YES Problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = 4x0 + 4, [ok](x0) = x0, [proper](x0) = x0, [nil] = 2, [s](x0) = x0, [length](x0) = 3x0, [U62](x0, x1) = 3x0 + 3x1, [isNat](x0) = 2x0, [U61](x0, x1, x2) = 4x0 + 3x1 + 6x2, [U52](x0) = x0, [isNatList](x0) = 2x0, [U51](x0, x1) = 2x0 + 4x1, [U42](x0) = x0, [isNatIList](x0) = 4x0, [U41](x0, x1) = 2x0 + 4x1, [U31](x0) = 2x0, [U21](x0) = x0, [U11](x0) = 3x0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = 2x0 + 4x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt())) = 0 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U31(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V2)) = 4V2 >= 4V2 = mark(U42(isNatIList(V2))) active(U42(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V2)) = 4V2 >= 2V2 = mark(U52(isNatList(V2))) active(U52(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L,N)) = 3L + 6N >= 3L + 6N = mark(U62(isNat(N),L)) active(U62(tt(),L)) = 3L >= 3L = mark(s(length(L))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 6V1 >= 6V1 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 2V1 >= 2V1 = mark(U21(isNat(V1))) active(isNatIList(V)) = 4V >= 4V = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 8V1 + 16V2 >= 4V1 + 4V2 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 4 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 4V1 + 8V2 >= 4V1 + 4V2 = mark(U51(isNat(V1),V2)) active(length(nil())) = 6 >= 0 = mark(0()) active(length(cons(N,L))) = 12L + 6N >= 11L + 6N = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = cons(active(X1),X2) active(U11(X)) = 3X >= 3X = U11(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X)) = 2X >= 2X = U31(active(X)) active(U41(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = U41(active(X1),X2) active(U42(X)) = X >= X = U42(active(X)) active(U51(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = U51(active(X1),X2) active(U52(X)) = X >= X = U52(active(X)) active(U61(X1,X2,X3)) = 4X1 + 3X2 + 6X3 >= 4X1 + 3X2 + 6X3 = U61(active(X1),X2,X3) active(U62(X1,X2)) = 3X1 + 3X2 >= 3X1 + 3X2 = U62(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = 3X >= 3X = length(active(X)) cons(mark(X1),X2) = 2X1 + 4X2 >= 2X1 + 4X2 = mark(cons(X1,X2)) U11(mark(X)) = 3X >= 3X = mark(U11(X)) U21(mark(X)) = X >= X = mark(U21(X)) U31(mark(X)) = 2X >= 2X = mark(U31(X)) U41(mark(X1),X2) = 2X1 + 4X2 >= 2X1 + 4X2 = mark(U41(X1,X2)) U42(mark(X)) = X >= X = mark(U42(X)) U51(mark(X1),X2) = 2X1 + 4X2 >= 2X1 + 4X2 = mark(U51(X1,X2)) U52(mark(X)) = X >= X = mark(U52(X)) U61(mark(X1),X2,X3) = 4X1 + 3X2 + 6X3 >= 4X1 + 3X2 + 6X3 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = 3X1 + 3X2 >= 3X1 + 3X2 = mark(U62(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = 3X >= 3X = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X)) = 3X >= 3X = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = X >= X = U21(proper(X)) proper(U31(X)) = 2X >= 2X = U31(proper(X)) proper(U41(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = U41(proper(X1),proper(X2)) proper(U42(X)) = X >= X = U42(proper(X)) proper(isNatIList(X)) = 4X >= 4X = isNatIList(proper(X)) proper(U51(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = U51(proper(X1),proper(X2)) proper(U52(X)) = X >= X = U52(proper(X)) proper(isNatList(X)) = 2X >= 2X = isNatList(proper(X)) proper(U61(X1,X2,X3)) = 4X1 + 3X2 + 6X3 >= 4X1 + 3X2 + 6X3 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = 3X1 + 3X2 >= 3X1 + 3X2 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 2X >= 2X = isNat(proper(X)) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = 3X >= 3X = length(proper(X)) proper(nil()) = 2 >= 2 = ok(nil()) cons(ok(X1),ok(X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = ok(cons(X1,X2)) U11(ok(X)) = 3X >= 3X = ok(U11(X)) U21(ok(X)) = X >= X = ok(U21(X)) U31(ok(X)) = 2X >= 2X = ok(U31(X)) U41(ok(X1),ok(X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = ok(U41(X1,X2)) U42(ok(X)) = X >= X = ok(U42(X)) isNatIList(ok(X)) = 4X >= 4X = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = ok(U51(X1,X2)) U52(ok(X)) = X >= X = ok(U52(X)) isNatList(ok(X)) = 2X >= 2X = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = 4X1 + 3X2 + 6X3 >= 4X1 + 3X2 + 6X3 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = 3X1 + 3X2 >= 3X1 + 3X2 = ok(U62(X1,X2)) isNat(ok(X)) = 2X >= 2X = ok(isNat(X)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = 3X >= 3X = ok(length(X)) top(mark(X)) = 4X + 4 >= 4X + 4 = top(proper(X)) top(ok(X)) = 4X + 4 >= 4X + 4 = top(active(X)) problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = 4x0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 0, [s](x0) = x0, [length](x0) = 3x0, [U62](x0, x1) = x0 + 5x1, [isNat](x0) = 4x0, [U61](x0, x1, x2) = x0 + 5x1 + 4x2, [U52](x0) = x0, [isNatList](x0) = x0, [U51](x0, x1) = x0 + x1, [U42](x0) = x0, [isNatIList](x0) = x0 + 2, [U41](x0, x1) = x0 + x1 + 2, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = 4x0 + 6x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt())) = 0 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U31(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V2)) = V2 + 2 >= V2 + 2 = mark(U42(isNatIList(V2))) active(U42(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V2)) = V2 >= V2 = mark(U52(isNatList(V2))) active(U52(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L,N)) = 5L + 4N >= 5L + 4N = mark(U62(isNat(N),L)) active(U62(tt(),L)) = 5L >= 3L = mark(s(length(L))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 12V1 >= V1 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 4V1 >= 4V1 = mark(U21(isNat(V1))) active(isNatIList(V)) = V + 2 >= V = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 2 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 4V1 + 6V2 + 2 >= 4V1 + V2 + 2 = mark(U41(isNat(V1),V2)) active(isNatList(cons(V1,V2))) = 4V1 + 6V2 >= 4V1 + V2 = mark(U51(isNat(V1),V2)) active(length(cons(N,L))) = 18L + 12N >= 6L + 4N = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = 4X1 + 6X2 >= 4X1 + 6X2 = cons(active(X1),X2) active(U11(X)) = X >= X = U11(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X)) = X >= X = U31(active(X)) active(U41(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = U41(active(X1),X2) active(U42(X)) = X >= X = U42(active(X)) active(U51(X1,X2)) = X1 + X2 >= X1 + X2 = U51(active(X1),X2) active(U52(X)) = X >= X = U52(active(X)) active(U61(X1,X2,X3)) = X1 + 5X2 + 4X3 >= X1 + 5X2 + 4X3 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X1 + 5X2 >= X1 + 5X2 = U62(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = 3X >= 3X = length(active(X)) cons(mark(X1),X2) = 4X1 + 6X2 >= 4X1 + 6X2 = mark(cons(X1,X2)) U11(mark(X)) = X >= X = mark(U11(X)) U21(mark(X)) = X >= X = mark(U21(X)) U31(mark(X)) = X >= X = mark(U31(X)) U41(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(U41(X1,X2)) U42(mark(X)) = X >= X = mark(U42(X)) U51(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(U51(X1,X2)) U52(mark(X)) = X >= X = mark(U52(X)) U61(mark(X1),X2,X3) = X1 + 5X2 + 4X3 >= X1 + 5X2 + 4X3 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X1 + 5X2 >= X1 + 5X2 = mark(U62(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = 3X >= 3X = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 4X1 + 6X2 >= 4X1 + 6X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X)) = X >= X = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = X >= X = U21(proper(X)) proper(U31(X)) = X >= X = U31(proper(X)) proper(U41(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = U41(proper(X1),proper(X2)) proper(U42(X)) = X >= X = U42(proper(X)) proper(isNatIList(X)) = X + 2 >= X + 2 = isNatIList(proper(X)) proper(U51(X1,X2)) = X1 + X2 >= X1 + X2 = U51(proper(X1),proper(X2)) proper(U52(X)) = X >= X = U52(proper(X)) proper(isNatList(X)) = X >= X = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X1 + 5X2 + 4X3 >= X1 + 5X2 + 4X3 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X1 + 5X2 >= X1 + 5X2 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 4X >= 4X = isNat(proper(X)) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = 3X >= 3X = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 4X1 + 6X2 >= 4X1 + 6X2 = ok(cons(X1,X2)) U11(ok(X)) = X >= X = ok(U11(X)) U21(ok(X)) = X >= X = ok(U21(X)) U31(ok(X)) = X >= X = ok(U31(X)) U41(ok(X1),ok(X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = ok(U41(X1,X2)) U42(ok(X)) = X >= X = ok(U42(X)) isNatIList(ok(X)) = X + 2 >= X + 2 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U51(X1,X2)) U52(ok(X)) = X >= X = ok(U52(X)) isNatList(ok(X)) = X >= X = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X1 + 5X2 + 4X3 >= X1 + 5X2 + 4X3 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X1 + 5X2 >= X1 + 5X2 = ok(U62(X1,X2)) isNat(ok(X)) = 4X >= 4X = ok(isNat(X)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = 3X >= 3X = ok(length(X)) top(mark(X)) = 4X >= 4X = top(proper(X)) top(ok(X)) = 4X >= 4X = top(active(X)) problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = 2x0 + 2, [ok](x0) = x0, [proper](x0) = x0, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U62](x0, x1) = x0 + x1, [isNat](x0) = x0, [U61](x0, x1, x2) = 4x0 + x1 + x2, [U52](x0) = 4x0, [isNatList](x0) = x0, [U51](x0, x1) = x0 + 4x1, [U42](x0) = 4x0, [isNatIList](x0) = x0, [U41](x0, x1) = x0 + 4x1, [U31](x0) = 4x0 + 4, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + 5x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt())) = 0 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U31(tt())) = 4 >= 0 = mark(tt()) active(U41(tt(),V2)) = 4V2 >= 4V2 = mark(U42(isNatIList(V2))) active(U42(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V2)) = 4V2 >= 4V2 = mark(U52(isNatList(V2))) active(U52(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L,N)) = L + N >= L + N = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L >= L = mark(s(length(L))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= V1 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = V1 >= V1 = mark(U21(isNat(V1))) active(isNatIList(cons(V1,V2))) = V1 + 5V2 >= V1 + 4V2 = mark(U41(isNat(V1),V2)) active(isNatList(cons(V1,V2))) = V1 + 5V2 >= V1 + 4V2 = mark(U51(isNat(V1),V2)) active(length(cons(N,L))) = 5L + N >= 5L + N = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X1 + 5X2 >= X1 + 5X2 = cons(active(X1),X2) active(U11(X)) = X >= X = U11(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X)) = 4X + 4 >= 4X + 4 = U31(active(X)) active(U41(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = U41(active(X1),X2) active(U42(X)) = 4X >= 4X = U42(active(X)) active(U51(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = U51(active(X1),X2) active(U52(X)) = 4X >= 4X = U52(active(X)) active(U61(X1,X2,X3)) = 4X1 + X2 + X3 >= 4X1 + X2 + X3 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X1 + X2 >= X1 + X2 = U62(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) cons(mark(X1),X2) = X1 + 5X2 >= X1 + 5X2 = mark(cons(X1,X2)) U11(mark(X)) = X >= X = mark(U11(X)) U21(mark(X)) = X >= X = mark(U21(X)) U31(mark(X)) = 4X + 4 >= 4X + 4 = mark(U31(X)) U41(mark(X1),X2) = X1 + 4X2 >= X1 + 4X2 = mark(U41(X1,X2)) U42(mark(X)) = 4X >= 4X = mark(U42(X)) U51(mark(X1),X2) = X1 + 4X2 >= X1 + 4X2 = mark(U51(X1,X2)) U52(mark(X)) = 4X >= 4X = mark(U52(X)) U61(mark(X1),X2,X3) = 4X1 + X2 + X3 >= 4X1 + X2 + X3 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(U62(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = X >= X = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + 5X2 >= X1 + 5X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X)) = X >= X = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = X >= X = U21(proper(X)) proper(U31(X)) = 4X + 4 >= 4X + 4 = U31(proper(X)) proper(U41(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = U41(proper(X1),proper(X2)) proper(U42(X)) = 4X >= 4X = U42(proper(X)) proper(isNatIList(X)) = X >= X = isNatIList(proper(X)) proper(U51(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = U51(proper(X1),proper(X2)) proper(U52(X)) = 4X >= 4X = U52(proper(X)) proper(isNatList(X)) = X >= X = isNatList(proper(X)) proper(U61(X1,X2,X3)) = 4X1 + X2 + X3 >= 4X1 + X2 + X3 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X1 + X2 >= X1 + X2 = U62(proper(X1),proper(X2)) proper(isNat(X)) = X >= X = isNat(proper(X)) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = X >= X = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 5X2 >= X1 + 5X2 = ok(cons(X1,X2)) U11(ok(X)) = X >= X = ok(U11(X)) U21(ok(X)) = X >= X = ok(U21(X)) U31(ok(X)) = 4X + 4 >= 4X + 4 = ok(U31(X)) U41(ok(X1),ok(X2)) = X1 + 4X2 >= X1 + 4X2 = ok(U41(X1,X2)) U42(ok(X)) = 4X >= 4X = ok(U42(X)) isNatIList(ok(X)) = X >= X = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X1 + 4X2 >= X1 + 4X2 = ok(U51(X1,X2)) U52(ok(X)) = 4X >= 4X = ok(U52(X)) isNatList(ok(X)) = X >= X = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = 4X1 + X2 + X3 >= 4X1 + X2 + X3 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U62(X1,X2)) isNat(ok(X)) = X >= X = ok(isNat(X)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) top(mark(X)) = 2X + 2 >= 2X + 2 = top(proper(X)) top(ok(X)) = 2X + 2 >= 2X + 2 = top(active(X)) problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(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, [s](x0) = x0, [length](x0) = 2x0 + 1, [U62](x0, x1) = x0 + 2x1 + 1, [isNat](x0) = 4x0, [U61](x0, x1, x2) = 2x0 + 2x1 + 4x2 + 1, [U52](x0) = x0, [isNatList](x0) = 4x0, [U51](x0, x1) = 2x0 + 4x1, [U42](x0) = 2x0, [isNatIList](x0) = 2x0, [U41](x0, x1) = x0 + 4x1, [U31](x0) = x0 + 2, [U21](x0) = x0, [U11](x0) = x0, [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())) = 0 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V2)) = 4V2 >= 4V2 = mark(U42(isNatIList(V2))) active(U42(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V2)) = 4V2 >= 4V2 = mark(U52(isNatList(V2))) active(U52(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L,N)) = 2L + 4N + 1 >= 2L + 4N + 1 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = 2L + 1 >= 2L + 1 = mark(s(length(L))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 8V1 + 4 >= 4V1 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 4V1 >= 4V1 = mark(U21(isNat(V1))) active(isNatIList(cons(V1,V2))) = 4V1 + 10V2 >= 4V1 + 4V2 = mark(U41(isNat(V1),V2)) active(isNatList(cons(V1,V2))) = 8V1 + 20V2 >= 8V1 + 4V2 = mark(U51(isNat(V1),V2)) active(length(cons(N,L))) = 10L + 4N + 1 >= 10L + 4N + 1 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = 2X1 + 5X2 >= 2X1 + 5X2 = cons(active(X1),X2) active(U11(X)) = X >= X = U11(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X)) = X + 2 >= X + 2 = U31(active(X)) active(U41(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = U41(active(X1),X2) active(U42(X)) = 2X >= 2X = U42(active(X)) active(U51(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = U51(active(X1),X2) active(U52(X)) = X >= X = U52(active(X)) active(U61(X1,X2,X3)) = 2X1 + 2X2 + 4X3 + 1 >= 2X1 + 2X2 + 4X3 + 1 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X1 + 2X2 + 1 >= X1 + 2X2 + 1 = U62(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = 2X + 1 >= 2X + 1 = length(active(X)) cons(mark(X1),X2) = 2X1 + 5X2 >= 2X1 + 5X2 = mark(cons(X1,X2)) U11(mark(X)) = X >= X = mark(U11(X)) U21(mark(X)) = X >= X = mark(U21(X)) U31(mark(X)) = X + 2 >= X + 2 = mark(U31(X)) U41(mark(X1),X2) = X1 + 4X2 >= X1 + 4X2 = mark(U41(X1,X2)) U42(mark(X)) = 2X >= 2X = mark(U42(X)) U51(mark(X1),X2) = 2X1 + 4X2 >= 2X1 + 4X2 = mark(U51(X1,X2)) U52(mark(X)) = X >= X = mark(U52(X)) U61(mark(X1),X2,X3) = 2X1 + 2X2 + 4X3 + 1 >= 2X1 + 2X2 + 4X3 + 1 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X1 + 2X2 + 1 >= X1 + 2X2 + 1 = mark(U62(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = 2X + 1 >= 2X + 1 = mark(length(X)) 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(X)) = X >= X = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = X >= X = U21(proper(X)) proper(U31(X)) = X + 2 >= X + 2 = U31(proper(X)) proper(U41(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = U41(proper(X1),proper(X2)) proper(U42(X)) = 2X >= 2X = U42(proper(X)) proper(isNatIList(X)) = 2X >= 2X = isNatIList(proper(X)) proper(U51(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = U51(proper(X1),proper(X2)) proper(U52(X)) = X >= X = U52(proper(X)) proper(isNatList(X)) = 4X >= 4X = isNatList(proper(X)) proper(U61(X1,X2,X3)) = 2X1 + 2X2 + 4X3 + 1 >= 2X1 + 2X2 + 4X3 + 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X1 + 2X2 + 1 >= X1 + 2X2 + 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 4X >= 4X = isNat(proper(X)) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = 2X + 1 >= 2X + 1 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 2X1 + 5X2 >= 2X1 + 5X2 = ok(cons(X1,X2)) U11(ok(X)) = X >= X = ok(U11(X)) U21(ok(X)) = X >= X = ok(U21(X)) U31(ok(X)) = X + 2 >= X + 2 = ok(U31(X)) U41(ok(X1),ok(X2)) = X1 + 4X2 >= X1 + 4X2 = ok(U41(X1,X2)) U42(ok(X)) = 2X >= 2X = ok(U42(X)) isNatIList(ok(X)) = 2X >= 2X = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = ok(U51(X1,X2)) U52(ok(X)) = X >= X = ok(U52(X)) isNatList(ok(X)) = 4X >= 4X = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = 2X1 + 2X2 + 4X3 + 1 >= 2X1 + 2X2 + 4X3 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X1 + 2X2 + 1 >= X1 + 2X2 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = 4X >= 4X = ok(isNat(X)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = 2X + 1 >= 2X + 1 = ok(length(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())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(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, [s](x0) = x0, [length](x0) = x0, [U62](x0, x1) = 2x0 + x1, [isNat](x0) = x0, [U61](x0, x1, x2) = x0 + x1 + 2x2, [U52](x0) = 2x0, [isNatList](x0) = x0, [U51](x0, x1) = 2x0 + 2x1, [U42](x0) = x0, [isNatIList](x0) = 2x0, [U41](x0, x1) = 2x0 + 4x1, [U31](x0) = 2x0, [U21](x0) = x0, [U11](x0) = 2x0 + 1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = 2x0 + 2x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt())) = 1 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V2)) = 4V2 >= 2V2 = mark(U42(isNatIList(V2))) active(U42(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V2)) = 2V2 >= 2V2 = mark(U52(isNatList(V2))) active(U52(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L,N)) = L + 2N >= L + 2N = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L >= L = mark(s(length(L))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(V1))) = V1 >= V1 = mark(U21(isNat(V1))) active(isNatIList(cons(V1,V2))) = 4V1 + 4V2 >= 2V1 + 4V2 = mark(U41(isNat(V1),V2)) active(isNatList(cons(V1,V2))) = 2V1 + 2V2 >= 2V1 + 2V2 = mark(U51(isNat(V1),V2)) active(length(cons(N,L))) = 2L + 2N >= 2L + 2N = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = cons(active(X1),X2) active(U11(X)) = 2X + 1 >= 2X + 1 = U11(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X)) = 2X >= 2X = U31(active(X)) active(U41(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = U41(active(X1),X2) active(U42(X)) = X >= X = U42(active(X)) active(U51(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = U51(active(X1),X2) active(U52(X)) = 2X >= 2X = U52(active(X)) active(U61(X1,X2,X3)) = X1 + X2 + 2X3 >= X1 + X2 + 2X3 = U61(active(X1),X2,X3) active(U62(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = U62(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) cons(mark(X1),X2) = 2X1 + 2X2 >= 2X1 + 2X2 = mark(cons(X1,X2)) U11(mark(X)) = 2X + 1 >= 2X + 1 = mark(U11(X)) U21(mark(X)) = X >= X = mark(U21(X)) U31(mark(X)) = 2X >= 2X = mark(U31(X)) U41(mark(X1),X2) = 2X1 + 4X2 >= 2X1 + 4X2 = mark(U41(X1,X2)) U42(mark(X)) = X >= X = mark(U42(X)) U51(mark(X1),X2) = 2X1 + 2X2 >= 2X1 + 2X2 = mark(U51(X1,X2)) U52(mark(X)) = 2X >= 2X = mark(U52(X)) U61(mark(X1),X2,X3) = X1 + X2 + 2X3 >= X1 + X2 + 2X3 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = 2X1 + X2 >= 2X1 + X2 = mark(U62(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = X >= X = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X)) = 2X + 1 >= 2X + 1 = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = X >= X = U21(proper(X)) proper(U31(X)) = 2X >= 2X = U31(proper(X)) proper(U41(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = U41(proper(X1),proper(X2)) proper(U42(X)) = X >= X = U42(proper(X)) proper(isNatIList(X)) = 2X >= 2X = isNatIList(proper(X)) proper(U51(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = U51(proper(X1),proper(X2)) proper(U52(X)) = 2X >= 2X = U52(proper(X)) proper(isNatList(X)) = X >= X = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X1 + X2 + 2X3 >= X1 + X2 + 2X3 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = U62(proper(X1),proper(X2)) proper(isNat(X)) = X >= X = isNat(proper(X)) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = X >= X = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = ok(cons(X1,X2)) U11(ok(X)) = 2X + 1 >= 2X + 1 = ok(U11(X)) U21(ok(X)) = X >= X = ok(U21(X)) U31(ok(X)) = 2X >= 2X = ok(U31(X)) U41(ok(X1),ok(X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = ok(U41(X1,X2)) U42(ok(X)) = X >= X = ok(U42(X)) isNatIList(ok(X)) = 2X >= 2X = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = ok(U51(X1,X2)) U52(ok(X)) = 2X >= 2X = ok(U52(X)) isNatList(ok(X)) = X >= X = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X1 + X2 + 2X3 >= X1 + X2 + 2X3 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = 2X1 + X2 >= 2X1 + X2 = ok(U62(X1,X2)) isNat(ok(X)) = X >= X = ok(isNat(X)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) top(mark(X)) = 2X + 1 >= 2X + 1 = top(proper(X)) top(ok(X)) = 2X + 1 >= 2X + 1 = top(active(X)) problem: active(zeros()) -> mark(cons(0(),zeros())) active(U21(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [top](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [ok](x0) = [0 0 0]x0 [0 0 1] , [proper](x0) = x0 , [1] [nil] = [0] [0], [1 0 1] [0] [s](x0) = [0 0 1]x0 + [1] [0 0 1] [0], [1 0 1] [length](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 1] [U62](x0, x1) = [0 0 1]x0 + [0 0 1]x1 [0 0 0] [0 0 0] , [1 0 0] [isNat](x0) = [0 0 1]x0 [0 0 1] , [1 0 0] [1 0 1] [1 0 0] [U61](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 0] [0 0 0] [0 0 0] , [1 0 0] [U52](x0) = [0 0 1]x0 [0 0 1] , [1 0 0] [isNatList](x0) = [1 1 0]x0 [0 0 0] , [1 0 0] [1 0 0] [U51](x0, x1) = [0 0 1]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [U42](x0) = [1 0 0]x0 [0 0 1] , [1 0 0] [isNatIList](x0) = [0 0 1]x0 [0 0 0] , [1 0 0] [1 0 0] [U41](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [0] [U31](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 1] [U21](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [U11](x0) = [0 0 1]x0 [0 0 0] , [0] [tt] = [0] [1], [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [1 0 1] [cons](x0, x1) = [0 1 1]x0 + [0 0 0]x1 [0 0 0] [1 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] active(U21(tt())) = [0] >= [0] = mark(tt()) [1] [1] [1 0 0] [1 0 0] active(U41(tt(),V2)) = [0 0 0]V2 >= [0 0 0]V2 = mark(U42(isNatIList(V2))) [0 0 0] [0 0 0] [0] [0] active(U42(tt())) = [0] >= [0] = mark(tt()) [1] [1] [1 0 0] [0] [1 0 0] active(U51(tt(),V2)) = [0 0 0]V2 + [1] >= [0 0 0]V2 = mark(U52(isNatList(V2))) [0 0 0] [0] [0 0 0] [0] [0] active(U52(tt())) = [1] >= [0] = mark(tt()) [1] [1] [1 0 1] [1 0 0] [1 0 1] [1 0 0] active(U61(tt(),L,N)) = [0 0 0]L + [0 0 0]N >= [0 0 0]L + [0 0 0]N = mark(U62(isNat(N),L)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [0] [1 0 1] active(U62(tt(),L)) = [0 0 1]L + [1] >= [0 0 0]L = mark(s(length(L))) [0 0 0] [0] [0 0 0] [0] [0] active(isNat(0())) = [1] >= [0] = mark(tt()) [1] [1] [1 0 1] [1 0 1] active(isNat(s(V1))) = [0 0 1]V1 >= [0 0 0]V1 = mark(U21(isNat(V1))) [0 0 1] [0 0 1] [1 0 0] [1 0 1] [1 0 0] [1 0 0] active(isNatIList(cons(V1,V2))) = [0 0 0]V1 + [1 0 0]V2 >= [0 0 0]V1 + [0 0 0]V2 = mark(U41(isNat(V1),V2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 1] [1 0 0] [1 0 0] active(isNatList(cons(V1,V2))) = [1 1 1]V1 + [1 0 1]V2 >= [0 0 0]V1 + [0 0 0]V2 = mark(U51(isNat(V1),V2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [2 0 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(U61(isNatList(L),L,N)) [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 1 1]X1 + [0 0 0]X2 >= [0 1 1]X1 + [0 0 0]X2 = cons(active(X1),X2) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] active(U11(X)) = [0 0 1]X >= [0 0 1]X = U11(active(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] active(U21(X)) = [0 0 0]X >= [0 0 0]X = U21(active(X)) [0 0 1] [0 0 1] [1 0 0] [0] [1 0 0] [0] active(U31(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = U31(active(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(U41(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U41(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(U42(X)) = [1 0 0]X >= [1 0 0]X = U42(active(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(U51(X1,X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 1]X1 + [0 0 0]X2 = U51(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(U52(X)) = [0 0 1]X >= [0 0 1]X = U52(active(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 1] [1 0 0] [1 0 0] [1 0 1] [1 0 0] active(U61(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = U61(active(X1),X2,X3) [0 0 0] [0 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(U62(X1,X2)) = [0 0 1]X1 + [0 0 1]X2 >= [0 0 1]X1 + [0 0 1]X2 = U62(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [0] [1 0 1] [0] active(s(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = s(active(X)) [0 0 1] [0] [0 0 1] [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 1] [1 0 0] [1 0 1] cons(mark(X1),X2) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(cons(X1,X2)) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] U11(mark(X)) = [0 0 1]X >= [0 0 0]X = mark(U11(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] U21(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(U21(X)) [0 0 1] [0 0 1] [1 0 0] [0] [1 0 0] U31(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X = mark(U31(X)) [0 0 0] [0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] U41(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(U41(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] U42(mark(X)) = [1 0 0]X >= [0 0 0]X = mark(U42(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] U51(mark(X1),X2) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(U51(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] U52(mark(X)) = [0 0 1]X >= [0 0 0]X = mark(U52(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 1] [1 0 0] [1 0 0] [1 0 1] [1 0 0] U61(mark(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = mark(U61(X1,X2,X3)) [0 0 0] [0 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] U62(mark(X1),X2) = [0 0 1]X1 + [0 0 1]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(U62(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [0] [1 0 1] s(mark(X)) = [0 0 1]X + [1] >= [0 0 0]X = mark(s(X)) [0 0 1] [0] [0 0 1] [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] [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 1 1]X1 + [0 0 0]X2 >= [0 1 1]X1 + [0 0 0]X2 = cons(proper(X1),proper(X2)) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [0] [0] proper(0()) = [1] >= [0] = ok(0()) [1] [1] [1 0 0] [1 0 0] proper(U11(X)) = [0 0 1]X >= [0 0 1]X = U11(proper(X)) [0 0 0] [0 0 0] [0] [0] proper(tt()) = [0] >= [0] = ok(tt()) [1] [1] [1 0 1] [1 0 1] proper(U21(X)) = [0 0 0]X >= [0 0 0]X = U21(proper(X)) [0 0 1] [0 0 1] [1 0 0] [0] [1 0 0] [0] proper(U31(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = U31(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(U41(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U41(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] proper(U42(X)) = [1 0 0]X >= [1 0 0]X = U42(proper(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] proper(isNatIList(X)) = [0 0 1]X >= [0 0 1]X = isNatIList(proper(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(U51(X1,X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 1]X1 + [0 0 0]X2 = U51(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] proper(U52(X)) = [0 0 1]X >= [0 0 1]X = U52(proper(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] proper(isNatList(X)) = [1 1 0]X >= [1 1 0]X = isNatList(proper(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 1] [1 0 0] [1 0 0] [1 0 1] [1 0 0] proper(U61(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = U61(proper(X1),proper(X2),proper(X3)) [0 0 0] [0 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] proper(U62(X1,X2)) = [0 0 1]X1 + [0 0 1]X2 >= [0 0 1]X1 + [0 0 1]X2 = U62(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 1]X >= [0 0 1]X = isNat(proper(X)) [0 0 1] [0 0 1] [1 0 1] [0] [1 0 1] [0] proper(s(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = s(proper(X)) [0 0 1] [0] [0 0 1] [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] [1] proper(nil()) = [0] >= [0] = ok(nil()) [0] [0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] cons(ok(X1),ok(X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(cons(X1,X2)) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] U11(ok(X)) = [0 0 1]X >= [0 0 0]X = ok(U11(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] U21(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(U21(X)) [0 0 1] [0 0 1] [1 0 0] [0] [1 0 0] U31(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X = ok(U31(X)) [0 0 0] [0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] U41(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(U41(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] U42(ok(X)) = [1 0 0]X >= [0 0 0]X = ok(U42(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] isNatIList(ok(X)) = [0 0 1]X >= [0 0 0]X = ok(isNatIList(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] U51(ok(X1),ok(X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(U51(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] U52(ok(X)) = [0 0 1]X >= [0 0 0]X = ok(U52(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] isNatList(ok(X)) = [1 0 0]X >= [0 0 0]X = ok(isNatList(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 1] [1 0 0] [1 0 0] [1 0 1] [1 0 0] U61(ok(X1),ok(X2),ok(X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = ok(U61(X1,X2,X3)) [0 0 0] [0 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] U62(ok(X1),ok(X2)) = [0 0 1]X1 + [0 0 1]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(U62(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] isNat(ok(X)) = [0 0 1]X >= [0 0 0]X = ok(isNat(X)) [0 0 1] [0 0 1] [1 0 1] [0] [1 0 1] s(ok(X)) = [0 0 1]X + [1] >= [0 0 0]X = ok(s(X)) [0 0 1] [0] [0 0 1] [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] top(mark(X)) = [0 0 0]X >= [0 0 0]X = top(proper(X)) [0 0 0] [0 0 0] [1 0 0] [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(zeros()) -> mark(cons(0(),zeros())) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [top](x0) = [0 0 0]x0 [0 0 0] , [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] [0], [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] [U62](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [0] [isNat](x0) = [0 1 0]x0 + [0] [1 0 0] [1], [1 0 0] [1 0 1] [1 0 0] [U61](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 0] [0 0 0] [0 0 0] , [1 0 1] [U52](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [isNatList](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 1] [U51](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [U42](x0) = x0 , [1 0 0] [0] [isNatIList](x0) = [0 1 0]x0 + [0] [1 0 0] [1], [1 0 0] [U41](x0, x1) = x0 + [0 0 0]x1 [1 0 0] , [1 0 0] [U31](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [U21](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [U11](x0) = [0 0 0]x0 [0 0 1] , [0] [tt] = [0] [1], [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [1 0 1] [cons](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 0 0] [1 0 0] , [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 0] [0] [1 0 0] [0] active(U41(tt(),V2)) = [0 0 0]V2 + [0] >= [0 0 0]V2 + [0] = mark(U42(isNatIList(V2))) [1 0 0] [1] [1 0 0] [1] [0] [0] active(U42(tt())) = [0] >= [0] = mark(tt()) [1] [1] [1 0 1] [1 0 0] active(U51(tt(),V2)) = [0 0 0]V2 >= [0 0 0]V2 = mark(U52(isNatList(V2))) [0 0 0] [0 0 0] [1] [0] active(U52(tt())) = [0] >= [0] = mark(tt()) [1] [1] [1 0 1] [1 0 0] [1 0 1] [1 0 0] active(U61(tt(),L,N)) = [0 0 0]L + [0 0 0]N >= [0 0 0]L + [0 0 0]N = mark(U62(isNat(N),L)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] active(U62(tt(),L)) = [0 0 0]L >= [0 0 0]L = mark(s(length(L))) [0 0 0] [0 0 0] [0] [0] active(isNat(0())) = [0] >= [0] = mark(tt()) [1] [1] [1 0 0] [0] [1 0 0] active(isNat(s(V1))) = [0 0 0]V1 + [0] >= [0 0 0]V1 = mark(U21(isNat(V1))) [1 0 0] [1] [0 0 0] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 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(U41(isNat(V1),V2)) [1 0 0] [1 0 1] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1 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(U51(isNat(V1),V2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [2 0 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(U61(isNatList(L),L,N)) [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 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] active(U11(X)) = [0 0 0]X >= [0 0 0]X = U11(active(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] active(U21(X)) = [0 0 0]X >= [0 0 0]X = U21(active(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(U31(X)) = [0 0 0]X >= [0 0 0]X = U31(active(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(U41(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U41(active(X1),X2) [0 0 1] [1 0 0] [0 0 1] [1 0 0] [1 0 0] [1 0 0] active(U42(X)) = [0 0 0]X >= [0 0 0]X = U42(active(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 1] [1 0 0] [1 0 1] active(U51(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U51(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] active(U52(X)) = [0 0 0]X >= [0 0 0]X = U52(active(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 1] [1 0 0] [1 0 0] [1 0 1] [1 0 0] active(U61(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = U61(active(X1),X2,X3) [0 0 0] [0 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(U62(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U62(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [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] [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 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 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] U11(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(U11(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] U21(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(U21(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] U31(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(U31(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] U41(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(U41(X1,X2)) [0 0 1] [1 0 0] [0 0 1] [1 0 0] [1 0 0] [1 0 0] U42(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(U42(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 1] [1 0 0] [1 0 1] U51(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(U51(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] U52(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(U52(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 1] [1 0 0] [1 0 0] [1 0 1] [1 0 0] U61(mark(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = mark(U61(X1,X2,X3)) [0 0 0] [0 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] U62(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(U62(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 = 