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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = x0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 0, [isNatIList](x0) = 2x0 + 2, [isNatList](x0) = 2x0, [isNat](x0) = 4x0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = 4x0, [U11](x0, x1) = x0 + 4x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = 4x0 + 2x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = 4L >= 4L = mark(s(length(L))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 16V1 >= 2V1 = mark(isNatList(V1)) active(isNat(s(V1))) = 4V1 >= 4V1 = mark(isNat(V1)) active(isNatIList(V)) = 2V + 2 >= 2V = mark(isNatList(V)) active(isNatIList(zeros())) = 2 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 8V1 + 4V2 + 2 >= 4V1 + 2V2 + 2 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 8V1 + 4V2 >= 4V1 + 2V2 = mark(and(isNat(V1),isNatList(V2))) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 8L + 16N >= 6L + 4N = mark(U11(and(isNatList(L),isNat(N)),L)) active(cons(X1,X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = U11(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = 4X >= 4X = length(active(X)) active(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(active(X1),X2) cons(mark(X1),X2) = 4X1 + 2X2 >= 4X1 + 2X2 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + 4X2 >= X1 + 4X2 = mark(U11(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = 4X >= 4X = mark(length(X)) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = 4X >= 4X = length(proper(X)) proper(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(proper(X1),proper(X2)) proper(isNat(X)) = 4X >= 4X = isNat(proper(X)) proper(isNatList(X)) = 2X >= 2X = isNatList(proper(X)) proper(isNatIList(X)) = 2X + 2 >= 2X + 2 = isNatIList(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 4X2 >= X1 + 4X2 = ok(U11(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = 4X >= 4X = ok(length(X)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNat(ok(X)) = 4X >= 4X = ok(isNat(X)) isNatList(ok(X)) = 2X >= 2X = ok(isNatList(X)) isNatIList(ok(X)) = 2X + 2 >= 2X + 2 = ok(isNatIList(X)) top(mark(X)) = X >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(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(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = 2x0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 1, [isNatIList](x0) = 4x0, [isNatList](x0) = 5x0, [isNat](x0) = x0, [and](x0, x1) = 2x0 + 5x1, [s](x0) = x0, [length](x0) = 5x0, [U11](x0, x1) = 2x0 + 5x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = 2x0 + 5x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = 5L >= 5L = mark(s(length(L))) active(and(tt(),X)) = 5X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 5V1 >= 5V1 = mark(isNatList(V1)) active(isNat(s(V1))) = V1 >= V1 = mark(isNat(V1)) active(isNatIList(cons(V1,V2))) = 8V1 + 20V2 >= 2V1 + 20V2 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 5 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 10V1 + 25V2 >= 2V1 + 25V2 = mark(and(isNat(V1),isNatList(V2))) active(length(nil())) = 5 >= 0 = mark(0()) active(length(cons(N,L))) = 25L + 10N >= 25L + 10N = mark(U11(and(isNatList(L),isNat(N)),L)) active(cons(X1,X2)) = 2X1 + 5X2 >= 2X1 + 5X2 = cons(active(X1),X2) active(U11(X1,X2)) = 2X1 + 5X2 >= 2X1 + 5X2 = U11(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = 5X >= 5X = length(active(X)) active(and(X1,X2)) = 2X1 + 5X2 >= 2X1 + 5X2 = and(active(X1),X2) cons(mark(X1),X2) = 2X1 + 5X2 >= 2X1 + 5X2 = mark(cons(X1,X2)) U11(mark(X1),X2) = 2X1 + 5X2 >= 2X1 + 5X2 = mark(U11(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = 5X >= 5X = mark(length(X)) and(mark(X1),X2) = 2X1 + 5X2 >= 2X1 + 5X2 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 2X1 + 5X2 >= 2X1 + 5X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 2X1 + 5X2 >= 2X1 + 5X2 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = 5X >= 5X = length(proper(X)) proper(and(X1,X2)) = 2X1 + 5X2 >= 2X1 + 5X2 = and(proper(X1),proper(X2)) proper(isNat(X)) = X >= X = isNat(proper(X)) proper(isNatList(X)) = 5X >= 5X = isNatList(proper(X)) proper(isNatIList(X)) = 4X >= 4X = isNatIList(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = 2X1 + 5X2 >= 2X1 + 5X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 2X1 + 5X2 >= 2X1 + 5X2 = ok(U11(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = 5X >= 5X = ok(length(X)) and(ok(X1),ok(X2)) = 2X1 + 5X2 >= 2X1 + 5X2 = ok(and(X1,X2)) isNat(ok(X)) = X >= X = ok(isNat(X)) isNatList(ok(X)) = 5X >= 5X = ok(isNatList(X)) isNatIList(ok(X)) = 4X >= 4X = ok(isNatIList(X)) top(mark(X)) = 2X >= 2X = top(proper(X)) top(ok(X)) = 2X >= 2X = top(active(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(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = x0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 1, [isNatIList](x0) = x0, [isNatList](x0) = x0, [isNat](x0) = x0, [and](x0, x1) = 2x0 + x1, [s](x0) = x0, [length](x0) = 4x0 + 1, [U11](x0, x1) = 2x0 + 4x1 + 1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = 5x0 + 2x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = 4L + 1 >= 4L + 1 = mark(s(length(L))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 4V1 + 1 >= V1 = mark(isNatList(V1)) active(isNat(s(V1))) = V1 >= V1 = mark(isNat(V1)) active(isNatIList(cons(V1,V2))) = 5V1 + 2V2 >= 2V1 + V2 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(cons(V1,V2))) = 5V1 + 2V2 >= 2V1 + V2 = mark(and(isNat(V1),isNatList(V2))) active(length(cons(N,L))) = 8L + 20N + 1 >= 8L + 2N + 1 = mark(U11(and(isNatList(L),isNat(N)),L)) active(cons(X1,X2)) = 5X1 + 2X2 >= 5X1 + 2X2 = cons(active(X1),X2) active(U11(X1,X2)) = 2X1 + 4X2 + 1 >= 2X1 + 4X2 + 1 = U11(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = 4X + 1 >= 4X + 1 = length(active(X)) active(and(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = and(active(X1),X2) cons(mark(X1),X2) = 5X1 + 2X2 >= 5X1 + 2X2 = mark(cons(X1,X2)) U11(mark(X1),X2) = 2X1 + 4X2 + 1 >= 2X1 + 4X2 + 1 = mark(U11(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = 4X + 1 >= 4X + 1 = mark(length(X)) and(mark(X1),X2) = 2X1 + X2 >= 2X1 + X2 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 5X1 + 2X2 >= 5X1 + 2X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 2X1 + 4X2 + 1 >= 2X1 + 4X2 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = 4X + 1 >= 4X + 1 = length(proper(X)) proper(and(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = and(proper(X1),proper(X2)) proper(isNat(X)) = X >= X = isNat(proper(X)) proper(isNatList(X)) = X >= X = isNatList(proper(X)) proper(isNatIList(X)) = X >= X = isNatIList(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = 5X1 + 2X2 >= 5X1 + 2X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 2X1 + 4X2 + 1 >= 2X1 + 4X2 + 1 = ok(U11(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = 4X + 1 >= 4X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = 2X1 + X2 >= 2X1 + X2 = ok(and(X1,X2)) isNat(ok(X)) = X >= X = ok(isNat(X)) isNatList(ok(X)) = X >= X = ok(isNatList(X)) isNatIList(ok(X)) = X >= X = ok(isNatIList(X)) top(mark(X)) = X >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [top](x0) = [0 0 1]x0 + [1] [0 0 1] [0], [ok](x0) = x0 , [proper](x0) = x0 , [0] [nil] = [1] [0], [1 1 0] [0] [isNatIList](x0) = [0 0 0]x0 + [0] [1 0 0] [1], [1 0 0] [isNatList](x0) = [0 0 0]x0 [1 1 0] , [1 0 0] [isNat](x0) = [0 0 0]x0 [1 0 0] , [1 0 0] [1 0 0] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [0 0 1] , [1 0 1] [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] [mark](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [1 1 1] [0] [cons](x0, x1) = [0 1 1]x0 + [0 1 0]x1 + [1] [0 0 0] [1 0 0] [0], [0] [0] = [1] [1], [active](x0) = x0 , [0] [zeros] = [0] [0] orientation: [0] [0] active(zeros()) = [0] >= [0] = mark(cons(0(),zeros())) [0] [0] [1 0 1] [1 0 1] active(U11(tt(),L)) = [0 0 0]L >= [0 0 0]L = mark(s(length(L))) [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(and(tt(),X)) = [0 0 0]X >= [0 0 0]X = mark(X) [0 0 1] [0 0 1] [0] [0] active(isNat(0())) = [0] >= [0] = mark(tt()) [0] [0] [1 0 1] [1 0 0] active(isNat(s(V1))) = [0 0 0]V1 >= [0 0 0]V1 = mark(isNat(V1)) [1 0 1] [1 0 0] [1 1 1] [1 2 1] [1] [1 0 0] [1 1 0] [0] 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))) [1 0 0] [1 1 1] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1 1 1] [0] [1 0 0] [1 0 0] 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))) [1 1 1] [1 2 1] [1] [1 0 0] [1 1 0] [2 1 1] [1 0 0] [2 0 1] [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] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] [0] active(cons(X1,X2)) = [0 1 1]X1 + [0 1 0]X2 + [1] >= [0 1 1]X1 + [0 1 0]X2 + [1] = cons(active(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] active(U11(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U11(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] active(s(X)) = [0 0 0]X >= [0 0 0]X = s(active(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] active(length(X)) = [0 0 0]X >= [0 0 0]X = length(active(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = and(active(X1),X2) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] cons(mark(X1),X2) = [0 0 1]X1 + [0 1 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 = mark(cons(X1,X2)) [0 0 0] [1 0 0] [0] [0 0 0] [1 0 0] [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 = mark(U11(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] s(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(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 = mark(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 = mark(and(X1,X2)) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0] [0] proper(zeros()) = [0] >= [0] = ok(zeros()) [0] [0] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] [0] proper(cons(X1,X2)) = [0 1 1]X1 + [0 1 0]X2 + [1] >= [0 1 1]X1 + [0 1 0]X2 + [1] = cons(proper(X1),proper(X2)) [0 0 0] [1 0 0] [0] [0 0 0] [1 0 0] [0] [0] [0] proper(0()) = [1] >= [1] = ok(0()) [1] [1] [1 0 0] [1 0 1] [1 0 0] [1 0 1] proper(U11(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U11(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] proper(tt()) = [0] >= [0] = ok(tt()) [0] [0] [1 0 1] [1 0 1] proper(s(X)) = [0 0 0]X >= [0 0 0]X = s(proper(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] proper(length(X)) = [0 0 0]X >= [0 0 0]X = length(proper(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = and(proper(X1),proper(X2)) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1 0 0] [1 0 0] proper(isNat(X)) = [0 0 0]X >= [0 0 0]X = isNat(proper(X)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(isNatList(X)) = [0 0 0]X >= [0 0 0]X = isNatList(proper(X)) [1 1 0] [1 1 0] [1 1 0] [0] [1 1 0] [0] proper(isNatIList(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = isNatIList(proper(X)) [1 0 0] [1] [1 0 0] [1] [0] [0] proper(nil()) = [1] >= [1] = ok(nil()) [0] [0] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] [0] cons(ok(X1),ok(X2)) = [0 1 1]X1 + [0 1 0]X2 + [1] >= [0 1 1]X1 + [0 1 0]X2 + [1] = ok(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(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(U11(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] s(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(s(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] length(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(length(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] and(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(and(X1,X2)) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1 0 0] [1 0 