] active(isNat(0())) = [3] >= [1] = mark(tt()) [1 0] [0] [1 0] [0] active(isNat(s(V1))) = [0 0]V1 + [3] >= [0 0]V1 + [3] = mark(isNat(V1)) [2 0] [4 4] [0] [1 0] [2 0] [0] active(isNatIList(cons(V1,V2))) = [0 0]V1 + [0 0]V2 + [1] >= [0 0]V1 + [0 0]V2 + [1] = mark(and(isNat(V1),isNatIList(V2))) [1 0] [2 2] [0] [1 0] [1 2] [0] active(isNatList(cons(V1,V2))) = [0 0]V1 + [0 0]V2 + [1] >= [0 0]V1 + [0 0]V2 + [1] = mark(and(isNat(V1),isNatList(V2))) [4 4] [2 0] [2] [4 4] [2 0] [2] active(length(cons(N,L))) = [0 0]L + [0 0]N + [1] >= [0 0]L + [0 0]N + [1] = mark(U11(and(isNatList(L),isNat(N)),L)) [0] [0] mark(zeros()) = [1] >= [1] = active(zeros()) [1 0] [2 2] [0] [1 0] [2 2] [0] mark(cons(X1,X2)) = [0 0]X1 + [0 0]X2 + [1] >= [0 0]X1 + [0 0]X2 + [1] = active(cons(mark(X1),X2)) [0] [0] mark(0()) = [1] >= [1] = active(0()) [2 0] [2 0] [2] [2 0] [2 0] [2] mark(U11(X1,X2)) = [0 0]X1 + [0 0]X2 + [1] >= [0 0]X1 + [0 0]X2 + [1] = active(U11(mark(X1),X2)) [0] [0] mark(tt()) = [1] >= [1] = active(tt()) [1 0] [0] [1 0] [0] mark(s(X)) = [0 0]X + [1] >= [0 0]X + [1] = active(s(mark(X))) [2 0] [2] [2 0] [2] mark(length(X)) = [0 0]X + [1] >= [0 0]X + [1] = active(length(mark(X))) [1 0] [0] [1 0] [0] mark(and(X1,X2)) = [0 0]X1 + X2 + [1] >= [0 0]X1 + X2 + [1] = active(and(mark(X1),X2)) [1 0] [0] [1 0] [0] mark(isNat(X)) = [0 0]X + [3] >= [0 0]X + [3] = active(isNat(X)) [1 2] [0] [1 2] [0] mark(isNatList(X)) = [0 0]X + [1] >= [0 0]X + [1] = active(isNatList(X)) [2 0] [0] [2 0] [0] mark(isNatIList(X)) = [0 0]X + [1] >= [0 0]X + [1] = active(isNatIList(X)) [1 0] [2 2] [1 0] [2 2] cons(mark(X1),X2) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = cons(X1,X2) [1 0] [2 2] [2] [1 0] [2 2] cons(X1,mark(X2)) = [0 0]X1 + [0 0]X2 + [0] >= [0 0]X1 + [0 0]X2 = cons(X1,X2) [1 0] [2 2] [1 0] [2 2] cons(active(X1),X2) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = cons(X1,X2) [1 0] [2 2] [2] [1 0] [2 2] cons(X1,active(X2)) = [0 0]X1 + [0 0]X2 + [0] >= [0 0]X1 + [0 0]X2 = cons(X1,X2) [2 0] [2 0] [2] [2 0] [2 0] [2] U11(mark(X1),X2) = [0 0]X1 + [0 0]X2 + [0] >= [0 0]X1 + [0 0]X2 + [0] = U11(X1,X2) [2 0] [2 0] [2] [2 0] [2 0] [2] U11(X1,mark(X2)) = [0 0]X1 + [0 0]X2 + [0] >= [0 0]X1 + [0 0]X2 + [0] = U11(X1,X2) [2 0] [2 0] [2] [2 0] [2 0] [2] U11(active(X1),X2) = [0 0]X1 + [0 0]X2 + [0] >= [0 0]X1 + [0 0]X2 + [0] = U11(X1,X2) [2 0] [2 0] [2] [2 0] [2 0] [2] U11(X1,active(X2)) = [0 0]X1 + [0 0]X2 + [0] >= [0 0]X1 + [0 0]X2 + [0] = U11(X1,X2) [1 0] [1 0] s(mark(X)) = [0 0]X >= [0 0]X = s(X) [1 0] [1 0] s(active(X)) = [0 0]X >= [0 0]X = s(X) [2 0] [2] [2 0] [2] length(mark(X)) = [0 0]X + [0] >= [0 0]X + [0] = length(X) [2 0] [2] [2 0] [2] length(active(X)) = [0 0]X + [0] >= [0 0]X + [0] = length(X) [1 0] [1 0] and(mark(X1),X2) = [0 0]X1 + X2 >= [0 0]X1 + X2 = and(X1,X2) [1 0] [1 0] and(active(X1),X2) = [0 0]X1 + X2 >= [0 0]X1 + X2 = and(X1,X2) [1 0] [0] [1 0] [0] isNat(mark(X)) = [0 0]X + [2] >= [0 0]X + [2] = isNat(X) [1 0] [0] [1 0] [0] isNat(active(X)) = [0 0]X + [2] >= [0 0]X + [2] = isNat(X) [1 2] [2] [1 2] isNatList(mark(X)) = [0 0]X + [0] >= [0 0]X = isNatList(X) [1 2] [2] [1 2] isNatList(active(X)) = [0 0]X + [0] >= [0 0]X = isNatList(X) problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) and(mark(X1),X2) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Matrix Interpretation Processor: dim=3 interpretation: [1 1 1] [1] [isNatIList](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [0] [isNatList](x0) = [0 1 0]x0 + [1] [0 1 0] [0], [1 0 1] [isNat](x0) = [0 0 0]x0 [1 0 0] , [1 0 1] [1 1 1] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 1] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [1] [length](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 1] [1 0 1] [0] [U11](x0, x1) = [0 0 0]x0 + [1 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [0] [tt] = [0] [0], [1 0 1] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [1 1 1] [0] [cons](x0, x1) = [1 0 0]x0 + [0 1 1]x1 + [1] [1 0 1] [1 0 0] [0], [0] [0] = [0] [0], [1 0 1] [active](x0) = [0 0 0]x0 [0 0 0] , [0] [zeros] = [0] [1] orientation: [1] [1] active(zeros()) = [0] >= [0] = mark(cons(0(),zeros())) [0] [0] [1 0 1] [1] [1 0 1] [1] active(U11(tt(),L)) = [0 0 0]L + [0] >= [0 0 0]L + [0] = mark(s(length(L))) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1 0 1] active(and(tt(),X)) = [0 0 0]X >= [0 0 0]X = mark(X) [0 0 0] [0 0 0] [0] [0] active(isNat(0())) = [0] >= [0] = mark(tt()) [0] [0] [2 0 2] [2 0 1] active(isNat(s(V1))) = [0 0 0]V1 >= [0 0 0]V1 = mark(isNat(V1)) [0 0 0] [0 0 0] [3 0 2] [2 2 2] [2] [2 0 1] [1 1 1] [1] active(isNatIList(cons(V1,V2))) = [0 0 0]V1 + [0 0 0]V2 + [0] >= [0 0 0]V1 + [0 0 0]V2 + [0] = mark(and(isNat(V1),isNatIList(V2))) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [2 0 1] [1 2 2] [1] [2 0 1] [1 2 0] [1] active(isNatList(cons(V1,V2))) = [0 0 0]V1 + [0 0 0]V2 + [0] >= [0 0 0]V1 + [0 0 0]V2 + [0] = mark(and(isNat(V1),isNatList(V2))) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [2 1 1] [2 0 2] [1] [2 1 1] [2 0 1] [1] active(length(cons(N,L))) = [0 0 0]L + [0 0 0]N + [0] >= [0 0 0]L + [0 0 0]N + [0] = mark(U11(and(isNatList(L),isNat(N)),L)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1] [1] mark(zeros()) = [0] >= [0] = active(zeros()) [0] [0] [2 0 2] [2 1 1] [2 0 2] [2 1 1] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(cons(mark(X1),X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] mark(0()) = [0] >= [0] = active(0()) [0] [0] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] mark(U11(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = active(U11(mark(X1),X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] [0] mark(tt()) = [0] >= [0] = active(tt()) [0] [0] [1 0 1] [1 0 1] mark(s(X)) = [0 0 0]X >= [0 0 0]X = active(s(mark(X))) [0 0 0] [0 0 0] [1 0 1] [1] [1 0 1] [1] mark(length(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(length(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1 1 1] [1 0 1] [1 1 1] mark(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(and(mark(X1),X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [2 0 1] [2 0 1] mark(isNat(X)) = [0 0 0]X >= [0 0 0]X = active(isNat(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] mark(isNatList(X)) = [0 0 0]X >= [0 0 0]X = active(isNatList(X)) [0 0 0] [0 0 0] [1 1 1] [1] [1 1 1] [1] mark(isNatIList(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(isNatIList(X)) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1 1 1] [0] [1 0 1] [1 1 1] [0] cons(mark(X1),X2) = [1 0 1]X1 + [0 1 1]X2 + [1] >= [1 0 0]X1 + [0 1 1]X2 + [1] = cons(X1,X2) [1 0 1] [1 0 0] [0] [1 0 1] [1 0 0] [0] [1 0 1] [1 1 1] [0] [1 0 1] [1 1 1] [0] cons(active(X1),X2) = [1 0 1]X1 + [0 1 1]X2 + [1] >= [1 0 0]X1 + [0 1 1]X2 + [1] = cons(X1,X2) [1 0 1] [1 0 0] [0] [1 0 1] [1 0 0] [0] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] [0] U11(mark(X1),X2) = [0 0 0]X1 + [1 0 0]X2 + [0] >= [0 0 0]X1 + [1 0 0]X2 + [0] = U11(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] [0] U11(X1,mark(X2)) = [0 0 0]X1 + [1 0 1]X2 + [0] >= [0 0 0]X1 + [1 0 0]X2 + [0] = U11(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] [0] U11(active(X1),X2) = [0 0 0]X1 + [1 0 0]X2 + [0] >= [0 0 0]X1 + [1 0 0]X2 + [0] = U11(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] [0] U11(X1,active(X2)) = [0 0 0]X1 + [1 0 1]X2 + [0] >= [0 0 0]X1 + [1 0 0]X2 + [0] = U11(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 1] [1 0 1] s(mark(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 0 1] [1 0 1] s(active(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 0 1] [1] [1 0 1] [1] length(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length(X) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1] [1 0 1] [1] length(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length(X) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1 1 1] [1 0 1] [1 1 1] and(mark(X1),X2) = [0 0 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 1] [1 1 1] [1 0 1] [1 1 1] and(active(X1),X2) = [0 0 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 1] [1 0 1] isNat(mark(X)) = [0 0 0]X >= [0 0 0]X = isNat(X) [1 0 1] [1 0 0] [1 0 1] [1 0 1] isNat(active(X)) = [0 0 0]X >= [0 0 0]X = isNat(X) [1 0 1] [1 0 0] problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) and(mark(X1),X2) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [isNatIList](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [0] [isNatList](x0) = [0 0 0]x0 + [0] [1 1 0] [1], [1 0 0] [1] [isNat](x0) = [0 0 0]x0 + [0] [1 0 0] [0], [1 0 0] [1 0 0] [0] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 1] [0 0 1] [1], [1 0 0] [s](x0) = [1 0 0]x0 [0 0 0] , [1 0 1] [1] [length](x0) = [0 0 0]x0 + [0] [1 0 0] [1], [1 0 0] [1 0 1] [U11](x0, x1) = [0 0 0]x0 + [1 0 0]x1 [1 0 0] [0 0 0] , [1] [tt] = [0] [0], [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [1 1 1] [0] [cons](x0, x1) = [1 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [1 0 0] [0], [0] [0] = [0] [0], [1 0 0] [active](x0) = [0 0 0]x0 [0 0 1] , [0] [zeros] = [0] [0] orientation: [0] [0] active(zeros()) = [0] >= [0] = mark(cons(0(),zeros())) [0] [0] [1 0 1] [1] [1 0 1] [1] active(U11(tt(),L)) = [0 0 0]L + [0] >= [0 0 0]L + [0] = mark(s(length(L))) [0 0 0] [1] [0 0 0] [0] [1 0 0] [1] [1 0 0] active(and(tt(),X)) = [0 0 0]X + [0] >= [0 0 0]X = mark(X) [0 0 1] [1] [0 0 1] [1] [1] active(isNat(0())) = [0] >= [0] = mark(tt()) [0] [0] [1 0 0] [1] [1 0 0] [1] active(isNat(s(V1))) = [0 0 0]V1 + [0] >= [0 0 0]V1 + [0] = mark(isNat(V1)) [1 0 0] [0] [1 0 0] [0] [2 0 0] [1 1 1] [1] [1 0 0] [1 1 0] [1] active(isNatList(cons(V1,V2))) = [0 0 0]V1 + [0 0 0]V2 + [0] >= [0 0 0]V1 + [0 0 0]V2 + [0] = mark(and(isNat(V1),isNatList(V2))) [2 0 0] [1 1 1] [2] [1 0 0] [1 1 0] [2] [2 1 1] [1 0 0] [1] [2 1 1] [1 0 0] [1] active(length(cons(N,L))) = [0 0 0]L + [0 0 0]N + [0] >= [0 0 0]L + [0 0 0]N + [0] = mark(U11(and(isNatList(L),isNat(N)),L)) [1 1 1] [1 0 0] [1] [1 1 0] [1 0 0] [1] [0] [0] mark(zeros()) = [0] >= [0] = active(zeros()) [0] [0] [1 0 0] [1 1 1] [1 0 0] [1 1 1] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(cons(mark(X1),X2)) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [0] [0] mark(0()) = [0] >= [0] = active(0()) [0] [0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] mark(U11(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(U11(mark(X1),X2)) [1 0 0] [0 0 0] [1 0 0] [0 0 0] [1] [1] mark(tt()) = [0] >= [0] = active(tt()) [0] [0] [1 0 0] [1 0 0] mark(s(X)) = [0 0 0]X >= [0 0 0]X = active(s(mark(X))) [0 0 0] [0 0 0] [1 0 1] [1] [1 0 1] [1] mark(length(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(length(mark(X))) [1 0 0] [1] [1 0 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] mark(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = active(and(mark(X1),X2)) [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] [1 0 0] [1] [1 0 0] [1] mark(isNat(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(isNat(X)) [1 0 0] [0] [1 0 0] [0] [1 1 0] [0] [1 1 0] [0] mark(isNatList(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(isNatList(X)) [1 1 0] [1] [1 1 0] [1] [1 0 0] [1 0 0] mark(isNatIList(X)) = [0 0 0]X >= [0 0 0]X = active(isNatIList(X)) [0 0 0] [0 0 0] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] [0] cons(mark(X1),X2) = [1 0 0]X1 + [0 0 0]X2 + [1] >= [1 0 0]X1 + [0 0 0]X2 + [1] = cons(X1,X2) [0 0 0] [1 0 0] [0] [0 0 0] [1 0 0] [0] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] [0] cons(active(X1),X2) = [1 0 0]X1 + [0 0 0]X2 + [1] >= [1 0 0]X1 + [0 0 0]X2 + [1] = cons(X1,X2) [0 0 0] [1 0 0] [0] [0 0 0] [1 0 0] [0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] U11(mark(X1),X2) = [0 0 0]X1 + [1 0 0]X2 >= [0 0 0]X1 + [1 0 0]X2 = U11(X1,X2) [1 0 0] [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] U11(X1,mark(X2)) = [0 0 0]X1 + [1 0 0]X2 >= [0 0 0]X1 + [1 0 0]X2 = U11(X1,X2) [1 0 0] [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] U11(active(X1),X2) = [0 0 0]X1 + [1 0 0]X2 >= [0 0 0]X1 + [1 0 0]X2 = U11(X1,X2) [1 0 0] [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] U11(X1,active(X2)) = [0 0 0]X1 + [1 0 0]X2 >= [0 0 0]X1 + [1 0 0]X2 = U11(X1,X2) [1 0 0] [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] s(mark(X)) = [1 0 0]X >= [1 0 0]X = s(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] s(active(X)) = [1 0 0]X >= [1 0 0]X = s(X) [0 0 0] [0 0 0] [1 0 1] [1] [1 0 1] [1] length(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length(X) [1 0 0] [1] [1 0 0] [1] [1 0 1] [1] [1 0 1] [1] length(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length(X) [1 0 0] [1] [1 0 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] and(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = and(X1,X2) [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] and(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = and(X1,X2) [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] [1 0 0] [1] [1 0 0] [1] isNat(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = isNat(X) [1 0 0] [0] [1 0 0] [0] [1 0 0] [1] [1 0 0] [1] isNat(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = isNat(X) [1 0 0] [0] [1 0 0] [0] problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) and(mark(X1),X2) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Matrix Interpretation Processor: dim=3 interpretation: [1 1 1] [isNatIList](x0) = [1 0 1]x0 [0 0 0] , [1 0 1] [isNatList](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [isNat](x0) = [0 1 1]x0 + [0] [0 0 0] [1], [1 0 0] [1 0 0] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [s](x0) = [0 0 0]x0 [0 1 1] , [1 1 1] [length](x0) = [0 0 0]x0 [0 0 0] , [1 1 1] [1 1 1] [U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [tt] = [1] [1], [1 0 0] [mark](x0) = [0 0 1]x0 [0 1 0] , [1 0 0] [1 0 1] [cons](x0, x1) = [1 0 0]x0 + [0 1 1]x1 [0 0 0] [1 0 0] , [0] [0] = [0] [1], [1 0 0] [active](x0) = [0 0 1]x0 [0 1 0] , [0] [zeros] = [0] [0] orientation: [0] [0] active(zeros()) = [0] >= [0] = mark(cons(0(),zeros())) [0] [0] [1 1 1] [2] [1 1 1] active(U11(tt(),L)) = [0 0 0]L + [0] >= [0 0 0]L = mark(s(length(L))) [0 0 0] [0] [0 0 0] [0] [0] active(isNat(0())) = [1] >= [1] = mark(tt()) [1] [1] [1 0 0] [0] [1 0 0] [0] active(isNat(s(V1))) = [0 0 0]V1 + [1] >= [0 0 0]V1 + [1] = mark(isNat(V1)) [0 1 1] [0] [0 1 1] [0] [1 0 0] [2 0 1] [1 0 0] [1 0 1] active(isNatList(cons(V1,V2))) = [0 0 0]V1 + [0 0 0]V2 >= [0 0 0]V1 + [0 0 0]V2 = mark(and(isNat(V1),isNatList(V2))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [2 1 2] [2 0 0] [2 1 2] [1 0 0] active(length(cons(N,L))) = [0 0 0]L + [0 0 0]N >= [0 0 0]L + [0 0 0]N = mark(U11(and(isNatList(L),isNat(N)),L)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] mark(zeros()) = [0] >= [0] = active(zeros()) [0] [0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] mark(cons(X1,X2)) = [0 0 0]X1 + [1 0 0]X2 >= [0 0 0]X1 + [1 0 0]X2 = active(cons(mark(X1),X2)) [1 0 0] [0 1 1] [1 0 0] [0 1 1] [0] [0] mark(0()) = [1] >= [1] = active(0()) [0] [0] [1 1 1] [1 1 1] [1 1 1] [1 1 1] mark(U11(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(U11(mark(X1),X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] mark(tt()) = [1] >= [1] = active(tt()) [1] [1] [1 0 0] [1 0 0] mark(s(X)) = [0 1 1]X >= [0 1 1]X = active(s(mark(X))) [0 0 0] [0 0 0] [1 1 1] [1 1 1] mark(length(X)) = [0 0 0]X >= [0 0 0]X = active(length(mark(X))) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] mark(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(and(mark(X1),X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] mark(isNat(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(isNat(X)) [0 1 1] [0] [0 1 1] [0] [1 0 1] [1 0 1] mark(isNatList(X)) = [0 0 0]X >= [0 0 0]X = active(isNatList(X)) [0 0 0] [0 0 0] [1 1 1] [1 1 1] mark(isNatIList(X)) = [0 0 0]X >= [0 0 0]X = active(isNatIList(X)) [1 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 0] [1 0 1] cons(mark(X1),X2) = [1 0 0]X1 + [0 1 1]X2 >= [1 0 0]X1 + [0 1 1]X2 = cons(X1,X2) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] cons(active(X1),X2) = [1 0 0]X1 + [0 1 1]X2 >= [1 0 0]X1 + [0 1 1]X2 = cons(X1,X2) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 1 1] [1 1 1] [1 1 1] [1 1 1] U11(mark(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 1] [1 1 1] [1 1 1] U11(X1,mark(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 1] [1 1 1] [1 1 1] U11(active(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 1] [1 1 1] [1 1 1] U11(X1,active(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 0 0] [1 0 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 1 1] [0 1 1] [1 0 0] [1 0 0] s(active(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 1 1] [0 1 1] [1 1 1] [1 1 1] length(mark(X)) = [0 0 0]X >= [0 0 0]X = length(X) [0 0 0] [0 0 0] [1 1 1] [1 1 1] length(active(X)) = [0 0 0]X >= [0 0 0]X = length(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] and(mark(X1),X2) = [0 0 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 0 0] [1 0 0] [1 0 0] and(active(X1),X2) = [0 0 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] [0] [1 0 0] [0] isNat(mark(X)) = [0 1 1]X + [0] >= [0 1 1]X + [0] = isNat(X) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] isNat(active(X)) = [0 1 1]X + [0] >= [0 1 1]X + [0] = isNat(X) [0 0 0] [1] [0 0 0] [1] problem: active(zeros()) -> mark(cons(0(),zeros())) active(isNat(0())) -> mark(tt()) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) and(mark(X1),X2) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Matrix Interpretation Processor: dim=1 interpretation: [isNatIList](x0) = x0 + 1, [isNatList](x0) = x0, [isNat](x0) = x0, [and](x0, x1) = x0 + x1, [s](x0) = 4x0, [length](x0) = 5x0, [U11](x0, x1) = x0 + 3x1, [tt] = 0, [mark](x0) = 4x0, [cons](x0, x1) = 6x0 + 4x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(V1))) = 4V1 >= 4V1 = mark(isNat(V1)) active(isNatList(cons(V1,V2))) = 6V1 + 4V2 >= 4V1 + 4V2 = mark(and(isNat(V1),isNatList(V2))) active(length(cons(N,L))) = 20L + 30N >= 16L + 4N = mark(U11(and(isNatList(L),isNat(N)),L)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 24X1 + 16X2 >= 24X1 + 4X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 4X1 + 12X2 >= 4X1 + 3X2 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(s(X)) = 16X >= 16X = active(s(mark(X))) mark(length(X)) = 20X >= 20X = active(length(mark(X))) mark(and(X1,X2)) = 4X1 + 4X2 >= 4X1 + X2 = active(and(mark(X1),X2)) mark(isNat(X)) = 4X >= X = active(isNat(X)) mark(isNatList(X)) = 4X >= X = active(isNatList(X)) mark(isNatIList(X)) = 4X + 4 >= X + 1 = active(isNatIList(X)) cons(mark(X1),X2) = 24X1 + 4X2 >= 6X1 + 4X2 = cons(X1,X2) cons(active(X1),X2) = 6X1 + 4X2 >= 6X1 + 4X2 = cons(X1,X2) U11(mark(X1),X2) = 4X1 + 3X2 >= X1 + 3X2 = U11(X1,X2) U11(X1,mark(X2)) = X1 + 12X2 >= X1 + 3X2 = U11(X1,X2) U11(active(X1),X2) = X1 + 3X2 >= X1 + 3X2 = U11(X1,X2) U11(X1,active(X2)) = X1 + 3X2 >= X1 + 3X2 = U11(X1,X2) s(mark(X)) = 16X >= 4X = s(X) s(active(X)) = 4X >= 4X = s(X) length(mark(X)) = 20X >= 5X = length(X) length(active(X)) = 5X >= 5X = length(X) and(mark(X1),X2) = 4X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) isNat(mark(X)) = 4X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) problem: active(zeros()) -> mark(cons(0(),zeros())) active(isNat(0())) -> mark(tt()) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) and(mark(X1),X2) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Matrix Interpretation Processor: dim=1 interpretation: [isNatList](x0) = 2x0, [isNat](x0) = x0, [and](x0, x1) = x0 + x1, [s](x0) = 4x0 + 4, [length](x0) = 4x0 + 4, [U11](x0, x1) = x0 + 2x1, [tt] = 0, [mark](x0) = 4x0, [cons](x0, x1) = 2x0 + 4x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(V1))) = 4V1 + 4 >= 4V1 = mark(isNat(V1)) active(isNatList(cons(V1,V2))) = 4V1 + 8V2 >= 4V1 + 8V2 = mark(and(isNat(V1),isNatList(V2))) active(length(cons(N,L))) = 16L + 8N + 4 >= 16L + 4N = mark(U11(and(isNatList(L),isNat(N)),L)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 8X1 + 16X2 >= 8X1 + 4X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 4X1 + 8X2 >= 4X1 + 2X2 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(s(X)) = 16X + 16 >= 16X + 4 = active(s(mark(X))) mark(length(X)) = 16X + 16 >= 16X + 4 = active(length(mark(X))) mark(and(X1,X2)) = 4X1 + 4X2 >= 4X1 + X2 = active(and(mark(X1),X2)) mark(isNat(X)) = 4X >= X = active(isNat(X)) mark(isNatList(X)) = 8X >= 2X = active(isNatList(X)) cons(mark(X1),X2) = 8X1 + 4X2 >= 2X1 + 4X2 = cons(X1,X2) cons(active(X1),X2) = 2X1 + 4X2 >= 2X1 + 4X2 = cons(X1,X2) U11(mark(X1),X2) = 4X1 + 2X2 >= X1 + 2X2 = U11(X1,X2) U11(X1,mark(X2)) = X1 + 8X2 >= X1 + 2X2 = U11(X1,X2) U11(active(X1),X2) = X1 + 2X2 >= X1 + 2X2 = U11(X1,X2) U11(X1,active(X2)) = X1 + 2X2 >= X1 + 2X2 = U11(X1,X2) s(mark(X)) = 16X + 4 >= 4X + 4 = s(X) s(active(X)) = 4X + 4 >= 4X + 4 = s(X) length(mark(X)) = 16X + 4 >= 4X + 4 = length(X) length(active(X)) = 4X + 4 >= 4X + 4 = length(X) and(mark(X1),X2) = 4X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) isNat(mark(X)) = 4X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) problem: active(zeros()) -> mark(cons(0(),zeros())) active(isNat(0())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) and(mark(X1),X2) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [isNatList](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [1] [isNat](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [1 0 0] [and](x0, x1) = [0 1 1]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1] [s](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [0] [length](x0) = [0 1 1]x0 + [1] [0 0 0] [0], [1 0 0] [1 0 0] [0] [U11](x0, x1) = [0 1 1]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [0], [0] [tt] = [0] [0], [1 1 1] [mark](x0) = [0 0 1]x0 [0 1 0] , [1 0 0] [1 0 0] [cons](x0, x1) = [0 1 1]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [0] [0] = [0] [0], [1 0 1] [active](x0) = [0 0 1]x0 [0 1 0] , [0] [zeros] = [0] [1] orientation: [1] [0] active(zeros()) = [1] >= [0] = mark(cons(0(),zeros())) [0] [0] [1] [0] active(isNat(0())) = [0] >= [0] = mark(tt()) [0] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] active(isNatList(cons(V1,V2))) = [0 0 0]V1 + [0 0 0]V2 + [1] >= [0 0 0]V1 + [0 0 0]V2 + [0] = mark(and(isNat(V1),isNatList(V2))) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1] [1] mark(zeros()) = [1] >= [1] = active(zeros()) [0] [0] [1 1 1] [1 1 0] [1 1 1] [1 0 0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(cons(mark(X1),X2)) [0 1 1] [0 1 0] [0 1 1] [0 1 0] [0] [0] mark(0()) = [0] >= [0] = active(0()) [0] [0] [1 1 1] [1 0 0] [1] [1 1 1] [1 0 0] [0] mark(U11(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = active(U11(mark(X1),X2)) [0 1 1] [0 0 0] [1] [0 1 1] [0 0 0] [1] [0] [0] mark(tt()) = [0] >= [0] = active(tt()) [0] [0] [1 1 1] [1 0 0] [1 1 1] [1 0 0] mark(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(and(mark(X1),X2)) [0 1 1] [0 0 0] [0 1 1] [0 0 0] [1 0 0] [1] [1 0 0] [1] mark(isNat(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(isNat(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] mark(isNatList(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(isNatList(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1 0 0] [1 0 0] [1 0 0] cons(mark(X1),X2) = [0 1 1]X1 + [0 1 0]X2 >= [0 1 1]X1 + [0 1 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1 0 0] [1 0 0] cons(active(X1),X2) = [0 1 1]X1 + [0 1 0]X2 >= [0 1 1]X1 + [0 1 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 0 0] [0] [1 0 0] [1 0 0] [0] U11(mark(X1),X2) = [0 1 1]X1 + [0 0 0]X2 + [1] >= [0 1 1]X1 + [0 0 0]X2 + [1] = U11(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 1 1] [0] [1 0 0] [1 0 0] [0] U11(X1,mark(X2)) = [0 1 1]X1 + [0 0 0]X2 + [1] >= [0 1 1]X1 + [0 0 0]X2 + [1] = U11(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 1] [1 0 0] [0] [1 0 0] [1 0 0] [0] U11(active(X1),X2) = [0 1 1]X1 + [0 0 0]X2 + [1] >= [0 1 1]X1 + [0 0 0]X2 + [1] = U11(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 0] [0] U11(X1,active(X2)) = [0 1 1]X1 + [0 0 0]X2 + [1] >= [0 1 1]X1 + [0 0 0]X2 + [1] = U11(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 1] [1] [1 0 0] [1] s(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(X) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1] [1 0 0] [1] s(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(X) [0 0 0] [0] [0 0 0] [0] [1 1 1] [0] [1 0 0] [0] length(mark(X)) = [0 1 1]X + [1] >= [0 1 1]X + [1] = length(X) [0 0 0] [0] [0 0 0] [0] [1 0 1] [0] [1 0 0] [0] length(active(X)) = [0 1 1]X + [1] >= [0 1 1]X + [1] = length(X) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1 0 0] [1 0 0] [1 0 0] and(mark(X1),X2) = [0 1 1]X1 + [0 0 0]X2 >= [0 1 1]X1 + [0 0 0]X2 = and(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1 0 0] [1 0 0] and(active(X1),X2) = [0 1 1]X1 + [0 0 0]X2 >= [0 1 1]X1 + [0 0 0]X2 = and(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1] [1 0 0] [1] isNat(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = isNat(X) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1] [1 0 0] [1] isNat(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = isNat(X) [0 0 0] [0] [0 0 0] [0] problem: active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(tt()) -> active(tt()) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) and(mark(X1),X2) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [isNatList](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [isNat](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 0] [1 0 0] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [length](x0) = [0 0 0]x0 [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] [tt] = [0] [0], [1 0 0] [mark](x0) = [0 0 0]x0 [0 1 0] , [1 0 0] [1 0 0] [0] [cons](x0, x1) = [0 0 0]x0 + [1 0 0]x1 + [0] [0 0 0] [0 1 1] [1], [1] [0] = [1] [0], [1 0 0] [active](x0) = [0 0 0]x0 [0 1 0] , [0] [zeros] = [0] [0] orientation: [1 0 0] [1 1 1] [1] [1 0 0] [1 0 1] active(isNatList(cons(V1,V2))) = [0 0 0]V1 + [0 0 0]V2 + [0] >= [0 0 0]V1 + [0 0 0]V2 = mark(and(isNat(V1),isNatList(V2))) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] mark(zeros()) = [0] >= [0] = active(zeros()) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(cons(mark(X1),X2)) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1] [1] mark(0()) = [0] >= [0] = active(0()) [1] [1] [0] [0] mark(tt()) = [0] >= [0] = active(tt()) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] mark(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(and(mark(X1),X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] mark(isNat(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(isNat(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1 0 1] mark(isNatList(X)) = [0 0 0]X >= [0 0 0]X = active(isNatList(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] cons(mark(X1),X2) = [0 0 0]X1 + [1 0 0]X2 + [0] >= [0 0 0]X1 + [1 0 0]X2 + [0] = cons(X1,X2) [0 0 0] [0 1 1] [1] [0 0 0] [0 1 1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] cons(active(X1),X2) = [0 0 0]X1 + [1 0 0]X2 + [0] >= [0 0 0]X1 + [1 0 0]X2 + [0] = cons(X1,X2) [0 0 0] [0 1 1] [1] [0 0 0] [0 1 1] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] U11(mark(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 0 0] [1 0 0] [1 0 0] [1 0 0] U11(X1,mark(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 0 0] [1 0 0] [1 0 0] [1 0 0] U11(active(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 0 0] [1 0 0] [1 0 0] [1 0 0] U11(X1,active(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 0 0] [1 0 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] s(active(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] length(mark(X)) = [0 0 0]X >= [0 0 0]X = length(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] length(active(X)) = [0 0 0]X >= [0 0 0]X = length(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] and(mark(X1),X2) = [0 0 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 0 0] [1 0 0] [1 0 0] and(active(X1),X2) = [0 0 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] [0] [1 0 0] [0] isNat(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = isNat(X) [0 0 0] [0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] isNat(active(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = isNat(X) [0 0 0] [0] [0 0 0] [0] problem: mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(tt()) -> active(tt()) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) and(mark(X1),X2) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Matrix Interpretation Processor: dim=1 interpretation: [isNatList](x0) = 4x0 + 1, [isNat](x0) = 2x0 + 4, [and](x0, x1) = 2x0 + x1 + 4, [s](x0) = 4x0 + 5, [length](x0) = x0 + 4, [U11](x0, x1) = x0 + 4x1, [tt] = 1, [mark](x0) = 2x0, [cons](x0, x1) = 2x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: mark(zeros()) = 2 >= 1 = active(zeros()) mark(cons(X1,X2)) = 4X1 + 2X2 >= 4X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(tt()) = 2 >= 1 = active(tt()) mark(and(X1,X2)) = 4X1 + 2X2 + 8 >= 4X1 + X2 + 4 = active(and(mark(X1),X2)) mark(isNat(X)) = 4X + 8 >= 2X + 4 = active(isNat(X)) mark(isNatList(X)) = 8X + 2 >= 4X + 1 = active(isNatList(X)) cons(mark(X1),X2) = 4X1 + X2 >= 2X1 + X2 = cons(X1,X2) cons(active(X1),X2) = 2X1 + X2 >= 2X1 + X2 = cons(X1,X2) U11(mark(X1),X2) = 2X1 + 4X2 >= X1 + 4X2 = U11(X1,X2) U11(X1,mark(X2)) = X1 + 8X2 >= X1 + 4X2 = U11(X1,X2) U11(active(X1),X2) = X1 + 4X2 >= X1 + 4X2 = U11(X1,X2) U11(X1,active(X2)) = X1 + 4X2 >= X1 + 4X2 = U11(X1,X2) s(mark(X)) = 8X + 5 >= 4X + 5 = s(X) s(active(X)) = 4X + 5 >= 4X + 5 = s(X) length(mark(X)) = 2X + 4 >= X + 4 = length(X) length(active(X)) = X + 4 >= X + 4 = length(X) and(mark(X1),X2) = 4X1 + X2 + 4 >= 2X1 + X2 + 4 = and(X1,X2) and(active(X1),X2) = 2X1 + X2 + 4 >= 2X1 + X2 + 4 = and(X1,X2) isNat(mark(X)) = 4X + 4 >= 2X + 4 = isNat(X) isNat(active(X)) = 2X + 4 >= 2X + 4 = isNat(X) problem: mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) cons(mark(X1),X2) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) and(mark(X1),X2) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) Matrix Interpretation Processor: dim=1 interpretation: [isNat](x0) = x0 + 1, [and](x0, x1) = 2x0 + 2x1 + 2, [s](x0) = 5x0, [length](x0) = 5x0, [U11](x0, x1) = 4x0 + x1 + 2, [mark](x0) = 4x0, [cons](x0, x1) = 4x0 + 4x1 + 4, [0] = 4, [active](x0) = x0 + 5 orientation: mark(cons(X1,X2)) = 16X1 + 16X2 + 16 >= 16X1 + 4X2 + 9 = active(cons(mark(X1),X2)) mark(0()) = 16 >= 9 = active(0()) cons(mark(X1),X2) = 16X1 + 4X2 + 4 >= 4X1 + 4X2 + 4 = cons(X1,X2) cons(active(X1),X2) = 4X1 + 4X2 + 24 >= 4X1 + 4X2 + 4 = cons(X1,X2) U11(mark(X1),X2) = 16X1 + X2 + 2 >= 4X1 + X2 + 2 = U11(X1,X2) U11(X1,mark(X2)) = 4X1 + 4X2 + 2 >= 4X1 + X2 + 2 = U11(X1,X2) U11(active(X1),X2) = 4X1 + X2 + 22 >= 4X1 + X2 + 2 = U11(X1,X2) U11(X1,active(X2)) = 4X1 + X2 + 7 >= 4X1 + X2 + 2 = U11(X1,X2) s(mark(X)) = 20X >= 5X = s(X) s(active(X)) = 5X + 25 >= 5X = s(X) length(mark(X)) = 20X >= 5X = length(X) length(active(X)) = 5X + 25 >= 5X = length(X) and(mark(X1),X2) = 8X1 + 2X2 + 2 >= 2X1 + 2X2 + 2 = and(X1,X2) and(active(X1),X2) = 2X1 + 2X2 + 12 >= 2X1 + 2X2 + 2 = and(X1,X2) isNat(mark(X)) = 4X + 1 >= X + 1 = isNat(X) isNat(active(X)) = X + 6 >= X + 1 = isNat(X) problem: cons(mark(X1),X2) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) length(mark(X)) -> length(X) and(mark(X1),X2) -> and(X1,X2) isNat(mark(X)) -> isNat(X) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [isNat](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [1 0 0] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 1] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [length](x0) = [1 0 0]x0 [0 0 0] , [1 1 0] [1 0 1] [0] [U11](x0, x1) = [1 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [0], [0] [mark](x0) = x0 + [1] [1], [1 0 1] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] orientation: [1 0 1] [1 0 0] [1] [1 0 1] [1 0 0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [1 1 0] [1 0 1] [1] [1 1 0] [1 0 1] [0] U11(mark(X1),X2) = [1 0 0]X1 + [0 0 0]X2 + [1] >= [1 0 0]X1 + [0 0 0]X2 + [1] = U11(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 0 1] [1] [1 1 0] [1 0 1] [0] U11(X1,mark(X2)) = [1 0 0]X1 + [0 0 0]X2 + [1] >= [1 0 0]X1 + [0 0 0]X2 + [1] = U11(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 1] [1] [1 0 1] s(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X = s(X) [0 0 0] [0] [0 0 0] [1 0 1] [1] [1 0 1] length(mark(X)) = [1 0 0]X + [0] >= [1 0 0]X = length(X) [0 0 0] [0] [0 0 0] [1 0 1] [1 0 0] [1] [1 0 1] [1 0 0] and(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = and(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [1 1 0] [1] [1 1 0] isNat(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X = isNat(X) [0 0 0] [0] [0 0 0] problem: Qed