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] [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 0] [1 0 0] [0 0 0] [1 0 0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 0 0] [1 0 0] proper(U11(X)) = [0 0 0]X >= [0 0 0]X = U11(proper(X)) [0 0 1] [0 0 1] [0] [0] proper(tt()) = [0] >= [0] = ok(tt()) [1] [1] [1 0 0] [1 0 0] proper(U21(X)) = [0 0 0]X >= [0 0 0]X = U21(proper(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] proper(U31(X)) = [0 0 0]X >= [0 0 0]X = U31(proper(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(U41(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U41(proper(X1),proper(X2)) [0 0 1] [1 0 0] [0 0 1] [1 0 0] [1 0 0] [1 0 0] proper(U42(X)) = [0 0 0]X >= [0 0 0]X = U42(proper(X)) [0 0 1] [0 0 1] [1 0 0] [0] [1 0 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] [1 0 0] [1 0 1] [1 0 0] [1 0 1] proper(U51(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U51(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] proper(U52(X)) = [0 0 0]X >= [0 0 0]X = U52(proper(X)) [0 0 1] [0 0 1] [1 0 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 0 0] [1 0 1] [1 0 0] [1 0 0] [1 0 1] [1 0 0] proper(U61(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = U61(proper(X1),proper(X2),proper(X3)) [0 0 0] [0 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] proper(U62(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U62(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [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 0] [1 0 0] 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] [0] [0] proper(nil()) = [0] >= [0] = ok(nil()) [0] [0] [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 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] U11(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(U11(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] U21(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(U21(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] U31(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(U31(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] U41(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(U41(X1,X2)) [0 0 1] [1 0 0] [0 0 1] [1 0 0] [1 0 0] [1 0 0] U42(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(U42(X)) [0 0 1] [0 0 1] [1 0 0] [0] [1 0 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] [1 0 1] [1 0 0] [1 0 1] U51(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(U51(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] U52(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(U52(X)) [0 0 1] [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 0 1] [1 0 0] [1 0 0] [1 0 1] [1 0 0] U61(ok(X1),ok(X2),ok(X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = ok(U61(X1,X2,X3)) [0 0 0] [0 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] U62(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(U62(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [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 0] [1 0 0] 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] top(mark(X)) = [0 0 0]X >= [0 0 0]X = top(proper(X)) [0 0 0] [0 0 0] [1 0 0] [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(zeros()) -> mark(cons(0(),zeros())) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [top](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [ok](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [proper](x0) = [0 0 1]x0 [0 0 1] , [0] [nil] = [0] [0], [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [0] [length](x0) = [1 0 0]x0 + [0] [1 0 0] [1], [1 0 0] [1 0 1] [U62](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [0] [isNat](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [1 0 1] [1 0 0] [0] [U61](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [1] [0 0 0] [0 0 0] [0 0 0] [1], [1 0 0] [U52](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [isNatList](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [U51](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 1] [U42](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [isNatIList](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [U41](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [U31](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [U21](x0) = [0 0 1]x0 [0 0 1] , [1 0 0] [U11](x0) = [0 0 0]x0 [0 0 0] , [0] [tt] = [0] [1], [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 1] , [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] [active](x0) = [0 0 1]x0 [0 0 1] , [0] [zeros] = [0] [0] orientation: [0] [0] active(zeros()) = [0] >= [0] = mark(cons(0(),zeros())) [0] [0] [1 0 0] [1 0 0] active(U41(tt(),V2)) = [0 0 0]V2 >= [0 0 0]V2 = mark(U42(isNatIList(V2))) [0 0 0] [0 0 0] [1] [0] active(U42(tt())) = [1] >= [0] = mark(tt()) [1] [1] [1 0 0] [1 0 0] active(U51(tt(),V2)) = [0 0 0]V2 >= [0 0 0]V2 = mark(U52(isNatList(V2))) [0 0 0] [0 0 0] [1 0 1] [1 0 0] [0] [1 0 1] [1 0 0] active(U61(tt(),L,N)) = [0 0 0]L + [0 0 0]N + [1] >= [0 0 0]L + [0 0 0]N = mark(U62(isNat(N),L)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1 0 1] [1 0 1] active(U62(tt(),L)) = [0 0 0]L >= [0 0 0]L = mark(s(length(L))) [0 0 0] [0 0 0] [0] [0] active(isNat(0())) = [1] >= [0] = 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 + [0] = mark(U21(isNat(V1))) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(isNatIList(cons(V1,V2))) = [0 0 0]V1 + [0 0 0]V2 >= [0 0 0]V1 + [0 0 0]V2 = mark(U41(isNat(V1),V2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(isNatList(cons(V1,V2))) = [0 0 0]V1 + [0 0 0]V2 >= [0 0 0]V1 + [0 0 0]V2 = mark(U51(isNat(V1),V2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [2 0 1] [1 0 0] [0] [2 0 1] [1 0 0] [0] active(length(cons(N,L))) = [1 0 0]L + [1 0 0]N + [1] >= [0 0 0]L + [0 0 0]N + [0] = mark(U61(isNatList(L),L,N)) [1 0 0] [1 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(cons(X1,X2)) = [0 0 0]X1 + [1 0 1]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 0] active(U11(X)) = [0 0 0]X >= [0 0 0]X = U11(active(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(U21(X)) = [0 0 1]X >= [0 0 1]X = U21(active(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] active(U31(X)) = [0 0 0]X >= [0 0 0]X = U31(active(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(U41(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U41(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] active(U42(X)) = [0 0 1]X >= [0 0 0]X = U42(active(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(U51(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U51(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(U52(X)) = [0 0 0]X >= [0 0 0]X = U52(active(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 1] [1 0 0] [0] [1 0 0] [1 0 1] [1 0 0] [0] active(U61(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] = U61(active(X1),X2,X3) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [1 0 0] [1 0 1] [1 0 0] [1 0 1] active(U62(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U62(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [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)) = [1 0 0]X + [1] >= [1 0 0]X + [0] = length(active(X)) [1 0 0] [1] [1 0 0] [1] [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 = mark(cons(X1,X2)) [0 0 0] [1 0 1] [0 0 0] [1 0 1] [1 0 0] [1 0 0] U11(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(U11(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] U21(mark(X)) = [0 0 1]X >= [0 0 0]X = mark(U21(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] U31(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(U31(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] U41(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(U41(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] U42(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(U42(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] U51(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(U51(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] U52(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(U52(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 1] [1 0 0] [0] [1 0 0] [1 0 1] [1 0 0] [0] U61(mark(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = mark(U61(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [1 0 0] [1 0 1] [1 0 0] [1 0 1] U62(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(U62(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 = mark(s(X)) [0 0 0] [0 0 0] [1 0 1] [0] [1 0 1] [0] length(mark(X)) = [1 0 0]X + [0] >= [0 0 0]X + [0] = mark(length(X)) [1 0 0] [1] [1 0 0] [1] [0] [0] proper(zeros()) = [0] >= [0] = ok(zeros()) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(cons(X1,X2)) = [0 0 0]X1 + [1 0 1]X2 >= [0 0 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()) = [0] >= [0] = ok(0()) [0] [0] [1 0 0] [1 0 0] proper(U11(X)) = [0 0 0]X >= [0 0 0]X = U11(proper(X)) [0 0 0] [0 0 0] [0] [0] proper(tt()) = [1] >= [0] = ok(tt()) [1] [1] [1 0 0] [1 0 0] proper(U21(X)) = [0 0 1]X >= [0 0 1]X = U21(proper(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] proper(U31(X)) = [0 0 0]X >= [0 0 0]X = U31(proper(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(U41(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U41(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] proper(U42(X)) = [0 0 1]X >= [0 0 0]X = U42(proper(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] proper(isNatIList(X)) = [0 0 0]X >= [0 0 0]X = isNatIList(proper(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(U51(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U51(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] proper(U52(X)) = [0 0 0]X >= [0 0 0]X = U52(proper(X)) [0 0 0] [0 0 0] [1 0 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 0 0] [1 0 1] [1 0 0] [0] [1 0 0] [1 0 1] [1 0 0] [0] proper(U61(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] = U61(proper(X1),proper(X2),proper(X3)) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [1 0 0] [1 0 1] [1 0 0] [1 0 1] proper(U62(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U62(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] proper(isNat(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = isNat(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] proper(s(X)) = [0 0 0]X >= [0 0 0]X = s(proper(X)) [0 0 0] [0 0 0] [1 0 1] [0] [1 0 1] [0] proper(length(X)) = [1 0 0]X + [1] >= [1 0 0]X + [0] = length(proper(X)) [1 0 0] [1] [1 0 0] [1] [0] [0] proper(nil()) = [0] >= [0] = ok(nil()) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] 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 0] [1 0 1] [0 0 0] [1 0 1] [1 0 0] [1 0 0] U11(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(U11(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] U21(ok(X)) = [0 0 1]X >= [0 0 0]X = ok(U21(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] U31(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(U31(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] U41(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(U41(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] U42(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(U42(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] isNatIList(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(isNatIList(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] U51(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(U51(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] U52(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(U52(X)) [0 0 0] [0 0 0] [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 0 1] [1 0 0] [0] [1 0 0] [1 0 1] [1 0 0] [0] U61(ok(X1),ok(X2),ok(X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = ok(U61(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [1 0 0] [1 0 1] [1 0 0] [1 0 1] U62(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(U62(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] isNat(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(isNat(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] s(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(s(X)) [0 