0] isNat(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(isNat(X)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] isNatList(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(isNatList(X)) [1 1 0] [1 1 0] [1 1 0] [0] [1 1 0] [0] isNatIList(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(isNatIList(X)) [1 0 0] [1] [1 0 0] [1] [1 0 0] [0] [1 0 0] [0] top(mark(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = top(proper(X)) [0 0 1] [0] [0 0 1] [0] [1 0 0] [0] [1 0 0] [0] top(ok(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = top(active(X)) [0 0 1] [0] [0 0 1] [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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [top](x0) = [1 0 1]x0 + [1] [0 0 0] [1], [ok](x0) = x0 , [proper](x0) = x0 , [0] [nil] = [0] [0], [1 0 0] [0] [isNatIList](x0) = [1 1 0]x0 + [0] [1 0 0] [1], [1 1 0] [isNatList](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [isNat](x0) = [0 0 0]x0 [0 1 0] , [1 0 1] [1 0 0] [and](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 0 0] [0 0 1] , [1 0 0] [s](x0) = [0 1 0]x0 [0 0 0] , [1 0 1] [0] [length](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [1 0 1] [0] [U11](x0, x1) = [0 1 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [0] [tt] = [0] [1], [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [1 1 0] [cons](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 0 0] [1 0 1] , [0] [0] = [1] [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] [0] [1 0 1] active(U11(tt(),L)) = [0 0 0]L + [0] >= [0 0 0]L = mark(s(length(L))) [0 0 0] [1] [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] [0] [0 0 1] [0] [0] active(isNat(0())) = [0] >= [0] = mark(tt()) [1] [1] [1 0 0] [1 0 0] active(isNat(s(V1))) = [0 0 0]V1 >= [0 0 0]V1 = mark(isNat(V1)) [0 1 0] [0 1 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] 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 1] [1 0 0] [0] [2 1 1] [1 0 0] [0] 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] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 1 0] [1 0 0] [1 1 0] active(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(active(X1),X2) [0 0 0] [1 0 1] [0 0 0] [1 0 1] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 1] [0] active(U11(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = U11(active(X1),X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 0 0] active(s(X)) = [0 0 0]X >= [0 0 0]X = s(active(X)) [0 0 0] [0 0 0] [1 0 1] [0] [1 0 1] [0] active(length(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length(active(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1 0 0] [1 0 1] [1 0 0] active(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = and(active(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] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(cons(X1,X2)) [0 0 0] [1 0 1] [0 0 0] [1 0 1] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 1] [0] U11(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = mark(U11(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 0 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(s(X)) [0 0 0] [0 0 0] [1 0 1] [0] [1 0 1] [0] length(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(length(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1 0 0] [1 0 1] [1 0 0] and(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(and(X1,X2)) [0 0 0] [0 0 1] [0 0 0] [0 0 1] [0] [0] proper(zeros()) = [0] >= [0] = ok(zeros()) [0] [0] [1 0 0] [1 1 0] [1 0 0] [1 1 0] proper(cons(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = cons(proper(X1),proper(X2)) [0 0 0] [1 0 1] [0 0 0] [1 0 1] [0] [0] proper(0()) = [1] >= [1] = ok(0()) [0] [0] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 1] [0] proper(U11(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 + [0] >= [0 1 0]X1 + [0 0 0]X2 + [0] = U11(proper(X1),proper(X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [0] [0] proper(tt()) = [0] >= [0] = ok(tt()) [1] [1] [1 0 0] [1 0 0] proper(s(X)) = [0 1 0]X >= [0 1 0]X = s(proper(X)) [0 0 0] [0 0 0] [1 0 1] [0] [1 0 1] [0] proper(length(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1 0 0] [1 0 1] [1 0 0] proper(and(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = and(proper(X1),proper(X2)) [0 0 0] [0 0 1] [0 0 0] [0 0 1] [1 0 0] [1 0 0] proper(isNat(X)) = [0 0 0]X >= [0 0 0]X = isNat(proper(X)) [0 1 0] [0 1 0] [1 1 0] [1 1 0] proper(isNatList(X)) = [0 1 0]X >= [0 1 0]X = isNatList(proper(X)) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] proper(isNatIList(X)) = [1 1 0]X + [0] >= [1 1 0]X + [0] = isNatIList(proper(X)) [1 0 0] [1] [1 0 0] [1] [0] [0] proper(nil()) = [0] >= [0] = ok(nil()) [0] [0] [1 0 0] [1 1 0] [1 0 0] [1 1 0] cons(ok(X1),ok(X2)) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = ok(cons(X1,X2)) [0 0 0] [1 0 1] [0 0 0] [1 0 1] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 1] [0] U11(ok(X1),ok(X2)) = [0 1 0]X1 + [0 0 0]X2 + [0] >= [0 1 0]X1 + [0 0 0]X2 + [0] = ok(U11(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 0 0] s(ok(X)) = [0 1 0]X >= [0 1 0]X = ok(s(X)) [0 0 0] [0 0 0] [1 0 1] [0] [1 0 1] [0] length(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(length(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1 0 0] [1 0 1] [1 0 0] and(ok(X1),ok(X2)) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = ok(and(X1,X2)) [0 0 0] [0 0 1] [0 0 0] [0 0 1] [1 0 0] [1 0 0] isNat(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(isNat(X)) [0 1 0] [0 1 0] [1 1 0] [1 1 0] isNatList(ok(X)) = [0 1 0]X >= [0 1 0]X = ok(isNatList(X)) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] isNatIList(ok(X)) = [1 1 0]X + [0] >= [1 1 0]X + [0] = ok(isNatIList(X)) [1 0 0] [1] [1 0 0] [1] [1 0 0] [0] [1 0 0] [0] top(mark(X)) = [1 0 1]X + [1] >= [1 0 1]X + [1] = top(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] top(ok(X)) = [1 