0 0] [0 0 0] [1 0 1] [0] [1 0 1] [0] length(ok(X)) = [1 0 0]X + [0] >= [0 0 0]X + [0] = ok(length(X)) [1 0 0] [1] [1 0 0] [1] [1 0 0] [1 0 0] top(mark(X)) = [0 0 0]X >= [0 0 0]X = top(proper(X)) [0 0 0] [0 0 0] [1 0 0] [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(zeros()) -> mark(cons(0(),zeros())) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [top](x0) = [0 0 1]x0 [0 0 0] , [ok](x0) = x0 , [proper](x0) = x0 , [0] [nil] = [0] [0], [1 0 0] [0] [s](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 1 1] [length](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [1 1 1] [1] [U62](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [0], [1 0 0] [0] [isNat](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [1 1 1] [1 1 1] [1 0 0] [U61](x0, x1, x2) = [0 1 0]x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 0] [0 0 0] [0 0 0] , [1 0 0] [U52](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [isNatList](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [U51](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [U42](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [0] [isNatIList](x0) = [0 0 0]x0 + [0] [0 1 0] [1], [1 0 0] [1 1 0] [U41](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [0 0 1] , [1 0 0] [U31](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [U21](x0) = [0 0 1]x0 [0 0 0] , [1 0 0] [U11](x0) = [0 0 0]x0 [0 0 0] , [0] [tt] = [1] [1], [mark](x0) = x0 , [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 1 1]x1 [0 0 0] [1 0 0] , [0] [0] = [1] [0], [active](x0) = x0 , [0] [zeros] = [0] [0] orientation: [0] [0] active(zeros()) = [0] >= [0] = mark(cons(0(),zeros())) [0] [0] [1 1 0] [0] [1 1 0] active(U41(tt(),V2)) = [0 0 0]V2 + [0] >= [0 0 0]V2 = mark(U42(isNatIList(V2))) [0 0 1] [1] [0 0 0] [1 0 0] [1 0 0] active(U51(tt(),V2)) = [0 0 0]V2 >= [0 0 0]V2 = mark(U52(isNatList(V2))) [0 0 0] [0 0 0] [1 1 1] [1 0 0] [2] [1 1 1] [1 0 0] [2] active(U61(tt(),L,N)) = [0 0 0]L + [0 0 0]N + [1] >= [0 0 0]L + [0 0 0]N + [1] = mark(U62(isNat(N),L)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 1] [2] [1 1 1] [0] active(U62(tt(),L)) = [0 0 0]L + [1] >= [0 0 0]L + [1] = mark(s(length(L))) [0 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(U21(isNat(V1))) [0 0 0] [1] [0 0 0] [0] [1 0 0] [1 1 1] [0] [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(U41(isNat(V1),V2)) [0 0 0] [0 1 1] [1] [0 0 0] [0 0 1] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(isNatList(cons(V1,V2))) = [0 0 0]V1 + [0 0 0]V2 >= [0 0 0]V1 + [0 0 0]V2 = mark(U51(isNat(V1),V2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [2 1 1] [1 0 0] [2 1 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(U61(isNatList(L),L,N)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(cons(X1,X2)) = [0 0 0]X1 + [0 1 1]X2 >= [0 0 0]X1 + [0 1 1]X2 = cons(active(X1),X2) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] active(U11(X)) = [0 0 0]X >= [0 0 0]X = U11(active(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(U21(X)) = [0 0 1]X >= [0 0 1]X = U21(active(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(U31(X)) = [0 0 0]X >= [0 0 0]X = U31(active(X)) [0 0 0] [0 0 0] [1 0 0] [1 1 0] [1 0 0] [1 1 0] active(U41(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U41(active(X1),X2) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1 0 0] [1 0 0] active(U42(X)) = [0 0 0]X >= [0 0 0]X = U42(active(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(U51(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U51(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(U52(X)) = [0 0 0]X >= [0 0 0]X = U52(active(X)) [0 0 0] [0 0 0] [1 1 1] [1 1 1] [1 0 0] [1 1 1] [1 1 1] [1 0 0] active(U61(X1,X2,X3)) = [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 = U61(active(X1),X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 1 1] [1] [1 0 1] [1 1 1] [1] active(U62(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = U62(active(X1),X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] active(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(active(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1 1 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] cons(mark(X1),X2) = [0 0 0]X1 + [0 1 1]X2 >= [0 0 0]X1 + [0 1 1]X2 = mark(cons(X1,X2)) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] U11(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(U11(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] U21(mark(X)) = [0 0 1]X >= [0 0 1]X = mark(U21(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] U31(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(U31(X)) [0 0 0] [0 0 0] [1 0 0] [1 1 0] [1 0 0] [1 1 0] U41(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(U41(X1,X2)) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1 0 0] [1 0 0] U42(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(U42(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] U51(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(U51(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] U52(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(U52(X)) [0 0 0] [0 0 0] [1 1 1] [1 1 1] [1 0 0] [1 1 1] [1 1 1] [1 0 0] U61(mark(X1),X2,X3) = [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 = mark(U61(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 1 1] [1] [1 0 1] [1 1 1] [1] U62(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = mark(U62(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] s(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = mark(s(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1 1 1] length(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(length(X)) [0 0 0] [0 0 0] [0] [0] proper(zeros()) = [0] >= [0] = ok(zeros()) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(cons(X1,X2)) = [0 0 0]X1 + [0 1 1]X2 >= [0 0 0]X1 + [0 1 1]X2 = cons(proper(X1),proper(X2)) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [0] [0] proper(0()) = [1] >= [1] = ok(0()) [0] [0] [1 0 0] [1 0 0] proper(U11(X)) = [0 0 0]X >= [0 0 0]X = U11(proper(X)) [0 0 0] [0 0 0] [0] [0] proper(tt()) = [1] >= [1] = ok(tt()) [1] [1] [1 0 0] [1 0 0] proper(U21(X)) = [0 0 1]X >= [0 0 1]X = U21(proper(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] proper(U31(X)) = [0 0 0]X >= [0 0 0]X = U31(proper(X)) [0 0 0] [0 0 0] [1 0 0] [1 1 0] [1 0 0] [1 1 0] proper(U41(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U41(proper(X1),proper(X2)) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1 0 0] [1 0 0] proper(U42(X)) = [0 0 0]X >= [0 0 0]X = U42(proper(X)) [0 0 0] [0 0 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)) [0 1 0] [1] [0 1 0] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(U51(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U51(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] proper(U52(X)) = [0 0 0]X >= [0 0 0]X = U52(proper(X)) [0 0 0] [0 0 0] [1 0 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 0] [1 1 1] [1 1 1] [1 0 0] proper(U61(X1,X2,X3)) = [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 = U61(proper(X1),proper(X2),proper(X3)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 1 1] [1] [1 0 1] [1 1 1] [1] proper(U62(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = U62(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] proper(isNat(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = isNat(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] proper(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1 1 1] proper(length(X)) = [0 0 0]X >= [0 0 0]X = length(proper(X)) [0 0 0] [0 0 0] [0] [0] proper(nil()) = [0] >= [0] = ok(nil()) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 1 1]X2 >= [0 0 0]X1 + [0 1 1]X2 = ok(cons(X1,X2)) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] U11(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(U11(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] U21(ok(X)) = [0 0 1]X >= [0 0 1]X = ok(U21(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] U31(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(U31(X)) [0 0 0] [0 0 0] [1 0 0] [1 1 0] [1 0 0] [1 1 0] U41(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(U41(X1,X2)) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1 0 0] [1 0 0] U42(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(U42(X)) [0 0 0] [0 0 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)) [0 1 0] [1] [0 1 0] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] U51(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(U51(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] U52(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(U52(X)) [0 0 0] [0 0 0] [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 1 1] [1 1 1] [1 0 0] [1 1 1] [1 1 1] [1 0 0] U61(ok(X1),ok(X2),ok(X3)) = [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 = ok(U61(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 1 1] [1] [1 0 1] [1 1 1] [1] U62(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = ok(U62(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] isNat(ok(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = ok(isNat(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] s(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(s(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1 1 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] top(mark(X)) = [0 0 1]X >= [0 0 1]X = top(proper(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] top(ok(X)) = [0 0 1]X >= [0 0 1]X = top(active(X)) [0 0 0] [0 0 0] problem: active(zeros()) -> mark(cons(0(),zeros())) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(isNat(0())) -> mark(tt()) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = x0 + 1, [ok](x0) = x0, [proper](x0) = x0, [nil] = 0, [s](x0) = 4x0 + 2, [length](x0) = 5x0 + 1, [U62](x0, x1) = x0 + 2x1, [isNat](x0) = x0, [U61](x0, x1, x2) = x0 + 4x1 + x2, [U52](x0) = x0, [isNatList](x0) = x0, [U51](x0, x1) = x0 + x1, [U42](x0) = x0, [isNatIList](x0) = 2x0, [U41](x0, x1) = x0 + 2x1, [U31](x0) = 4x0, [U21](x0) = 2x0, [U11](x0) = x0 + 6, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U41(tt(),V2)) = 2V2 >= 2V2 = mark(U42(isNatIList(V2))) active(U51(tt(),V2)) = V2 >= V2 = mark(U52(isNatList(V2))) active(U61(tt(),L,N)) = 4L + N >= 2L + N = mark(U62(isNat(N),L)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(V1))) = 4V1 + 2 >= 2V1 = mark(U21(isNat(V1))) active(isNatIList(cons(V1,V2))) = 2V1 + 2V2 >= V1 + 2V2 = mark(U41(isNat(V1),V2)) active(isNatList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U51(isNat(V1),V2)) active(length(cons(N,L))) = 5L + 5N + 1 >= 5L + N = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(active(X1),X2) active(U11(X)) = X + 6 >= X + 6 = U11(active(X)) active(U21(X)) = 2X >= 2X = U21(active(X)) active(U31(X)) = 4X >= 4X = U31(active(X)) active(U41(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = U41(active(X1),X2) active(U42(X)) = X >= X = U42(active(X)) active(U51(X1,X2)) = X1 + X2 >= X1 + X2 = U51(active(X1),X2) active(U52(X)) = X >= X = U52(active(X)) active(U61(X1,X2,X3)) = X1 + 4X2 + X3 >= X1 + 4X2 + X3 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = U62(active(X1),X2) active(s(X)) = 4X + 2 >= 4X + 2 = s(active(X)) active(length(X)) = 5X + 1 >= 5X + 1 = length(active(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(cons(X1,X2)) U11(mark(X)) = X + 6 >= X + 6 = mark(U11(X)) U21(mark(X)) = 2X >= 2X = mark(U21(X)) U31(mark(X)) = 4X >= 4X = mark(U31(X)) U41(mark(X1),X2) = X1 + 2X2 >= X1 + 2X2 = mark(U41(X1,X2)) U42(mark(X)) = X >= X = mark(U42(X)) U51(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(U51(X1,X2)) U52(mark(X)) = X >= X = mark(U52(X)) U61(mark(X1),X2,X3) = X1 + 4X2 + X3 >= X1 + 4X2 + X3 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X1 + 2X2 >= X1 + 2X2 = mark(U62(X1,X2)) s(mark(X)) = 4X + 2 >= 4X + 2 = mark(s(X)) length(mark(X)) = 5X + 1 >= 5X + 1 = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X)) = X + 6 >= X + 6 = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = 2X >= 2X = U21(proper(X)) proper(U31(X)) = 4X >= 4X = U31(proper(X)) proper(U41(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = U41(proper(X1),proper(X2)) proper(U42(X)) = X >= X = U42(proper(X)) proper(isNatIList(X)) = 2X >= 2X = isNatIList(proper(X)) proper(U51(X1,X2)) = X1 + X2 >= X1 + X2 = U51(proper(X1),proper(X2)) proper(U52(X)) = X >= X = U52(proper(X)) proper(isNatList(X)) = X >= X = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X1 + 4X2 + X3 >= X1 + 4X2 + X3 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = U62(proper(X1),proper(X2)) proper(isNat(X)) = X >= X = isNat(proper(X)) proper(s(X)) = 4X + 2 >= 4X + 2 = s(proper(X)) proper(length(X)) = 5X + 1 >= 5X + 1 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X)) = X + 6 >= X + 6 = ok(U11(X)) U21(ok(X)) = 2X >= 2X = ok(U21(X)) U31(ok(X)) = 4X >= 4X = ok(U31(X)) U41(ok(X1),ok(X2)) = X1 + 2X2 >= X1 + 2X2 = ok(U41(X1,X2)) U42(ok(X)) = X >= X = ok(U42(X)) isNatIList(ok(X)) = 2X >= 2X = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U51(X1,X2)) U52(ok(X)) = X >= X = ok(U52(X)) isNatList(ok(X)) = X >= X = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X1 + 4X2 + X3 >= X1 + 4X2 + X3 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X1 + 2X2 >= X1 + 2X2 = ok(U62(X1,X2)) isNat(ok(X)) = X >= X = ok(isNat(X)) s(ok(X)) = 4X + 2 >= 4X + 2 = ok(s(X)) length(ok(X)) = 5X + 1 >= 5X + 1 = ok(length(X)) top(mark(X)) = X + 1 >= X + 1 = top(proper(X)) top(ok(X)) = X + 1 >= X + 1 = top(active(X)) problem: active(zeros()) -> mark(cons(0(),zeros())) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(isNat(0())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(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, [s](x0) = 4x0, [length](x0) = x0, [U62](x0, x1) = x0 + x1, [isNat](x0) = x0, [U61](x0, x1, x2) = 2x0 + x1 + x2 + 2, [U52](x0) = x0, [isNatList](x0) = x0, [U51](x0, x1) = x0 + x1, [U42](x0) = 2x0, [isNatIList](x0) = 2x0, [U41](x0, x1) = x0 + 4x1, [U31](x0) = 4x0, [U21](x0) = 2x0, [U11](x0) = x0, [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(U41(tt(),V2)) = 4V2 >= 4V2 = mark(U42(isNatIList(V2))) active(U51(tt(),V2)) = V2 >= V2 = mark(U52(isNatList(V2))) active(U61(tt(),L,N)) = L + N + 2 >= L + N = mark(U62(isNat(N),L)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 8V1 + 8V2 >= V1 + 4V2 = mark(U41(isNat(V1),V2)) active(isNatList(cons(V1,V2))) = 4V1 + 4V2 >= V1 + V2 = mark(U51(isNat(V1),V2)) active(cons(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = cons(active(X1),X2) active(U11(X)) = X >= X = U11(active(X)) active(U21(X)) = 2X >= 2X = U21(active(X)) active(U31(X)) = 4X >= 4X = U31(active(X)) active(U41(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = U41(active(X1),X2) active(U42(X)) = 2X >= 2X = U42(active(X)) active(U51(X1,X2)) = X1 + X2 >= X1 + X2 = U51(active(X1),X2) active(U52(X)) = X >= X = U52(active(X)) active(U61(X1,X2,X3)) = 2X1 + X2 + X3 + 2 >= 2X1 + X2 + X3 + 2 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X1 + X2 >= X1 + X2 = U62(active(X1),X2) active(s(X)) = 4X >= 4X = s(active(X)) active(length(X)) = X >= X = length(active(X)) cons(mark(X1),X2) = 4X1 + 4X2 >= 4X1 + 4X2 = mark(cons(X1,X2)) U11(mark(X)) = X >= X = mark(U11(X)) U21(mark(X)) = 2X >= 2X = mark(U21(X)) U31(mark(X)) = 4X >= 4X = mark(U31(X)) U41(mark(X1),X2) = X1 + 4X2 >= X1 + 4X2 = mark(U41(X1,X2)) U42(mark(X)) = 2X >= 2X = mark(U42(X)) U51(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(U51(X1,X2)) U52(mark(X)) = X >= X = mark(U52(X)) U61(mark(X1),X2,X3) = 2X1 + X2 + X3 + 2 >= 2X1 + X2 + X3 + 2 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(U62(X1,X2)) s(mark(X)) = 4X >= 4X = mark(s(X)) length(mark(X)) = X >= X = mark(length(X)) 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(X)) = X >= X = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = 2X >= 2X = U21(proper(X)) proper(U31(X)) = 4X >= 4X = U31(proper(X)) proper(U41(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = U41(proper(X1),proper(X2)) proper(U42(X)) = 2X >= 2X = U42(proper(X)) proper(isNatIList(X)) = 2X >= 2X = isNatIList(proper(X)) proper(U51(X1,X2)) = X1 + X2 >= X1 + X2 = U51(proper(X1),proper(X2)) proper(U52(X)) = X >= X = U52(proper(X)) proper(isNatList(X)) = X >= X = isNatList(proper(X)) proper(U61(X1,X2,X3)) = 2X1 + X2 + X3 + 2 >= 2X1 + X2 + X3 + 2 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X1 + X2 >= X1 + X2 = U62(proper(X1),proper(X2)) proper(isNat(X)) = X >= X = isNat(proper(X)) proper(s(X)) = 4X >= 4X = s(proper(X)) proper(length(X)) = X >= X = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = ok(cons(X1,X2)) U11(ok(X)) = X >= X = ok(U11(X)) U21(ok(X)) = 2X >= 2X = ok(U21(X)) U31(ok(X)) = 4X >= 4X = ok(U31(X)) U41(ok(X1),ok(X2)) = X1 + 4X2 >= X1 + 4X2 = ok(U41(X1,X2)) U42(ok(X)) = 2X >= 2X = ok(U42(X)) isNatIList(ok(X)) = 2X >= 2X = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U51(X1,X2)) U52(ok(X)) = X >= X = ok(U52(X)) isNatList(ok(X)) = X >= X = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = 2X1 + X2 + X3 + 2 >= 2X1 + X2 + X3 + 2 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U62(X1,X2)) isNat(ok(X)) = X >= X = ok(isNat(X)) s(ok(X)) = 4X >= 4X = ok(s(X)) length(ok(X)) = X >= X = ok(length(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(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(isNat(0())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [top](x0) = [0 0 0]x0 [0 0 0] , [ok](x0) = x0 , [proper](x0) = x0 , [0] [nil] = [0] [0], [1 0 0] [0] [s](x0) = [0 0 0]x0 + [1] [0 0 0] [1], [1 0 0] [0] [length](x0) = [0 0 0]x0 + [1] [0 0 0] [1], [1 0 0] [1 0 0] [0] [U62](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [1], [1 0 1] [isNat](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [1 0 0] [0] [U61](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 1 0]x2 + [1] [0 0 0] [0 0 0] [0 1 0] [1], [1 0 0] [0] [U52](x0) = [0 0 0]x0 + [1] [0 0 0] [1], [1 0 1] [0] [isNatList](x0) = [1 0 1]x0 + [1] [0 0 0] [0], [1 0 0] [1 0 1] [U51](x0, x1) = [1 0 0]x0 + [0 0 0]x1 [1 0 0] [0 0 0] , [1 0 0] [0] [U42](x0) = [0 0 0]x0 + [1] [0 0 0] [1], [1 0 1] [1] [isNatIList](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 0] [1 0 1] [0] [U41](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [1], [1 0 0] [0] [U31](x0) = [0 0 0]x0 + [1] [0 0 0] [1], [1 0 0] [0] [U21](x0) = [0 0 0]x0 + [1] [0 0 0] [1], [1 0 0] [0] [U11](x0) = [1 1 0]x0 + [1] [1 0 1] [1], [1] [tt] = [0] [0], [1 0 0] [mark](x0) = [0 0 0]x0 [0 1 0] , [1 0 1] [cons](x0, x1) = x0 + [0 0 0]x1 [0 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 0 1] [1] [1 0 1] [1] active(U41(tt(),V2)) = [0 0 0]V2 + [1] >= [0 0 0]V2 + [0] = mark(U42(isNatIList(V2))) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1] [1 0 1] [0] active(U51(tt(),V2)) = [0 0 0]V2 + [1] >= [0 0 0]V2 + [0] = mark(U52(isNatList(V2))) [0 0 0] [1] [0 0 0] [1] [1] [1] active(isNat(0())) = [0] >= [0] = mark(tt()) [0] [0] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [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(U41(isNat(V1),V2)) [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] active(isNatList(cons(V1,V2))) = [0 0 0]V1 + [0 0 0]V2 + [0] >= [0 0 0]V1 + [0 0 0]V2 = mark(U51(isNat(V1),V2)) [1 0 1] [1 0 1] [1] [1 0 1] [0 0 0] [1 0 0] [1 0 1] [1 0 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) [0 1 0] [0 0 0] [0 1 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] active(U11(X)) = [1 0 1]X + [1] >= [1 0 1]X + [1] = U11(active(X)) [1 1 0] [1] [1 1 0] [1] [1 0 0] [0] [1 0 0] [0] active(U21(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = U21(active(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] active(U31(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = U31(active(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 1] [0] active(U41(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = U41(active(X1),X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] active(U42(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = U42(active(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 1] [1 0 0] [1 0 1] active(U51(X1,X2)) = [1 0 0]X1 + [0 0 0]X2 >= [1 0 0]X1 + [0 0 0]X2 = U51(active(X1),X2) [1 0 0] [0 0 0] [1 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] active(U52(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = U52(active(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] active(U61(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 1 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 1 0]X3 + [1] = U61(active(X1),X2,X3) [0 0 0] [0 0 0] [0 1 0] [1] [0 0 0] [0 0 0] [0 1 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] active(U62(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = U62(active(X1),X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] active(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(active(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] active(length(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = length(active(X)) [0 0 0] [1] [0 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 = mark(cons(X1,X2)) [0 1 0] [0 0 0] [0 1 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] U11(mark(X)) = [1 0 0]X + [1] >= [0 0 0]X + [0] = mark(U11(X)) [1 1 0] [1] [1 1 0] [1] [1 0 0] [0] [1 0 0] [0] U21(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = mark(U21(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] U31(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = mark(U31(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 1] [0] U41(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0] = mark(U41(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] U42(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = mark(U42(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 1] [1 0 0] [1 0 1] U51(mark(X1),X2) = [1 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(U51(X1,X2)) [1 0 0] [0 0 0] [1 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] U52(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = mark(U52(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] U61(mark(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 1 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = mark(U61(X1,X2,X3)) [0 0 0] [0 0 0] [0 1 0] [1] [0 0 0] [0 0 0] [0 1 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] U62(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0] = mark(U62(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] s(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = mark(s(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] length(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = mark(length(X)) [0 0 0] [1] [0 0 0] [1] [0] [0] proper(zeros()) = [0] >= [0] = ok(zeros()) [0] [0] [1 0 1] [1 0 1] proper(cons(X1,X2)) = X1 + [0 0 0]X2 >= X1 + [0 0 0]X2 = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [1] [1] [1 0 0] [0] [1 0 0] [0] proper(U11(X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = U11(proper(X)) [1 0 1] [1] [1 0 1] [1] [1] [1] proper(tt()) = [0] >= [0] = ok(tt()) [0] [0] [1 0 0] [0] [1 0 0] [0] proper(U21(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = U21(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] proper(U31(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = U31(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 1] [0] proper(U41(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = U41(proper(X1),proper(X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] proper(U42(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = U42(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1] [1 0 1] [1] proper(isNatIList(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = isNatIList(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] proper(U51(X1,X2)) = [1 