0 1]X + [1] >= [1 0 1]X + [1] = top(active(X)) [0 0 0] [1] [0 0 0] [1] 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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [top](x0) = [0 0 1]x0 + [0] [0 0 0] [1], [1 0 0] [ok](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [proper](x0) = [0 0 0]x0 [0 0 1] , [0] [nil] = [0] [1], [1 0 0] [0] [isNatIList](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 1] [isNatList](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [0] [isNat](x0) = [0 0 0]x0 + [0] [1 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] [0] [s](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 1] [1] [length](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 1] [1 0 1] [1] [U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 1] [0 0 0] [0], [0] [tt] = [0] [1], [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 1] , [1 0 1] [cons](x0, x1) = x0 + [0 0 0]x1 [1 0 1] , [0] [0] = [1] [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] [2] [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] [1] [0] [0] active(isNat(0())) = [0] >= [0] = mark(tt()) [1] [1] [1 0 1] [1] [1 0 1] [0] active(isNat(s(V1))) = [0 0 0]V1 + [0] >= [0 0 0]V1 + [0] = mark(isNat(V1)) [1 0 0] [1] [1 0 0] [1] [1 0 1] [2 0 2] [1 0 1] [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 0 2] [1 0 1] [1] [2 0 2] [1 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 0 0] [1 0 1] [1 0 0] [1 0 1] active(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(active(X1),X2) [0 0 1] [1 0 1] [0 0 1] [1 0 1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] active(U11(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = U11(active(X1),X2) [0 0 1] [0 0 0] [0] [0 0 1] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] active(s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(active(X)) [0 0 1] [1] [0 0 1] [1] [1 0 1] [1] [1 0 1] [1] active(length(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length(active(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = and(active(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] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(cons(X1,X2)) [0 0 1] [1 0 1] [0 0 1] [1 0 1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] U11(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = mark(U11(X1,X2)) [0 0 1] [0 0 0] [0] [0 0 1] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] s(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(s(X)) [0 0 1] [1] [0 0 1] [1] [1 0 1] [1] [1 0 1] [1] length(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(length(X)) [0 0 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 = mark(and(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] proper(zeros()) = [0] >= [0] = ok(zeros()) [0] [0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] proper(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(proper(X1),proper(X2)) [0 0 1] [1 0 1] [0 0 1] [1 0 1] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] proper(U11(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = U11(proper(X1),proper(X2)) [0 0 1] [0 0 0] [0] [0 0 1] [0 0 0] [0] [0] [0] proper(tt()) = [0] >= [0] = ok(tt()) [1] [1] [1 0 0] [0] [1 0 0] [0] proper(s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(proper(X)) [0 0 1] [1] [0 0 1] [1] [1 0 1] [1] [1 0 1] [1] proper(length(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = length(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = and(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [0] [1 0 1] [0] proper(isNat(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = isNat(proper(X)) [1 0 0] [1] [1 0 0] [1] [1 0 1] [1 0 1] proper(isNatList(X)) = [0 0 0]X >= [0 0 0]X = isNatList(proper(X)) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] proper(isNatIList(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = isNatIList(proper(X)) [0 0 0] [1] [0 0 0] [1] [0] [0] proper(nil()) = [0] >= [0] = ok(nil()) [1] [1] [1 0 0] [1 0 1] [1 0 0] [1 0 1] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(cons(X1,X2)) [0 0 1] [1 0 1] [0 0 1] [1 0 1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] U11(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = ok(U11(X1,X2)) [0 0 1] [0 0 0] [0] [0 0 1] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] s(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(s(X)) [0 0 1] [1] [0 0 1] [1] [1 0 1] [1] [1 0 1] [1] length(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(length(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] and(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(and(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [0] [1 0 1] [0] isNat(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(isNat(X)) [1 0 0] [1] [1 0 0] [1] [1 0 1] [1 0 1] isNatList(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(isNatList(X)) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] isNatIList(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(isNatIList(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] top(mark(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = top(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] top(ok(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = top(active(X)) [0 0 0] [1] [0 0 0] [1] problem: active(zeros()) -> mark(cons(0(),zeros())) active(isNat(0())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = 2x0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 0, [isNatIList](x0) = 4x0, [isNatList](x0) = 4x0, [isNat](x0) = x0, [and](x0, x1) = 2x0 + x1, [s](x0) = 2x0 + 4, [length](x0) = 5x0 + 2, [U11](x0, x1) = 2x0 + 4x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = 4x0 + 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(isNatList(cons(V1,V2))) = 16V1 + 16V2 >= 2V1 + 4V2 = mark(and(isNat(V1),isNatList(V2))) active(length(cons(N,L))) = 20L + 20N + 2 >= 20L + 2N = mark(U11(and(isNatList(L),isNat(N)),L)) active(cons(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = cons(active(X1),X2) active(U11(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = U11(active(X1),X2) active(s(X)) = 2X + 4 >= 2X + 4 = s(active(X)) active(length(X)) = 5X + 2 >= 5X + 2 = length(active(X)) active(and(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = and(active(X1),X2) cons(mark(X1),X2) = 4X1 + 4X2 >= 4X1 + 4X2 = mark(cons(X1,X2)) U11(mark(X1),X2) = 2X1 + 4X2 >= 2X1 + 4X2 = mark(U11(X1,X2)) s(mark(X)) = 2X + 4 >= 2X + 4 = mark(s(X)) length(mark(X)) = 5X + 2 >= 5X + 2 = mark(length(X)) and(mark(X1),X2) = 2X1 + X2 >= 2X1 + X2 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = 2X + 4 >= 2X + 4 = s(proper(X)) proper(length(X)) = 5X + 2 >= 5X + 2 = length(proper(X)) proper(and(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = and(proper(X1),proper(X2)) proper(isNat(X)) = X >= X = isNat(proper(X)) proper(isNatList(X)) = 4X >= 4X = isNatList(proper(X)) proper(isNatIList(X)) = 4X >= 4X = isNatIList(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = ok(U11(X1,X2)) s(ok(X)) = 2X + 4 >= 2X + 4 = ok(s(X)) length(ok(X)) = 5X + 2 >= 5X + 2 = ok(length(X)) and(ok(X1),ok(X2)) = 2X1 + X2 >= 2X1 + X2 = ok(and(X1,X2)) isNat(ok(X)) = X >= X = ok(isNat(X)) isNatList(ok(X)) = 4X >= 4X = ok(isNatList(X)) isNatIList(ok(X)) = 4X >= 4X = ok(isNatIList(X)) top(mark(X)) = 2X >= 2X = top(proper(X)) top(ok(X)) = 2X >= 2X = top(active(X)) problem: active(zeros()) -> mark(cons(0(),zeros())) active(isNat(0())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [top](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [ok](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [proper](x0) = [0 1 0]x0 [0 0 0] , [0] [nil] = [0] [0], [1 0 1] [isNatIList](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [1] [isNatList](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [isNat](x0) = [0 0 0]x0 [0 0 0] , [1 1 1] [and](x0, x1) = x0 + [0 0 0]x1 [0 0 0] , [s](x0) = x0 , [length](x0) = x0 , [1 0 0] [U11](x0, x1) = x0 + [0 0 0]x1 [0 0 0] , [0] [tt] = [0] [0], [1 0 0] [mark](x0) = [0 1 1]x0 [0 0 0] , [1 0 1] [cons](x0, x1) = x0 + [0 0 0]x1 [0 0 0] , [0] [0] = [0] [0], [1 1 0] [active](x0) = [0 0 1]x0 [1 1 0] , [0] [zeros] = [1] [0] orientation: [1] [0] active(zeros()) = [0] >= [0] = mark(cons(0(),zeros())) [1] [0] [0] [0] active(isNat(0())) = [0] >= [0] = mark(tt()) [0] [0] [1 0 0] [1 0 1] [1] [1 0 0] [1 0 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))) [1 0 0] [1 0 1] [1] [0 0 0] [0 0 0] [0] [1 1 0] [1 0 1] [1 1 0] [1 0 1] active(cons(X1,X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 1]X1 + [0 0 0]X2 = cons(active(X1),X2) [1 1 0] [1 0 1] [1 1 0] [0 0 0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] active(U11(X1,X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 1]X1 + [0 0 0]X2 = U11(active(X1),X2) [1 1 0] [1 0 0] [1 1 0] [0 0 0] [1 1 0] [1 1 0] active(s(X)) = [0 0 1]X >= [0 0 1]X = s(active(X)) [1 1 0] [1 1 0] [1 1 0] [1 1 0] active(length(X)) = [0 0 1]X >= [0 0 1]X = length(active(X)) [1 1 0] [1 1 0] [1 1 0] [1 1 1] [1 1 0] [1 1 1] active(and(X1,X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 1]X1 + [0 0 0]X2 = and(active(X1),X2) [1 1 0] [1 1 1] [1 1 0] [0 0 0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] cons(mark(X1),X2) = [0 1 1]X1 + [0 0 0]X2 >= [0 1 1]X1 + [0 0 0]X2 = mark(cons(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(mark(X1),X2) = [0 1 1]X1 + [0 0 0]X2 >= [0 1 1]X1 + [0 0 0]X2 = mark(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 1 1]X >= [0 1 1]X = mark(s(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] length(mark(X)) = [0 1 1]X >= [0 1 1]X = mark(length(X)) [0 0 0] [0 0 0] [1 0 0] [1 1 1] [1 0 0] [1 1 1] and(mark(X1),X2) = [0 1 1]X1 + [0 0 0]X2 >= [0 1 1]X1 + [0 0 0]X2 = mark(and(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] proper(zeros()) = [1] >= [1] = ok(zeros()) [0] [0] [1 0 0] [1 0 1] [1 0 0] [1 0 0] proper(cons(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(U11(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = U11(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] proper(tt()) = [0] >= [0] = ok(tt()) [0] [0] [1 0 0] [1 0 0] proper(s(X)) = [0 1 0]X >= [0 1 0]X = s(proper(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] proper(length(X)) = [0 1 0]X >= [0 1 0]X = length(proper(X)) [0 0 0] [0 0 0] [1 0 0] [1 1 1] [1 0 0] [1 1 0] proper(and(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = and(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] proper(isNat(X)) = [0 0 0]X >= [0 0 0]X = isNat(proper(X)) [0 0 0] [0 0 0] [1 0 0] [1] [1 0 0] [1] proper(isNatList(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = isNatList(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1 0 0] proper(isNatIList(X)) = [0 1 0]X >= [0 1 0]X = isNatIList(proper(X)) [0 0 0] [0 0 0] [0] [0] proper(nil()) = [0] >= [0] = ok(nil()) [0] [0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] cons(ok(X1),ok(X2)) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = ok(cons(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 0] U11(ok(X1),ok(X2)) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = ok(U11(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] s(ok(X)) = [0 1 0]X >= [0 1 0]X = ok(s(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] length(ok(X)) = [0 1 0]X >= [0 1 0]X = ok(length(X)) [0 0 0] [0 0 0] [1 0 1] [1 1 1] [1 0 1] [1 1 1] and(ok(X1),ok(X2)) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = ok(and(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 0] isNat(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(isNat(X)) [0 0 0] [0 0 0] [1 0 1] [1] [1 0 0] [1] isNatList(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(isNatList(X)) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1 0 1] isNatIList(ok(X)) = [0 1 0]X >= [0 1 0]X = ok(isNatIList(X)) [0 0 0] [0 0 0] [1 1 1] [1 1 0] top(mark(X)) = [0 0 0]X >= [0 0 0]X = top(proper(X)) [0 0 0] [0 0 0] [1 1 1] [1 1 1] top(ok(X)) = [0 0 0]X >= [0 0 0]X = top(active(X)) [0 0 0] [0 0 0] problem: active(isNat(0())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = x0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 1, [isNatIList](x0) = 2x0, [isNatList](x0) = x0, [isNat](x0) = x0, [and](x0, x1) = x0 + 2x1 + 2, [s](x0) = 2x0, [length](x0) = x0, [U11](x0, x1) = 4x0 + x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + 2x1 + 2, [0] = 4, [active](x0) = x0, [zeros] = 0 orientation: active(isNat(0())) = 4 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 + 2V2 + 2 >= V1 + 2V2 + 2 = mark(and(isNat(V1),isNatList(V2))) active(cons(X1,X2)) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = cons(active(X1),X2) active(U11(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = U11(active(X1),X2) active(s(X)) = 2X >= 2X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(and(X1,X2)) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = and(active(X1),X2) cons(mark(X1),X2) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = mark(cons(X1,X2)) U11(mark(X1),X2) = 4X1 + X2 >= 4X1 + X2 = mark(U11(X1,X2)) s(mark(X)) = 2X >= 2X = mark(s(X)) length(mark(X)) = X >= X = mark(length(X)) and(mark(X1),X2) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = cons(proper(X1),proper(X2)) proper(0()) = 4 >= 4 = ok(0()) proper(U11(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = 2X >= 2X = s(proper(X)) proper(length(X)) = X >= X = length(proper(X)) proper(and(X1,X2)) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = and(proper(X1),proper(X2)) proper(isNat(X)) = X >= X = isNat(proper(X)) proper(isNatList(X)) = X >= X = isNatList(proper(X)) proper(isNatIList(X)) = 2X >= 2X = isNatIList(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 4X1 + X2 >= 4X1 + X2 = ok(U11(X1,X2)) s(ok(X)) = 2X >= 2X = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) and(ok(X1),ok(X2)) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = ok(and(X1,X2)) isNat(ok(X)) = X >= X = ok(isNat(X)) isNatList(ok(X)) = X >= X = ok(isNatList(X)) isNatIList(ok(X)) = 2X >= 2X = ok(isNatIList(X)) top(mark(X)) = X >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(X)) problem: active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = x0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 0, [isNatIList](x0) = 4x0, [isNatList](x0) = x0, [isNat](x0) = x0, [and](x0, x1) = 4x0 + x1, [s](x0) = 4x0, [length](x0) = x0, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = 4x0 + 2x1 + 2, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: active(isNatList(cons(V1,V2))) = 4V1 + 2V2 + 2 >= 4V1 + V2 = mark(and(isNat(V1),isNatList(V2))) active(cons(X1,X2)) = 4X1 + 2X2 + 2 >= 4X1 + 2X2 + 2 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(active(X1),X2) active(s(X)) = 4X >= 4X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(and(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = and(active(X1),X2) cons(mark(X1),X2) = 4X1 + 2X2 + 2 >= 4X1 + 2X2 + 2 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(U11(X1,X2)) s(mark(X)) = 4X >= 4X = mark(s(X)) length(mark(X)) = X >= X = mark(length(X)) and(mark(X1),X2) = 4X1 + X2 >= 4X1 + X2 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 4X1 + 2X2 + 2 >= 4X1 + 2X2 + 2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(s(X)) = 4X >= 4X = s(proper(X)) proper(length(X)) = X >= X = length(proper(X)) proper(and(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = and(proper(X1),proper(X2)) proper(isNat(X)) = X >= X = isNat(proper(X)) proper(isNatList(X)) = X >= X = isNatList(proper(X)) proper(isNatIList(X)) = 4X >= 4X = isNatIList(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 4X1 + 2X2 + 2 >= 4X1 + 2X2 + 2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U11(X1,X2)) s(ok(X)) = 4X >= 4X = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) and(ok(X1),ok(X2)) = 4X1 + X2 >= 4X1 + X2 = ok(and(X1,X2)) isNat(ok(X)) = X >= X = ok(isNat(X)) isNatList(ok(X)) = X >= X = ok(isNatList(X)) isNatIList(ok(X)) = 4X >= 4X = ok(isNatIList(X)) top(mark(X)) = X >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(X)) problem: active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = 2x0 + 1, [ok](x0) = x0, [proper](x0) = x0, [nil] = 0, [isNatIList](x0) = x0, [isNatList](x0) = 2x0 + 4, [isNat](x0) = x0 + 4, [and](x0, x1) = 2x0 + 2x1 + 4, [s](x0) = 4x0 + 4, [length](x0) = 2x0, [U11](x0, x1) = 2x0 + x1 + 1, [tt] = 2, [mark](x0) = x0 + 1, [cons](x0, x1) = 4x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: active(cons(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = cons(active(X1),X2) active(U11(X1,X2)) = 2X1 + X2 + 1 >= 2X1 + X2 + 1 = U11(active(X1),X2) active(s(X)) = 4X + 4 >= 4X + 4 = s(active(X)) active(length(X)) = 2X >= 2X = length(active(X)) active(and(X1,X2)) = 2X1 + 2X2 + 4 >= 2X1 + 2X2 + 4 = and(active(X1),X2) cons(mark(X1),X2) = 4X1 + X2 + 4 >= 4X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = 2X1 + X2 + 3 >= 2X1 + X2 + 2 = mark(U11(X1,X2)) s(mark(X)) = 4X + 8 >= 4X + 5 = mark(s(X)) length(mark(X)) = 2X + 2 >= 2X + 1 = mark(length(X)) and(mark(X1),X2) = 2X1 + 2X2 + 6 >= 2X1 + 2X2 + 5 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 2X1 + X2 + 1 >= 2X1 + X2 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 2 >= 2 = ok(tt()) proper(s(X)) = 4X + 4 >= 4X + 4 = s(proper(X)) proper(length(X)) = 2X >= 2X = length(proper(X)) proper(and(X1,X2)) = 2X1 + 2X2 + 4 >= 2X1 + 2X2 + 4 = and(proper(X1),proper(X2)) proper(isNat(X)) = X + 4 >= X + 4 = isNat(proper(X)) proper(isNatList(X)) = 2X + 4 >= 2X + 4 = isNatList(proper(X)) proper(isNatIList(X)) = X >= X = isNatIList(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 4X1 + X2 >= 4X1 + X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 2X1 + X2 + 1 >= 2X1 + X2 + 1 = ok(U11(X1,X2)) s(ok(X)) = 4X + 4 >= 4X + 4 = ok(s(X)) length(ok(X)) = 2X >= 2X = ok(length(X)) and(ok(X1),ok(X2)) = 2X1 + 2X2 + 4 >= 2X1 + 2X2 + 4 = ok(and(X1,X2)) isNat(ok(X)) = X + 4 >= X + 4 = ok(isNat(X)) isNatList(ok(X)) = 2X + 4 >= 2X + 4 = ok(isNatList(X)) isNatIList(ok(X)) = X >= X = ok(isNatIList(X)) top(mark(X)) = 2X + 3 >= 2X + 1 = top(proper(X)) top(ok(X)) = 2X + 1 >= 2X + 1 = top(active(X)) problem: active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [top](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [ok](x0) = [0 0 0]x0 [0 0 1] , [1 1 0] [proper](x0) = [0 1 0]x0 [0 0 0] , [1] [nil] = [0] [0], [1 0 0] [1] [isNatIList](x0) = [0 1 0]x0 + [0] [0 0 0] [0], [1 0 0] [isNatList](x0) = [0 1 0]x0 [0 0 0] , [0] [isNat](x0) = x0 + [1] [0], [1 0 0] [1 0 0] [and](x0, x1) = [0 1 0]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [1 0 0] [0] [s](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 0] [length](x0) = [0 1 0]x0 [0 0 0] , [1 0 1] [1 0 0] [U11](x0, x1) = [0 1 0]x0 + [0 1 1]x1 [0 0 0] [0 0 0] , [0] [tt] = [0] [0], [1 0 1] [1 0 0] [0] [cons](x0, x1) = [0 1 0]x0 + [0 1 0]x1 + [1] [0 0 0] [0 0 0] [0], [0] [0] = [0] [0], [1 0 0] [active](x0) = [0 1 0]x0 [0 0 0] , [0] [zeros] = [0] [0] orientation: [1 0 1] [1 0 0] [0] [1 0 0] [1 0 0] [0] active(cons(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = cons(active(X1),X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 1] [1 0 0] [1 0 0] [1 0 0] active(U11(X1,X2)) = [0 1 0]X1 + [0 1 1]X2 >= [0 1 0]X1 + [0 1 1]X2 = U11(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] active(s(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = s(active(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] active(length(X)) = [0 1 0]X >= [0 1 0]X = length(active(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(and(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = and(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] proper(zeros()) = [0] >= [0] = ok(zeros()) [0] [0] [1 1 1] [1 1 0] [1] [1 1 0] [1 1 0] [0] proper(cons(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 1 1] [1 1 1] [1 1 0] [1 1 0] proper(U11(X1,X2)) = [0 1 0]X1 + [0 1 1]X2 >= [0 1 0]X1 + [0 1 0]X2 = U11(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] proper(tt()) = [0] >= [0] = ok(tt()) [0] [0] [1 1 0] [1] [1 1 0] [0] proper(s(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = s(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 1 0] proper(length(X)) = [0 1 0]X >= [0 1 0]X = length(proper(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] proper(and(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = and(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1] [1 1 0] [0] proper(isNat(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = isNat(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 1 0] proper(isNatList(X)) = [0 1 0]X >= [0 1 0]X = isNatList(proper(X)) [0 0 0] [0 0 0] [1 1 0] [1] [1 1 0] [1] proper(isNatIList(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = isNatIList(proper(X)) [0 0 0] [0] [0 0 0] [0] [1] [1] proper(nil()) = [0] >= [0] = ok(nil()) [0] [0] [1 0 1] [1 0 0] [0] [1 0 1] [1 0 0] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 = ok(cons(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1 0 1] [1 0 0] U11(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 1]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(U11(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] s(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X = ok(s(X)) [0 0 0] [0] [0 0 0] [1 0 0] [1 0 0] length(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(length(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] and(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(and(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] isNat(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X = ok(isNat(X)) [0 0 1] [0] [0 0 1] [1 0 0] [1 0 0] isNatList(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(isNatList(X)) [0 0 0] [0 0 0] [1 0 0] [1] [1 0 0] [1] isNatIList(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(isNatIList(X)) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1 0 0] top(ok(X)) = [0 0 0]X >= [0 0 0]X = top(active(X)) [0 0 0] [0 0 0] problem: active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) proper(zeros()) -> ok(zeros()) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = x0 + 1, [ok](x0) = x0 + 1, [proper](x0) = 4x0, [nil] = 4, [isNatIList](x0) = x0 + 6, [isNatList](x0) = x0, [isNat](x0) = 4x0, [and](x0, x1) = 6x0 + x1, [s](x0) = x0 + 1, [length](x0) = x0 + 4, [U11](x0, x1) = x0 + x1, [tt] = 4, [cons](x0, x1) = 4x0 + 2x1 + 5, [0] = 1, [active](x0) = x0, [zeros] = 6 orientation: active(cons(X1,X2)) = 4X1 + 2X2 + 5 >= 4X1 + 2X2 + 5 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 4 >= X + 4 = length(active(X)) active(and(X1,X2)) = 6X1 + X2 >= 6X1 + X2 = and(active(X1),X2) proper(zeros()) = 24 >= 7 = ok(zeros()) proper(0()) = 4 >= 2 = ok(0()) proper(U11(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = U11(proper(X1),proper(X2)) proper(tt()) = 16 >= 5 = ok(tt()) proper(length(X)) = 4X + 16 >= 4X + 4 = length(proper(X)) proper(and(X1,X2)) = 24X1 + 4X2 >= 24X1 + 4X2 = and(proper(X1),proper(X2)) proper(isNatList(X)) = 4X >= 4X = isNatList(proper(X)) proper(isNatIList(X)) = 4X + 24 >= 4X + 6 = isNatIList(proper(X)) proper(nil()) = 16 >= 5 = ok(nil()) cons(ok(X1),ok(X2)) = 4X1 + 2X2 + 11 >= 4X1 + 2X2 + 6 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + X2 + 2 >= X1 + X2 + 1 = ok(U11(X1,X2)) s(ok(X)) = X + 2 >= X + 2 = ok(s(X)) length(ok(X)) = X + 5 >= X + 5 = ok(length(X)) and(ok(X1),ok(X2)) = 6X1 + X2 + 7 >= 6X1 + X2 + 1 = ok(and(X1,X2)) isNat(ok(X)) = 4X + 4 >= 4X + 1 = ok(isNat(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) isNatIList(ok(X)) = X + 7 >= X + 7 = ok(isNatIList(X)) top(ok(X)) = X + 2 >= X + 1 = top(active(X)) problem: active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatList(X)) -> isNatList(proper(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [0] [ok](x0) = [0 0 1]x0 + [1] [0 1 0] [1], [1 0 0] [proper](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [isNatIList](x0) = [0 0 1]x0 [0 1 0] , [1 1 0] [isNatList](x0) = [0 0 1]x0 [0 1 0] , [1 0 0] [1 0 0] [1] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [0], [1 1 0] [s](x0) = [0 0 1]x0 [0 1 0] , [1 1 0] [length](x0) = [0 0 1]x0 [0 1 0] , [1 0 0] [1 0 0] [1] [U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [0], [1 0 0] [1 1 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [active](x0) = [0 0 0]x0 [0 0 0] orientation: [1 0 0] [1 1 0] [1 0 0] [1 1 0] active(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] active(U11(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = U11(active(X1),X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 0 0] active(s(X)) = [0 0 0]X >= [0 0 0]X = s(active(X)) [0 0 0] [0 0 0] [1 1 0] [1 0 0] active(length(X)) = [0 0 0]X >= [0 0 0]X = length(active(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] active(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = and(active(X1),X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] proper(U11(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = U11(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] proper(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = and(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 0 0] proper(isNatList(X)) = [0 0 0]X >= [0 0 0]X = isNatList(proper(X)) [0 0 0] [0 0 0] [1 1 1] [1] [1 1 1] [0] s(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(s(X)) [0 0 1] [1] [0 0 1] [1] [1 1 1] [1] [1 1 1] [0] length(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(length(X)) [0 0 1] [1] [0 0 1] [1] [1 1 1] [1] [1 1 1] [0] isNatList(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(isNatList(X)) [0 0 1] [1] [0 0 1] [1] [1 1 1] [1] [1 1 1] [0] isNatIList(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(isNatIList(X)) [0 0 1] [1] [0 0 1] [1] problem: active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatList(X)) -> isNatList(proper(X)) Matrix Interpretation Processor: dim=1 interpretation: [proper](x0) = 4x0 + 1, [isNatList](x0) = 6x0 + 2, [and](x0, x1) = 6x0 + 4x1 + 3, [s](x0) = x0, [length](x0) = 2x0, [U11](x0, x1) = 4x0 + 3x1 + 2, [cons](x0, x1) = x0 + 2x1, [active](x0) = x0 orientation: active(cons(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = cons(active(X1),X2) active(U11(X1,X2)) = 4X1 + 3X2 + 2 >= 4X1 + 3X2 + 2 = U11(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = 2X >= 2X = length(active(X)) active(and(X1,X2)) = 6X1 + 4X2 + 3 >= 6X1 + 4X2 + 3 = and(active(X1),X2) proper(U11(X1,X2)) = 16X1 + 12X2 + 9 >= 16X1 + 12X2 + 9 = U11(proper(X1),proper(X2)) proper(and(X1,X2)) = 24X1 + 16X2 + 13 >= 24X1 + 16X2 + 13 = and(proper(X1),proper(X2)) proper(isNatList(X)) = 24X + 9 >= 24X + 8 = isNatList(proper(X)) problem: active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) Matrix Interpretation Processor: dim=1 interpretation: [proper](x0) = 3x0 + 1, [and](x0, x1) = 6x0 + 4x1 + 6, [s](x0) = 2x0, [length](x0) = 2x0, [U11](x0, x1) = 3x0 + 2x1 + 2, [cons](x0, x1) = 4x0 + x1, [active](x0) = x0 orientation: active(cons(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = cons(active(X1),X2) active(U11(X1,X2)) = 3X1 + 2X2 + 2 >= 3X1 + 2X2 + 2 = U11(active(X1),X2) active(s(X)) = 2X >= 2X = s(active(X)) active(length(X)) = 2X >= 2X = length(active(X)) active(and(X1,X2)) = 6X1 + 4X2 + 6 >= 6X1 + 4X2 + 6 = and(active(X1),X2) proper(U11(X1,X2)) = 9X1 + 6X2 + 7 >= 9X1 + 6X2 + 7 = U11(proper(X1),proper(X2)) proper(and(X1,X2)) = 18X1 + 12X2 + 19 >= 18X1 + 12X2 + 16 = and(proper(X1),proper(X2)) problem: active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) Matrix Interpretation Processor: dim=1 interpretation: [proper](x0) = 3x0, [and](x0, x1) = x0 + 2x1, [s](x0) = 2x0 + 2, [length](x0) = 2x0 + 4, [U11](x0, x1) = x0 + 4x1 + 2, [cons](x0, x1) = x0 + x1 + 2, [active](x0) = x0 orientation: active(cons(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 4X2 + 2 >= X1 + 4X2 + 2 = U11(active(X1),X2) active(s(X)) = 2X + 2 >= 2X + 2 = s(active(X)) active(length(X)) = 2X + 4 >= 2X + 4 = length(active(X)) active(and(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = and(active(X1),X2) proper(U11(X1,X2)) = 3X1 + 12X2 + 6 >= 3X1 + 12X2 + 2 = U11(proper(X1),proper(X2)) problem: active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) Matrix Interpretation Processor: dim=1 interpretation: [and](x0, x1) = 2x0 + 2x1 + 1, [s](x0) = x0 + 2, [length](x0) = 4x0 + 2, [U11](x0, x1) = 4x0 + 4x1 + 4, [cons](x0, x1) = x0 + x1, [active](x0) = 6x0 + 2 orientation: active(cons(X1,X2)) = 6X1 + 6X2 + 2 >= 6X1 + X2 + 2 = cons(active(X1),X2) active(U11(X1,X2)) = 24X1 + 24X2 + 26 >= 24X1 + 4X2 + 12 = U11(active(X1),X2) active(s(X)) = 6X + 14 >= 6X + 4 = s(active(X)) active(length(X)) = 24X + 14 >= 24X + 10 = length(active(X)) active(and(X1,X2)) = 12X1 + 12X2 + 8 >= 12X1 + 2X2 + 5 = and(active(X1),X2) problem: active(cons(X1,X2)) -> cons(active(X1),X2) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [1 0 0] [0] [cons](x0, x1) = [0 0 1]x0 + [0 0 0]x1 + [1] [0 1 0] [0 0 0] [1], [1 0 1] [active](x0) = [0 0 1]x0 [0 1 0] orientation: [1 1 1] [1 0 0] [1] [1 1 1] [1 0 0] [0] active(cons(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [1] = cons(active(X1),X2) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] problem: Qed