0 0]X1 + [0 0 0]X2 >= [1 0 0]X1 + [0 0 0]X2 = U51(proper(X1),proper(X2)) [1 0 0] [0 0 0] [1 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] proper(U52(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = U52(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [0] [1 0 1] [0] proper(isNatList(X)) = [1 0 1]X + [1] >= [1 0 1]X + [1] = isNatList(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] proper(U61(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 1 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 1 0]X3 + [1] = U61(proper(X1),proper(X2),proper(X3)) [0 0 0] [0 0 0] [0 1 0] [1] [0 0 0] [0 0 0] [0 1 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] proper(U62(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = U62(proper(X1),proper(X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 1] [1 0 1] proper(isNat(X)) = [0 0 0]X >= [0 0 0]X = isNat(proper(X)) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] proper(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] proper(length(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = length(proper(X)) [0 0 0] [1] [0 0 0] [1] [0] [0] proper(nil()) = [0] >= [0] = ok(nil()) [0] [0] [1 0 1] [1 0 1] cons(ok(X1),ok(X2)) = X1 + [0 0 0]X2 >= X1 + [0 0 0]X2 = ok(cons(X1,X2)) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] U11(ok(X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = ok(U11(X)) [1 0 1] [1] [1 0 1] [1] [1 0 0] [0] [1 0 0] [0] U21(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(U21(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] U31(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(U31(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 1] [0] U41(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = ok(U41(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] U42(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(U42(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1] [1 0 1] [1] isNatIList(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(isNatIList(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] U51(ok(X1),ok(X2)) = [1 0 0]X1 + [0 0 0]X2 >= [1 0 0]X1 + [0 0 0]X2 = ok(U51(X1,X2)) [1 0 0] [0 0 0] [1 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] U52(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(U52(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [0] [1 0 1] [0] isNatList(ok(X)) = [1 0 1]X + [1] >= [1 0 1]X + [1] = ok(isNatList(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] U61(ok(X1),ok(X2),ok(X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 1 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 1 0]X3 + [1] = ok(U61(X1,X2,X3)) [0 0 0] [0 0 0] [0 1 0] [1] [0 0 0] [0 0 0] [0 1 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] U62(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = ok(U62(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 1] [1 0 1] isNat(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(isNat(X)) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] s(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(s(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] length(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(length(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] top(mark(X)) = [0 0 0]X >= [0 0 0]X = top(proper(X)) [0 0 0] [0 0 0] [1 0 0] [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(zeros()) -> mark(cons(0(),zeros())) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(isNat(0())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(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, [s](x0) = 2x0 + 3, [length](x0) = x0, [U62](x0, x1) = 4x0 + x1, [isNat](x0) = x0 + 1, [U61](x0, x1, x2) = 2x0 + x1 + x2 + 1, [U52](x0) = 4x0, [isNatList](x0) = 4x0 + 1, [U51](x0, x1) = x0 + 4x1, [U42](x0) = 2x0, [isNatIList](x0) = x0, [U41](x0, x1) = 4x0 + 2x1, [U31](x0) = 4x0, [U21](x0) = 4x0 + 4, [U11](x0) = 4x0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U41(tt(),V2)) = 2V2 >= 2V2 = mark(U42(isNatIList(V2))) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 4V1 + 4V2 + 1 >= V1 + 4V2 + 1 = mark(U51(isNat(V1),V2)) active(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(active(X1),X2) active(U11(X)) = 4X >= 4X = U11(active(X)) active(U21(X)) = 4X + 4 >= 4X + 4 = U21(active(X)) active(U31(X)) = 4X >= 4X = U31(active(X)) active(U41(X1,X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = U41(active(X1),X2) active(U42(X)) = 2X >= 2X = U42(active(X)) active(U51(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = U51(active(X1),X2) active(U52(X)) = 4X >= 4X = U52(active(X)) active(U61(X1,X2,X3)) = 2X1 + X2 + X3 + 1 >= 2X1 + X2 + X3 + 1 = U61(active(X1),X2,X3) active(U62(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = U62(active(X1),X2) active(s(X)) = 2X + 3 >= 2X + 3 = s(active(X)) active(length(X)) = X >= X = length(active(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(cons(X1,X2)) U11(mark(X)) = 4X >= 4X = mark(U11(X)) U21(mark(X)) = 4X + 4 >= 4X + 4 = mark(U21(X)) U31(mark(X)) = 4X >= 4X = mark(U31(X)) U41(mark(X1),X2) = 4X1 + 2X2 >= 4X1 + 2X2 = mark(U41(X1,X2)) U42(mark(X)) = 2X >= 2X = mark(U42(X)) U51(mark(X1),X2) = X1 + 4X2 >= X1 + 4X2 = mark(U51(X1,X2)) U52(mark(X)) = 4X >= 4X = mark(U52(X)) U61(mark(X1),X2,X3) = 2X1 + X2 + X3 + 1 >= 2X1 + X2 + X3 + 1 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = 4X1 + X2 >= 4X1 + X2 = mark(U62(X1,X2)) s(mark(X)) = 2X + 3 >= 2X + 3 = mark(s(X)) length(mark(X)) = X >= X = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X)) = 4X >= 4X = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = 4X + 4 >= 4X + 4 = U21(proper(X)) proper(U31(X)) = 4X >= 4X = U31(proper(X)) proper(U41(X1,X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = U41(proper(X1),proper(X2)) proper(U42(X)) = 2X >= 2X = U42(proper(X)) proper(isNatIList(X)) = X >= X = isNatIList(proper(X)) proper(U51(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = U51(proper(X1),proper(X2)) proper(U52(X)) = 4X >= 4X = U52(proper(X)) proper(isNatList(X)) = 4X + 1 >= 4X + 1 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = 2X1 + X2 + X3 + 1 >= 2X1 + X2 + X3 + 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = U62(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(s(X)) = 2X + 3 >= 2X + 3 = s(proper(X)) proper(length(X)) = X >= X = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X)) = 4X >= 4X = ok(U11(X)) U21(ok(X)) = 4X + 4 >= 4X + 4 = ok(U21(X)) U31(ok(X)) = 4X >= 4X = ok(U31(X)) U41(ok(X1),ok(X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = ok(U41(X1,X2)) U42(ok(X)) = 2X >= 2X = ok(U42(X)) isNatIList(ok(X)) = X >= X = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X1 + 4X2 >= X1 + 4X2 = ok(U51(X1,X2)) U52(ok(X)) = 4X >= 4X = ok(U52(X)) isNatList(ok(X)) = 4X + 1 >= 4X + 1 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = 2X1 + X2 + X3 + 1 >= 2X1 + X2 + X3 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = 4X1 + X2 >= 4X1 + X2 = ok(U62(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = 2X + 3 >= 2X + 3 = ok(s(X)) length(ok(X)) = X >= X = ok(length(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(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = 4x0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 0, [s](x0) = x0 + 6, [length](x0) = 4x0, [U62](x0, x1) = 2x0 + 2x1, [isNat](x0) = 2x0 + 2, [U61](x0, x1, x2) = x0 + 2x1 + 2x2 + 6, [U52](x0) = x0, [isNatList](x0) = x0 + 5, [U51](x0, x1) = 2x0 + x1, [U42](x0) = x0 + 2, [isNatIList](x0) = x0 + 6, [U41](x0, x1) = x0 + x1 + 5, [U31](x0) = x0, [U21](x0) = 4x0, [U11](x0) = x0 + 4, [tt] = 4, [mark](x0) = x0, [cons](x0, x1) = 4x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 4 orientation: active(zeros()) = 4 >= 4 = mark(cons(0(),zeros())) active(U41(tt(),V2)) = V2 + 9 >= V2 + 8 = mark(U42(isNatIList(V2))) active(isNatList(cons(V1,V2))) = 4V1 + V2 + 5 >= 4V1 + V2 + 4 = mark(U51(isNat(V1),V2)) active(cons(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = cons(active(X1),X2) active(U11(X)) = X + 4 >= X + 4 = U11(active(X)) active(U21(X)) = 4X >= 4X = U21(active(X)) active(U31(X)) = X >= X = U31(active(X)) active(U41(X1,X2)) = X1 + X2 + 5 >= X1 + X2 + 5 = U41(active(X1),X2) active(U42(X)) = X + 2 >= X + 2 = U42(active(X)) active(U51(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = U51(active(X1),X2) active(U52(X)) = X >= X = U52(active(X)) active(U61(X1,X2,X3)) = X1 + 2X2 + 2X3 + 6 >= X1 + 2X2 + 2X3 + 6 = U61(active(X1),X2,X3) active(U62(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = U62(active(X1),X2) active(s(X)) = X + 6 >= X + 6 = s(active(X)) active(length(X)) = 4X >= 4X = length(active(X)) cons(mark(X1),X2) = 4X1 + X2 >= 4X1 + X2 = mark(cons(X1,X2)) U11(mark(X)) = X + 4 >= X + 4 = mark(U11(X)) U21(mark(X)) = 4X >= 4X = mark(U21(X)) U31(mark(X)) = X >= X = mark(U31(X)) U41(mark(X1),X2) = X1 + X2 + 5 >= X1 + X2 + 5 = mark(U41(X1,X2)) U42(mark(X)) = X + 2 >= X + 2 = mark(U42(X)) U51(mark(X1),X2) = 2X1 + X2 >= 2X1 + X2 = mark(U51(X1,X2)) U52(mark(X)) = X >= X = mark(U52(X)) U61(mark(X1),X2,X3) = X1 + 2X2 + 2X3 + 6 >= X1 + 2X2 + 2X3 + 6 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = 2X1 + 2X2 >= 2X1 + 2X2 = mark(U62(X1,X2)) s(mark(X)) = X + 6 >= X + 6 = mark(s(X)) length(mark(X)) = 4X >= 4X = mark(length(X)) proper(zeros()) = 4 >= 4 = ok(zeros()) proper(cons(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X)) = X + 4 >= X + 4 = U11(proper(X)) proper(tt()) = 4 >= 4 = ok(tt()) proper(U21(X)) = 4X >= 4X = U21(proper(X)) proper(U31(X)) = X >= X = U31(proper(X)) proper(U41(X1,X2)) = X1 + X2 + 5 >= X1 + X2 + 5 = U41(proper(X1),proper(X2)) proper(U42(X)) = X + 2 >= X + 2 = U42(proper(X)) proper(isNatIList(X)) = X + 6 >= X + 6 = isNatIList(proper(X)) proper(U51(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = U51(proper(X1),proper(X2)) proper(U52(X)) = X >= X = U52(proper(X)) proper(isNatList(X)) = X + 5 >= X + 5 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X1 + 2X2 + 2X3 + 6 >= X1 + 2X2 + 2X3 + 6 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 2X + 2 >= 2X + 2 = isNat(proper(X)) proper(s(X)) = X + 6 >= X + 6 = s(proper(X)) proper(length(X)) = 4X >= 4X = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 4X1 + X2 >= 4X1 + X2 = ok(cons(X1,X2)) U11(ok(X)) = X + 4 >= X + 4 = ok(U11(X)) U21(ok(X)) = 4X >= 4X = ok(U21(X)) U31(ok(X)) = X >= X = ok(U31(X)) U41(ok(X1),ok(X2)) = X1 + X2 + 5 >= X1 + X2 + 5 = ok(U41(X1,X2)) U42(ok(X)) = X + 2 >= X + 2 = ok(U42(X)) isNatIList(ok(X)) = X + 6 >= X + 6 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = 2X1 + X2 >= 2X1 + X2 = ok(U51(X1,X2)) U52(ok(X)) = X >= X = ok(U52(X)) isNatList(ok(X)) = X + 5 >= X + 5 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X1 + 2X2 + 2X3 + 6 >= X1 + 2X2 + 2X3 + 6 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = ok(U62(X1,X2)) isNat(ok(X)) = 2X + 2 >= 2X + 2 = ok(isNat(X)) s(ok(X)) = X + 6 >= X + 6 = ok(s(X)) length(ok(X)) = 4X >= 4X = ok(length(X)) top(mark(X)) = 4X >= 4X = top(proper(X)) top(ok(X)) = 4X >= 4X = top(active(X)) problem: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) DP Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(U11(X)) -> active#(X) active#(U11(X)) -> U11#(active(X)) active#(U21(X)) -> active#(X) active#(U21(X)) -> U21#(active(X)) active#(U31(X)) -> active#(X) active#(U31(X)) -> U31#(active(X)) active#(U41(X1,X2)) -> active#(X1) active#(U41(X1,X2)) -> U41#(active(X1),X2) active#(U42(X)) -> active#(X) active#(U42(X)) -> U42#(active(X)) active#(U51(X1,X2)) -> active#(X1) active#(U51(X1,X2)) -> U51#(active(X1),X2) active#(U52(X)) -> active#(X) active#(U52(X)) -> U52#(active(X)) active#(U61(X1,X2,X3)) -> active#(X1) active#(U61(X1,X2,X3)) -> U61#(active(X1),X2,X3) active#(U62(X1,X2)) -> active#(X1) active#(U62(X1,X2)) -> U62#(active(X1),X2) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(length(X)) -> active#(X) active#(length(X)) -> length#(active(X)) cons#(mark(X1),X2) -> cons#(X1,X2) U11#(mark(X)) -> U11#(X) U21#(mark(X)) -> U21#(X) U31#(mark(X)) -> U31#(X) U41#(mark(X1),X2) -> U41#(X1,X2) U42#(mark(X)) -> U42#(X) U51#(mark(X1),X2) -> U51#(X1,X2) U52#(mark(X)) -> U52#(X) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U62#(mark(X1),X2) -> U62#(X1,X2) s#(mark(X)) -> s#(X) length#(mark(X)) -> length#(X) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(U11(X)) -> proper#(X) proper#(U11(X)) -> U11#(proper(X)) proper#(U21(X)) -> proper#(X) proper#(U21(X)) -> U21#(proper(X)) proper#(U31(X)) -> proper#(X) proper#(U31(X)) -> U31#(proper(X)) proper#(U41(X1,X2)) -> proper#(X2) proper#(U41(X1,X2)) -> proper#(X1) proper#(U41(X1,X2)) -> U41#(proper(X1),proper(X2)) proper#(U42(X)) -> proper#(X) proper#(U42(X)) -> U42#(proper(X)) proper#(isNatIList(X)) -> proper#(X) proper#(isNatIList(X)) -> isNatIList#(proper(X)) proper#(U51(X1,X2)) -> proper#(X2) proper#(U51(X1,X2)) -> proper#(X1) proper#(U51(X1,X2)) -> U51#(proper(X1),proper(X2)) proper#(U52(X)) -> proper#(X) proper#(U52(X)) -> U52#(proper(X)) proper#(isNatList(X)) -> proper#(X) proper#(isNatList(X)) -> isNatList#(proper(X)) proper#(U61(X1,X2,X3)) -> proper#(X3) proper#(U61(X1,X2,X3)) -> proper#(X2) proper#(U61(X1,X2,X3)) -> proper#(X1) proper#(U61(X1,X2,X3)) -> U61#(proper(X1),proper(X2),proper(X3)) proper#(U62(X1,X2)) -> proper#(X2) proper#(U62(X1,X2)) -> proper#(X1) proper#(U62(X1,X2)) -> U62#(proper(X1),proper(X2)) proper#(isNat(X)) -> proper#(X) proper#(isNat(X)) -> isNat#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) U11#(ok(X)) -> U11#(X) U21#(ok(X)) -> U21#(X) U31#(ok(X)) -> U31#(X) U41#(ok(X1),ok(X2)) -> U41#(X1,X2) U42#(ok(X)) -> U42#(X) isNatIList#(ok(X)) -> isNatIList#(X) U51#(ok(X1),ok(X2)) -> U51#(X1,X2) U52#(ok(X)) -> U52#(X) isNatList#(ok(X)) -> isNatList#(X) U61#(ok(X1),ok(X2),ok(X3)) -> U61#(X1,X2,X3) U62#(ok(X1),ok(X2)) -> U62#(X1,X2) isNat#(ok(X)) -> isNat#(X) s#(ok(X)) -> s#(X) length#(ok(X)) -> length#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(U11(X)) -> active#(X) active#(U11(X)) -> U11#(active(X)) active#(U21(X)) -> active#(X) active#(U21(X)) -> U21#(active(X)) active#(U31(X)) -> active#(X) active#(U31(X)) -> U31#(active(X)) active#(U41(X1,X2)) -> active#(X1) active#(U41(X1,X2)) -> U41#(active(X1),X2) active#(U42(X)) -> active#(X) active#(U42(X)) -> U42#(active(X)) active#(U51(X1,X2)) -> active#(X1) active#(U51(X1,X2)) -> U51#(active(X1),X2) active#(U52(X)) -> active#(X) active#(U52(X)) -> U52#(active(X)) active#(U61(X1,X2,X3)) -> active#(X1) active#(U61(X1,X2,X3)) -> U61#(active(X1),X2,X3) active#(U62(X1,X2)) -> active#(X1) active#(U62(X1,X2)) -> U62#(active(X1),X2) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(length(X)) -> active#(X) active#(length(X)) -> length#(active(X)) cons#(mark(X1),X2) -> cons#(X1,X2) U11#(mark(X)) -> U11#(X) U21#(mark(X)) -> U21#(X) U31#(mark(X)) -> U31#(X) U41#(mark(X1),X2) -> U41#(X1,X2) U42#(mark(X)) -> U42#(X) U51#(mark(X1),X2) -> U51#(X1,X2) U52#(mark(X)) -> U52#(X) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U62#(mark(X1),X2) -> U62#(X1,X2) s#(mark(X)) -> s#(X) length#(mark(X)) -> length#(X) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(U11(X)) -> proper#(X) proper#(U11(X)) -> U11#(proper(X)) proper#(U21(X)) -> proper#(X) proper#(U21(X)) -> U21#(proper(X)) proper#(U31(X)) -> proper#(X) proper#(U31(X)) -> U31#(proper(X)) proper#(U41(X1,X2)) -> proper#(X2) proper#(U41(X1,X2)) -> proper#(X1) proper#(U41(X1,X2)) -> U41#(proper(X1),proper(X2)) proper#(U42(X)) -> proper#(X) proper#(U42(X)) -> U42#(proper(X)) proper#(isNatIList(X)) -> proper#(X) proper#(isNatIList(X)) -> isNatIList#(proper(X)) proper#(U51(X1,X2)) -> proper#(X2) proper#(U51(X1,X2)) -> proper#(X1) proper#(U51(X1,X2)) -> U51#(proper(X1),proper(X2)) proper#(U52(X)) -> proper#(X) proper#(U52(X)) -> U52#(proper(X)) proper#(isNatList(X)) -> proper#(X) proper#(isNatList(X)) -> isNatList#(proper(X)) proper#(U61(X1,X2,X3)) -> proper#(X3) proper#(U61(X1,X2,X3)) -> proper#(X2) proper#(U61(X1,X2,X3)) -> proper#(X1) proper#(U61(X1,X2,X3)) -> U61#(proper(X1),proper(X2),proper(X3)) proper#(U62(X1,X2)) -> proper#(X2) proper#(U62(X1,X2)) -> proper#(X1) proper#(U62(X1,X2)) -> U62#(proper(X1),proper(X2)) proper#(isNat(X)) -> proper#(X) proper#(isNat(X)) -> isNat#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) U11#(ok(X)) -> U11#(X) U21#(ok(X)) -> U21#(X) U31#(ok(X)) -> U31#(X) U41#(ok(X1),ok(X2)) -> U41#(X1,X2) U42#(ok(X)) -> U42#(X) isNatIList#(ok(X)) -> isNatIList#(X) U51#(ok(X1),ok(X2)) -> U51#(X1,X2) U52#(ok(X)) -> U52#(X) isNatList#(ok(X)) -> isNatList#(X) U61#(ok(X1),ok(X2),ok(X3)) -> U61#(X1,X2,X3) U62#(ok(X1),ok(X2)) -> U62#(X1,X2) isNat#(ok(X)) -> isNat#(X) s#(ok(X)) -> s#(X) length#(ok(X)) -> length#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: ... SCC Processor: #sccs: 18 #rules: 62 #arcs: 1229/8464 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Usable Rule Processor: DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U11(ok(X)) -> ok(U11(X)) U21(mark(X)) -> mark(U21(X)) U21(ok(X)) -> ok(U21(X)) U31(mark(X)) -> mark(U31(X)) U31(ok(X)) -> ok(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U42(ok(X)) -> ok(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U52(ok(X)) -> ok(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) length(mark(X)) -> mark(length(X)) length(ok(X)) -> ok(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) Semantic Labeling Processor: dimension: 1 usable rules: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U11(ok(X)) -> ok(U11(X)) U21(mark(X)) -> mark(U21(X)) U21(ok(X)) -> ok(U21(X)) U31(mark(X)) -> mark(U31(X)) U31(ok(X)) -> ok(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U42(ok(X)) -> ok(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U52(ok(X)) -> ok(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) length(mark(X)) -> mark(length(X)) length(ok(X)) -> ok(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) interpretation: [ok](x0) = 1, [proper](x0) = 0, [nil] = 0, [s](x0) = 0, [length](x0) = 0, [U62](x0, x1) = 0, [isNat](x0) = 0, [U61](x0, x1, x2) = 0, [U52](x0) = 0, [isNatList](x0) = 0, [U51](x0, x1) = 1, [U42](x0) = 1, [isNatIList](x0) = 0, [U41](x0, x1) = 0, [U31](x0) = 0, [U21](x0) = 0, [U11](x0) = 0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 0, [0] = 0, [active](x0) = 1, [zeros] = 0 labeled: zeros 0 usable (for model): zeros 0 argument filtering: pi(zeros) = [] pi(active) = 0 pi(0) = [] pi(cons) = 0 pi(mark) = [0] pi(tt) = [] pi(U11) = 0 pi(U21) = 0 pi(U31) = 0 pi(U41) = 0 pi(isNatIList) = 0 pi(U42) = 0 pi(U51) = 0 pi(isNatList) = 0 pi(U52) = 0 pi(U61) = 0 pi(isNat) = 0 pi(U62) = 0 pi(length) = 0 pi(s) = 0 pi(nil) = [] pi(proper) = 0 pi(ok) = 0 pi(top#) = 0 precedence: zeros > 0 > top# ~ ok ~ proper ~ nil ~ s ~ length ~ U62 ~ isNat ~ U61 ~ U52 ~ isNatList ~ U51 ~ U42 ~ isNatIList ~ U41 ~ U31 ~ U21 ~ U11 ~ tt ~ mark ~ cons ~ active problem: DPs: top#(ok(X)) -> top#(active(X)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U11(ok(X)) -> ok(U11(X)) U21(mark(X)) -> mark(U21(X)) U21(ok(X)) -> ok(U21(X)) U31(mark(X)) -> mark(U31(X)) U31(ok(X)) -> ok(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U42(ok(X)) -> ok(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U52(ok(X)) -> ok(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) length(mark(X)) -> mark(length(X)) length(ok(X)) -> ok(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) isNatIList(ok(X)) -> ok(isNatIList(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) Restore Modifier: DPs: top#(ok(X)) -> top#(active(X)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Bounds Processor: bound: 0 enrichment: match-dp automaton: final states: {1} transitions: U310(3) -> 3* U310(6) -> 6* U210(3) -> 3* U210(6) -> 6* f1960() -> 2* U110(3) -> 3* U110(6) -> 6* cons0(6,2) -> 6* cons0(3,2) -> 3* cons0(5,4) -> 6* top{#,0}(3) -> 1* mark0(6) -> 3* active0(2) -> 3* length0(3) -> 3* length0(6) -> 6* 00() -> 5* s0(3) -> 3* s0(6) -> 6* zeros0() -> 4* U620(6,2) -> 6* U620(3,2) -> 3* U610(3,2,2) -> 3* U610(6,2,2) -> 6* U520(3) -> 3* U520(6) -> 6* U510(6,2) -> 6* U510(3,2) -> 3* U420(3) -> 3* U420(6) -> 6* U410(6,2) -> 6* U410(3,2) -> 3* problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: active#(cons(X1,X2)) -> active#(X1) active#(U11(X)) -> active#(X) active#(U21(X)) -> active#(X) active#(U31(X)) -> active#(X) active#(U41(X1,X2)) -> active#(X1) active#(U42(X)) -> active#(X) active#(U51(X1,X2)) -> active#(X1) active#(U52(X)) -> active#(X) active#(U61(X1,X2,X3)) -> active#(X1) active#(U62(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) active#(length(X)) -> active#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(active#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(U11(X)) -> proper#(X) proper#(U21(X)) -> proper#(X) proper#(U31(X)) -> proper#(X) proper#(U41(X1,X2)) -> proper#(X2) proper#(U41(X1,X2)) -> proper#(X1) proper#(U42(X)) -> proper#(X) proper#(isNatIList(X)) -> proper#(X) proper#(U51(X1,X2)) -> proper#(X2) proper#(U51(X1,X2)) -> proper#(X1) proper#(U52(X)) -> proper#(X) proper#(isNatList(X)) -> proper#(X) proper#(U61(X1,X2,X3)) -> proper#(X3) proper#(U61(X1,X2,X3)) -> proper#(X2) proper#(U61(X1,X2,X3)) -> proper#(X1) proper#(U62(X1,X2)) -> proper#(X2) proper#(U62(X1,X2)) -> proper#(X1) proper#(isNat(X)) -> proper#(X) proper#(s(X)) -> proper#(X) proper#(length(X)) -> proper#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(proper#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: length#(mark(X)) -> length#(X) length#(ok(X)) -> length#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(length#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: s#(mark(X)) -> s#(X) s#(ok(X)) -> s#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(s#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: isNat#(ok(X)) -> isNat#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(isNat#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U62#(mark(X1),X2) -> U62#(X1,X2) U62#(ok(X1),ok(X2)) -> U62#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U62#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U61#(ok(X1),ok(X2),ok(X3)) -> U61#(X1,X2,X3) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U61#) = 1 problem: DPs: U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U61#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: isNatList#(ok(X)) -> isNatList#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(isNatList#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U52#(mark(X)) -> U52#(X) U52#(ok(X)) -> U52#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U52#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U51#(mark(X1),X2) -> U51#(X1,X2) U51#(ok(X1),ok(X2)) -> U51#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U51#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: isNatIList#(ok(X)) -> isNatIList#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(isNatIList#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U42#(mark(X)) -> U42#(X) U42#(ok(X)) -> U42#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U42#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U41#(mark(X1),X2) -> U41#(X1,X2) U41#(ok(X1),ok(X2)) -> U41#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U41#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U31#(mark(X)) -> U31#(X) U31#(ok(X)) -> U31#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U31#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U21#(mark(X)) -> U21#(X) U21#(ok(X)) -> U21#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U21#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U11#(mark(X)) -> U11#(X) U11#(ok(X)) -> U11#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U11#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: cons#(mark(X1),X2) -> cons#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(cons#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed