MAYBE Time: 2.456442 TRS: { active and(X1, X2) -> and(X1, active X2), active and(X1, X2) -> and(active X1, X2), active and(tt(), T) -> mark T, active isNatList cons(N, L) -> mark and(isNat N, isNatList L), active isNatList nil() -> mark tt(), active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL), active isNatIList IL -> mark isNatList IL, active isNatIList zeros() -> mark tt(), active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL), active isNat 0() -> mark tt(), active isNat s N -> mark isNat N, active isNat length L -> mark isNatList L, active s X -> s active X, active length X -> length active X, active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L), active zeros() -> mark cons(0(), zeros()), active cons(X1, X2) -> cons(active X1, X2), active take(X1, X2) -> take(X1, active X2), active take(X1, X2) -> take(active X1, X2), active take(0(), IL) -> mark uTake1 isNatIList IL, active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), active uTake1 X -> uTake1 active X, active uTake1 tt() -> mark nil(), active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4), active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)), active uLength(X1, X2) -> uLength(active X1, X2), active uLength(tt(), L) -> mark s length L, and(X1, mark X2) -> mark and(X1, X2), and(mark X1, X2) -> mark and(X1, X2), and(ok X1, ok X2) -> ok and(X1, X2), isNatList ok X -> ok isNatList X, isNatIList ok X -> ok isNatIList X, isNat ok X -> ok isNat X, s mark X -> mark s X, s ok X -> ok s X, length mark X -> mark length X, length ok X -> ok length X, cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), take(X1, mark X2) -> mark take(X1, X2), take(mark X1, X2) -> mark take(X1, X2), take(ok X1, ok X2) -> ok take(X1, X2), uTake1 mark X -> mark uTake1 X, uTake1 ok X -> ok uTake1 X, uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4), uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4), uLength(mark X1, X2) -> mark uLength(X1, X2), uLength(ok X1, ok X2) -> ok uLength(X1, X2), proper and(X1, X2) -> and(proper X1, proper X2), proper tt() -> ok tt(), proper isNatList X -> isNatList proper X, proper isNatIList X -> isNatIList proper X, proper isNat X -> isNat proper X, proper 0() -> ok 0(), proper s X -> s proper X, proper length X -> length proper X, proper zeros() -> ok zeros(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper nil() -> ok nil(), proper take(X1, X2) -> take(proper X1, proper X2), proper uTake1 X -> uTake1 proper X, proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4), proper uLength(X1, X2) -> uLength(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} DP: DP: { active# and(X1, X2) -> active# X1, active# and(X1, X2) -> active# X2, active# and(X1, X2) -> and#(X1, active X2), active# and(X1, X2) -> and#(active X1, X2), active# isNatList cons(N, L) -> and#(isNat N, isNatList L), active# isNatList cons(N, L) -> isNatList# L, active# isNatList cons(N, L) -> isNat# N, active# isNatList take(N, IL) -> and#(isNat N, isNatIList IL), active# isNatList take(N, IL) -> isNatIList# IL, active# isNatList take(N, IL) -> isNat# N, active# isNatIList IL -> isNatList# IL, active# isNatIList cons(N, IL) -> and#(isNat N, isNatIList IL), active# isNatIList cons(N, IL) -> isNatIList# IL, active# isNatIList cons(N, IL) -> isNat# N, active# isNat s N -> isNat# N, active# isNat length L -> isNatList# L, active# s X -> active# X, active# s X -> s# active X, active# length X -> active# X, active# length X -> length# active X, active# length cons(N, L) -> and#(isNat N, isNatList L), active# length cons(N, L) -> isNatList# L, active# length cons(N, L) -> isNat# N, active# length cons(N, L) -> uLength#(and(isNat N, isNatList L), L), active# zeros() -> cons#(0(), zeros()), active# cons(X1, X2) -> active# X1, active# cons(X1, X2) -> cons#(active X1, X2), active# take(X1, X2) -> active# X1, active# take(X1, X2) -> active# X2, active# take(X1, X2) -> take#(X1, active X2), active# take(X1, X2) -> take#(active X1, X2), active# take(0(), IL) -> isNatIList# IL, active# take(0(), IL) -> uTake1# isNatIList IL, active# take(s M, cons(N, IL)) -> and#(isNat N, isNatIList IL), active# take(s M, cons(N, IL)) -> and#(isNat M, and(isNat N, isNatIList IL)), active# take(s M, cons(N, IL)) -> isNatIList# IL, active# take(s M, cons(N, IL)) -> isNat# N, active# take(s M, cons(N, IL)) -> isNat# M, active# take(s M, cons(N, IL)) -> uTake2#(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), active# uTake1 X -> active# X, active# uTake1 X -> uTake1# active X, active# uTake2(X1, X2, X3, X4) -> active# X1, active# uTake2(X1, X2, X3, X4) -> uTake2#(active X1, X2, X3, X4), active# uTake2(tt(), M, N, IL) -> cons#(N, take(M, IL)), active# uTake2(tt(), M, N, IL) -> take#(M, IL), active# uLength(X1, X2) -> active# X1, active# uLength(X1, X2) -> uLength#(active X1, X2), active# uLength(tt(), L) -> s# length L, active# uLength(tt(), L) -> length# L, and#(X1, mark X2) -> and#(X1, X2), and#(mark X1, X2) -> and#(X1, X2), and#(ok X1, ok X2) -> and#(X1, X2), isNatList# ok X -> isNatList# X, isNatIList# ok X -> isNatIList# X, isNat# ok X -> isNat# X, s# mark X -> s# X, s# ok X -> s# X, length# mark X -> length# X, length# ok X -> length# X, cons#(mark X1, X2) -> cons#(X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2), take#(X1, mark X2) -> take#(X1, X2), take#(mark X1, X2) -> take#(X1, X2), take#(ok X1, ok X2) -> take#(X1, X2), uTake1# mark X -> uTake1# X, uTake1# ok X -> uTake1# X, uTake2#(mark X1, X2, X3, X4) -> uTake2#(X1, X2, X3, X4), uTake2#(ok X1, ok X2, ok X3, ok X4) -> uTake2#(X1, X2, X3, X4), uLength#(mark X1, X2) -> uLength#(X1, X2), uLength#(ok X1, ok X2) -> uLength#(X1, X2), proper# and(X1, X2) -> and#(proper X1, proper X2), proper# and(X1, X2) -> proper# X1, proper# and(X1, X2) -> proper# X2, proper# isNatList X -> isNatList# proper X, proper# isNatList X -> proper# X, proper# isNatIList X -> isNatIList# proper X, proper# isNatIList X -> proper# X, proper# isNat X -> isNat# proper X, proper# isNat X -> proper# X, proper# s X -> s# proper X, proper# s X -> proper# X, proper# length X -> length# proper X, proper# length X -> proper# X, proper# cons(X1, X2) -> cons#(proper X1, proper X2), proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# take(X1, X2) -> take#(proper X1, proper X2), proper# take(X1, X2) -> proper# X1, proper# take(X1, X2) -> proper# X2, proper# uTake1 X -> uTake1# proper X, proper# uTake1 X -> proper# X, proper# uTake2(X1, X2, X3, X4) -> uTake2#(proper X1, proper X2, proper X3, proper X4), proper# uTake2(X1, X2, X3, X4) -> proper# X1, proper# uTake2(X1, X2, X3, X4) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> proper# X3, proper# uTake2(X1, X2, X3, X4) -> proper# X4, proper# uLength(X1, X2) -> uLength#(proper X1, proper X2), proper# uLength(X1, X2) -> proper# X1, proper# uLength(X1, X2) -> proper# X2, top# mark X -> proper# X, top# mark X -> top# proper X, top# ok X -> active# X, top# ok X -> top# active X} TRS: { active and(X1, X2) -> and(X1, active X2), active and(X1, X2) -> and(active X1, X2), active and(tt(), T) -> mark T, active isNatList cons(N, L) -> mark and(isNat N, isNatList L), active isNatList nil() -> mark tt(), active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL), active isNatIList IL -> mark isNatList IL, active isNatIList zeros() -> mark tt(), active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL), active isNat 0() -> mark tt(), active isNat s N -> mark isNat N, active isNat length L -> mark isNatList L, active s X -> s active X, active length X -> length active X, active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L), active zeros() -> mark cons(0(), zeros()), active cons(X1, X2) -> cons(active X1, X2), active take(X1, X2) -> take(X1, active X2), active take(X1, X2) -> take(active X1, X2), active take(0(), IL) -> mark uTake1 isNatIList IL, active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), active uTake1 X -> uTake1 active X, active uTake1 tt() -> mark nil(), active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4), active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)), active uLength(X1, X2) -> uLength(active X1, X2), active uLength(tt(), L) -> mark s length L, and(X1, mark X2) -> mark and(X1, X2), and(mark X1, X2) -> mark and(X1, X2), and(ok X1, ok X2) -> ok and(X1, X2), isNatList ok X -> ok isNatList X, isNatIList ok X -> ok isNatIList X, isNat ok X -> ok isNat X, s mark X -> mark s X, s ok X -> ok s X, length mark X -> mark length X, length ok X -> ok length X, cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), take(X1, mark X2) -> mark take(X1, X2), take(mark X1, X2) -> mark take(X1, X2), take(ok X1, ok X2) -> ok take(X1, X2), uTake1 mark X -> mark uTake1 X, uTake1 ok X -> ok uTake1 X, uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4), uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4), uLength(mark X1, X2) -> mark uLength(X1, X2), uLength(ok X1, ok X2) -> ok uLength(X1, X2), proper and(X1, X2) -> and(proper X1, proper X2), proper tt() -> ok tt(), proper isNatList X -> isNatList proper X, proper isNatIList X -> isNatIList proper X, proper isNat X -> isNat proper X, proper 0() -> ok 0(), proper s X -> s proper X, proper length X -> length proper X, proper zeros() -> ok zeros(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper nil() -> ok nil(), proper take(X1, X2) -> take(proper X1, proper X2), proper uTake1 X -> uTake1 proper X, proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4), proper uLength(X1, X2) -> uLength(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} EDG: { (active# cons(X1, X2) -> cons#(active X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2)) (active# cons(X1, X2) -> cons#(active X1, X2), cons#(mark X1, X2) -> cons#(X1, X2)) (active# uLength(X1, X2) -> uLength#(active X1, X2), uLength#(ok X1, ok X2) -> uLength#(X1, X2)) (active# uLength(X1, X2) -> uLength#(active X1, X2), uLength#(mark X1, X2) -> uLength#(X1, X2)) (and#(mark X1, X2) -> and#(X1, X2), and#(ok X1, ok X2) -> and#(X1, X2)) (and#(mark X1, X2) -> and#(X1, X2), and#(mark X1, X2) -> and#(X1, X2)) (and#(mark X1, X2) -> and#(X1, X2), and#(X1, mark X2) -> and#(X1, X2)) (cons#(mark X1, X2) -> cons#(X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2)) (cons#(mark X1, X2) -> cons#(X1, X2), cons#(mark X1, X2) -> cons#(X1, X2)) (take#(X1, mark X2) -> take#(X1, X2), take#(ok X1, ok X2) -> take#(X1, X2)) (take#(X1, mark X2) -> take#(X1, X2), take#(mark X1, X2) -> take#(X1, X2)) (take#(X1, mark X2) -> take#(X1, X2), take#(X1, mark X2) -> take#(X1, X2)) (take#(ok X1, ok X2) -> take#(X1, X2), take#(ok X1, ok X2) -> take#(X1, X2)) (take#(ok X1, ok X2) -> take#(X1, X2), take#(mark X1, X2) -> take#(X1, X2)) (take#(ok X1, ok X2) -> take#(X1, X2), take#(X1, mark X2) -> take#(X1, X2)) (uLength#(ok X1, ok X2) -> uLength#(X1, X2), uLength#(ok X1, ok X2) -> uLength#(X1, X2)) (uLength#(ok X1, ok X2) -> uLength#(X1, X2), uLength#(mark X1, X2) -> uLength#(X1, X2)) (active# take(s M, cons(N, IL)) -> and#(isNat M, and(isNat N, isNatIList IL)), and#(ok X1, ok X2) -> and#(X1, X2)) (active# take(s M, cons(N, IL)) -> and#(isNat M, and(isNat N, isNatIList IL)), and#(mark X1, X2) -> and#(X1, X2)) (active# take(s M, cons(N, IL)) -> and#(isNat M, and(isNat N, isNatIList IL)), and#(X1, mark X2) -> and#(X1, X2)) (active# isNatList take(N, IL) -> isNat# N, isNat# ok X -> isNat# X) (active# take(s M, cons(N, IL)) -> isNat# N, isNat# ok X -> isNat# X) (active# uTake2(tt(), M, N, IL) -> cons#(N, take(M, IL)), cons#(ok X1, ok X2) -> cons#(X1, X2)) (active# uTake2(tt(), M, N, IL) -> cons#(N, take(M, IL)), cons#(mark X1, X2) -> cons#(X1, X2)) (active# uTake2(X1, X2, X3, X4) -> uTake2#(active X1, X2, X3, X4), uTake2#(ok X1, ok X2, ok X3, ok X4) -> uTake2#(X1, X2, X3, X4)) (active# uTake2(X1, X2, X3, X4) -> uTake2#(active X1, X2, X3, X4), uTake2#(mark X1, X2, X3, X4) -> uTake2#(X1, X2, X3, X4)) (uTake2#(ok X1, ok X2, ok X3, ok X4) -> uTake2#(X1, X2, X3, X4), uTake2#(ok X1, ok X2, ok X3, ok X4) -> uTake2#(X1, X2, X3, X4)) (uTake2#(ok X1, ok X2, ok X3, ok X4) -> uTake2#(X1, X2, X3, X4), uTake2#(mark X1, X2, X3, X4) -> uTake2#(X1, X2, X3, X4)) (active# and(X1, X2) -> active# X2, active# uLength(tt(), L) -> length# L) (active# and(X1, X2) -> active# X2, active# uLength(tt(), L) -> s# length L) (active# and(X1, X2) -> active# X2, active# uLength(X1, X2) -> uLength#(active X1, X2)) (active# and(X1, X2) -> active# X2, active# uLength(X1, X2) -> active# X1) (active# and(X1, X2) -> active# X2, active# uTake2(tt(), M, N, IL) -> take#(M, IL)) (active# and(X1, X2) -> active# X2, active# uTake2(tt(), M, N, IL) -> cons#(N, take(M, IL))) (active# and(X1, X2) -> active# X2, active# uTake2(X1, X2, X3, X4) -> uTake2#(active X1, X2, X3, X4)) (active# and(X1, X2) -> active# X2, active# uTake2(X1, X2, X3, X4) -> active# X1) (active# and(X1, X2) -> active# X2, active# uTake1 X -> uTake1# active X) (active# and(X1, X2) -> active# X2, active# uTake1 X -> active# X) (active# and(X1, X2) -> active# X2, active# take(s M, cons(N, IL)) -> uTake2#(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL)) (active# and(X1, X2) -> active# X2, active# take(s M, cons(N, IL)) -> isNat# M) (active# and(X1, X2) -> active# X2, active# take(s M, cons(N, IL)) -> isNat# N) (active# and(X1, X2) -> active# X2, active# take(s M, cons(N, IL)) -> isNatIList# IL) (active# and(X1, X2) -> active# X2, active# take(s M, cons(N, IL)) -> and#(isNat M, and(isNat N, isNatIList IL))) (active# and(X1, X2) -> active# X2, active# take(s M, cons(N, IL)) -> and#(isNat N, isNatIList IL)) (active# and(X1, X2) -> active# X2, active# take(0(), IL) -> uTake1# isNatIList IL) (active# and(X1, X2) -> active# X2, active# take(0(), IL) -> isNatIList# IL) (active# and(X1, X2) -> active# X2, active# take(X1, X2) -> take#(active X1, X2)) (active# and(X1, X2) -> active# X2, active# take(X1, X2) -> take#(X1, active X2)) (active# and(X1, X2) -> active# X2, active# take(X1, X2) -> active# X2) (active# and(X1, X2) -> active# X2, active# take(X1, X2) -> active# X1) (active# and(X1, X2) -> active# X2, active# cons(X1, X2) -> cons#(active X1, X2)) (active# and(X1, X2) -> active# X2, active# cons(X1, X2) -> active# X1) (active# and(X1, X2) -> active# X2, active# zeros() -> cons#(0(), zeros())) (active# and(X1, X2) -> active# X2, active# length cons(N, L) -> uLength#(and(isNat N, isNatList L), L)) (active# and(X1, X2) -> active# X2, active# length cons(N, L) -> isNat# N) (active# and(X1, X2) -> active# X2, active# length cons(N, L) -> isNatList# L) (active# and(X1, X2) -> active# X2, active# length cons(N, L) -> and#(isNat N, isNatList L)) (active# and(X1, X2) -> active# X2, active# length X -> length# active X) (active# and(X1, X2) -> active# X2, active# length X -> active# X) (active# and(X1, X2) -> active# X2, active# s X -> s# active X) (active# and(X1, X2) -> active# X2, active# s X -> active# X) (active# and(X1, X2) -> active# X2, active# isNat length L -> isNatList# L) (active# and(X1, X2) -> active# X2, active# isNat s N -> isNat# N) (active# and(X1, X2) -> active# X2, active# isNatIList cons(N, IL) -> isNat# N) (active# and(X1, X2) -> active# X2, active# isNatIList cons(N, IL) -> isNatIList# IL) (active# and(X1, X2) -> active# X2, active# isNatIList cons(N, IL) -> and#(isNat N, isNatIList IL)) (active# and(X1, X2) -> active# X2, active# isNatIList IL -> isNatList# IL) (active# and(X1, X2) -> active# X2, active# isNatList take(N, IL) -> isNat# N) (active# and(X1, X2) -> active# X2, active# isNatList take(N, IL) -> isNatIList# IL) (active# and(X1, X2) -> active# X2, active# isNatList take(N, IL) -> and#(isNat N, isNatIList IL)) (active# and(X1, X2) -> active# X2, active# isNatList cons(N, L) -> isNat# N) (active# and(X1, X2) -> active# X2, active# isNatList cons(N, L) -> isNatList# L) (active# and(X1, X2) -> active# X2, active# isNatList cons(N, L) -> and#(isNat N, isNatList L)) (active# and(X1, X2) -> active# X2, active# and(X1, X2) -> and#(active X1, X2)) (active# and(X1, X2) -> active# X2, active# and(X1, X2) -> and#(X1, active X2)) (active# and(X1, X2) -> active# X2, active# and(X1, X2) -> active# X2) (active# and(X1, X2) -> active# X2, active# and(X1, X2) -> active# X1) (proper# and(X1, X2) -> proper# X2, proper# uLength(X1, X2) -> proper# X2) (proper# and(X1, X2) -> proper# X2, proper# uLength(X1, X2) -> proper# X1) (proper# and(X1, X2) -> proper# X2, proper# uLength(X1, X2) -> uLength#(proper X1, proper X2)) (proper# and(X1, X2) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> proper# X4) (proper# and(X1, X2) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> proper# X3) (proper# and(X1, X2) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> proper# X2) (proper# and(X1, X2) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> proper# X1) (proper# and(X1, X2) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> uTake2#(proper X1, proper X2, proper X3, proper X4)) (proper# and(X1, X2) -> proper# X2, proper# uTake1 X -> proper# X) (proper# and(X1, X2) -> proper# X2, proper# uTake1 X -> uTake1# proper X) (proper# and(X1, X2) -> proper# X2, proper# take(X1, X2) -> proper# X2) (proper# and(X1, X2) -> proper# X2, proper# take(X1, X2) -> proper# X1) (proper# and(X1, X2) -> proper# X2, proper# take(X1, X2) -> take#(proper X1, proper X2)) (proper# and(X1, X2) -> proper# X2, proper# cons(X1, X2) -> proper# X2) (proper# and(X1, X2) -> proper# X2, proper# cons(X1, X2) -> proper# X1) (proper# and(X1, X2) -> proper# X2, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (proper# and(X1, X2) -> proper# X2, proper# length X -> proper# X) (proper# and(X1, X2) -> proper# X2, proper# length X -> length# proper X) (proper# and(X1, X2) -> proper# X2, proper# s X -> proper# X) (proper# and(X1, X2) -> proper# X2, proper# s X -> s# proper X) (proper# and(X1, X2) -> proper# X2, proper# isNat X -> proper# X) (proper# and(X1, X2) -> proper# X2, proper# isNat X -> isNat# proper X) (proper# and(X1, X2) -> proper# X2, proper# isNatIList X -> proper# X) (proper# and(X1, X2) -> proper# X2, proper# isNatIList X -> isNatIList# proper X) (proper# and(X1, X2) -> proper# X2, proper# isNatList X -> proper# X) (proper# and(X1, X2) -> proper# X2, proper# isNatList X -> isNatList# proper X) (proper# and(X1, X2) -> proper# X2, proper# and(X1, X2) -> proper# X2) (proper# and(X1, X2) -> proper# X2, proper# and(X1, X2) -> proper# X1) (proper# and(X1, X2) -> proper# X2, proper# and(X1, X2) -> and#(proper X1, proper X2)) (proper# take(X1, X2) -> proper# X2, proper# uLength(X1, X2) -> proper# X2) (proper# take(X1, X2) -> proper# X2, proper# uLength(X1, X2) -> proper# X1) (proper# take(X1, X2) -> proper# X2, proper# uLength(X1, X2) -> uLength#(proper X1, proper X2)) (proper# take(X1, X2) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> proper# X4) (proper# take(X1, X2) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> proper# X3) (proper# take(X1, X2) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> proper# X2) (proper# take(X1, X2) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> proper# X1) (proper# take(X1, X2) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> uTake2#(proper X1, proper X2, proper X3, proper X4)) (proper# take(X1, X2) -> proper# X2, proper# uTake1 X -> proper# X) (proper# take(X1, X2) -> proper# X2, proper# uTake1 X -> uTake1# proper X) (proper# take(X1, X2) -> proper# X2, proper# take(X1, X2) -> proper# X2) (proper# take(X1, X2) -> proper# X2, proper# take(X1, X2) -> proper# X1) (proper# take(X1, X2) -> proper# X2, proper# take(X1, X2) -> take#(proper X1, proper X2)) (proper# take(X1, X2) -> proper# X2, proper# cons(X1, X2) -> proper# X2) (proper# take(X1, X2) -> proper# X2, proper# cons(X1, X2) -> proper# X1) (proper# take(X1, X2) -> proper# X2, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (proper# take(X1, X2) -> proper# X2, proper# length X -> proper# X) (proper# take(X1, X2) -> proper# X2, proper# length X -> length# proper X) (proper# take(X1, X2) -> proper# X2, proper# s X -> proper# X) (proper# take(X1, X2) -> proper# X2, proper# s X -> s# proper X) (proper# take(X1, X2) -> proper# X2, proper# isNat X -> proper# X) (proper# take(X1, X2) -> proper# X2, proper# isNat X -> isNat# proper X) (proper# take(X1, X2) -> proper# X2, proper# isNatIList X -> proper# X) (proper# take(X1, X2) -> proper# X2, proper# isNatIList X -> isNatIList# proper X) (proper# take(X1, X2) -> proper# X2, proper# isNatList X -> proper# X) (proper# take(X1, X2) -> proper# X2, proper# isNatList X -> isNatList# proper X) (proper# take(X1, X2) -> proper# X2, proper# and(X1, X2) -> proper# X2) (proper# take(X1, X2) -> proper# X2, proper# and(X1, X2) -> proper# X1) (proper# take(X1, X2) -> proper# X2, proper# and(X1, X2) -> and#(proper X1, proper X2)) (proper# uLength(X1, X2) -> proper# X2, proper# uLength(X1, X2) -> proper# X2) (proper# uLength(X1, X2) -> proper# X2, proper# uLength(X1, X2) -> proper# X1) (proper# uLength(X1, X2) -> proper# X2, proper# uLength(X1, X2) -> uLength#(proper X1, proper X2)) (proper# uLength(X1, X2) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> proper# X4) (proper# uLength(X1, X2) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> proper# X3) (proper# uLength(X1, X2) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> proper# X2) (proper# uLength(X1, X2) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> proper# X1) (proper# uLength(X1, X2) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> uTake2#(proper X1, proper X2, proper X3, proper X4)) (proper# uLength(X1, X2) -> proper# X2, proper# uTake1 X -> proper# X) (proper# uLength(X1, X2) -> proper# X2, proper# uTake1 X -> uTake1# proper X) (proper# uLength(X1, X2) -> proper# X2, proper# take(X1, X2) -> proper# X2) (proper# uLength(X1, X2) -> proper# X2, proper# take(X1, X2) -> proper# X1) (proper# uLength(X1, X2) -> proper# X2, proper# take(X1, X2) -> take#(proper X1, proper X2)) (proper# uLength(X1, X2) -> proper# X2, proper# cons(X1, X2) -> proper# X2) (proper# uLength(X1, X2) -> proper# X2, proper# cons(X1, X2) -> proper# X1) (proper# uLength(X1, X2) -> proper# X2, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (proper# uLength(X1, X2) -> proper# X2, proper# length X -> proper# X) (proper# uLength(X1, X2) -> proper# X2, proper# length X -> length# proper X) (proper# uLength(X1, X2) -> proper# X2, proper# s X -> proper# X) (proper# uLength(X1, X2) -> proper# X2, proper# s X -> s# proper X) (proper# uLength(X1, X2) -> proper# X2, proper# isNat X -> proper# X) (proper# uLength(X1, X2) -> proper# X2, proper# isNat X -> isNat# proper X) (proper# uLength(X1, X2) -> proper# X2, proper# isNatIList X -> proper# X) (proper# uLength(X1, X2) -> proper# X2, proper# isNatIList X -> isNatIList# proper X) (proper# uLength(X1, X2) -> proper# X2, proper# isNatList X -> proper# X) (proper# uLength(X1, X2) -> proper# X2, proper# isNatList X -> isNatList# proper X) (proper# uLength(X1, X2) -> proper# X2, proper# and(X1, X2) -> proper# X2) (proper# uLength(X1, X2) -> proper# X2, proper# and(X1, X2) -> proper# X1) (proper# uLength(X1, X2) -> proper# X2, proper# and(X1, X2) -> and#(proper X1, proper X2)) (active# s X -> s# active X, s# ok X -> s# X) (active# s X -> s# active X, s# mark X -> s# X) (active# uTake1 X -> uTake1# active X, uTake1# ok X -> uTake1# X) (active# uTake1 X -> uTake1# active X, uTake1# mark X -> uTake1# X) (proper# isNatIList X -> isNatIList# proper X, isNatIList# ok X -> isNatIList# X) (proper# s X -> s# proper X, s# ok X -> s# X) (proper# s X -> s# proper X, s# mark X -> s# X) (proper# uTake1 X -> uTake1# proper X, uTake1# ok X -> uTake1# X) (proper# uTake1 X -> uTake1# proper X, uTake1# mark X -> uTake1# X) (top# ok X -> top# active X, top# ok X -> top# active X) (top# ok X -> top# active X, top# ok X -> active# X) (top# ok X -> top# active X, top# mark X -> top# proper X) (top# ok X -> top# active X, top# mark X -> proper# X) (active# take(0(), IL) -> uTake1# isNatIList IL, uTake1# ok X -> uTake1# X) (active# take(0(), IL) -> uTake1# isNatIList IL, uTake1# mark X -> uTake1# X) (active# isNatList take(N, IL) -> isNatIList# IL, isNatIList# ok X -> isNatIList# X) (active# isNatIList cons(N, IL) -> isNatIList# IL, isNatIList# ok X -> isNatIList# X) (active# take(s M, cons(N, IL)) -> isNatIList# IL, isNatIList# ok X -> isNatIList# X) (active# uLength(tt(), L) -> length# L, length# ok X -> length# X) (active# uLength(tt(), L) -> length# L, length# mark X -> length# X) (active# cons(X1, X2) -> active# X1, active# uLength(tt(), L) -> length# L) (active# cons(X1, X2) -> active# X1, active# uLength(tt(), L) -> s# length L) (active# cons(X1, X2) -> active# X1, active# uLength(X1, X2) -> uLength#(active X1, X2)) (active# cons(X1, X2) -> active# X1, active# uLength(X1, X2) -> active# X1) (active# cons(X1, X2) -> active# X1, active# uTake2(tt(), M, N, IL) -> take#(M, IL)) (active# cons(X1, X2) -> active# X1, active# uTake2(tt(), M, N, IL) -> cons#(N, take(M, IL))) (active# cons(X1, X2) -> active# X1, active# uTake2(X1, X2, X3, X4) -> uTake2#(active X1, X2, X3, X4)) (active# cons(X1, X2) -> active# X1, active# uTake2(X1, X2, X3, X4) -> active# X1) (active# cons(X1, X2) -> active# X1, active# uTake1 X -> uTake1# active X) (active# cons(X1, X2) -> active# X1, active# uTake1 X -> active# X) (active# cons(X1, X2) -> active# X1, active# take(s M, cons(N, IL)) -> uTake2#(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL)) (active# cons(X1, X2) -> active# X1, active# take(s M, cons(N, IL)) -> isNat# M) (active# cons(X1, X2) -> active# X1, active# take(s M, cons(N, IL)) -> isNat# N) (active# cons(X1, X2) -> active# X1, active# take(s M, cons(N, IL)) -> isNatIList# IL) (active# cons(X1, X2) -> active# X1, active# take(s M, cons(N, IL)) -> and#(isNat M, and(isNat N, isNatIList IL))) (active# cons(X1, X2) -> active# X1, active# take(s M, cons(N, IL)) -> and#(isNat N, isNatIList IL)) (active# cons(X1, X2) -> active# X1, active# take(0(), IL) -> uTake1# isNatIList IL) (active# cons(X1, X2) -> active# X1, active# take(0(), IL) -> isNatIList# IL) (active# cons(X1, X2) -> active# X1, active# take(X1, X2) -> take#(active X1, X2)) (active# cons(X1, X2) -> active# X1, active# take(X1, X2) -> take#(X1, active X2)) (active# cons(X1, X2) -> active# X1, active# take(X1, X2) -> active# X2) (active# cons(X1, X2) -> active# X1, active# take(X1, X2) -> active# X1) (active# cons(X1, X2) -> active# X1, active# cons(X1, X2) -> cons#(active X1, X2)) (active# cons(X1, X2) -> active# X1, active# cons(X1, X2) -> active# X1) (active# cons(X1, X2) -> active# X1, active# zeros() -> cons#(0(), zeros())) (active# cons(X1, X2) -> active# X1, active# length cons(N, L) -> uLength#(and(isNat N, isNatList L), L)) (active# cons(X1, X2) -> active# X1, active# length cons(N, L) -> isNat# N) (active# cons(X1, X2) -> active# X1, active# length cons(N, L) -> isNatList# L) (active# cons(X1, X2) -> active# X1, active# length cons(N, L) -> and#(isNat N, isNatList L)) (active# cons(X1, X2) -> active# X1, active# length X -> length# active X) (active# cons(X1, X2) -> active# X1, active# length X -> active# X) (active# cons(X1, X2) -> active# X1, active# s X -> s# active X) (active# cons(X1, X2) -> active# X1, active# s X -> active# X) (active# cons(X1, X2) -> active# X1, active# isNat length L -> isNatList# L) (active# cons(X1, X2) -> active# X1, active# isNat s N -> isNat# N) (active# cons(X1, X2) -> active# X1, active# isNatIList cons(N, IL) -> isNat# N) (active# cons(X1, X2) -> active# X1, active# isNatIList cons(N, IL) -> isNatIList# IL) (active# cons(X1, X2) -> active# X1, active# isNatIList cons(N, IL) -> and#(isNat N, isNatIList IL)) (active# cons(X1, X2) -> active# X1, active# isNatIList IL -> isNatList# IL) (active# cons(X1, X2) -> active# X1, active# isNatList take(N, IL) -> isNat# N) (active# cons(X1, X2) -> active# X1, active# isNatList take(N, IL) -> isNatIList# IL) (active# cons(X1, X2) -> active# X1, active# isNatList take(N, IL) -> and#(isNat N, isNatIList IL)) (active# cons(X1, X2) -> active# X1, active# isNatList cons(N, L) -> isNat# N) (active# cons(X1, X2) -> active# X1, active# isNatList cons(N, L) -> isNatList# L) (active# cons(X1, X2) -> active# X1, active# isNatList cons(N, L) -> and#(isNat N, isNatList L)) (active# cons(X1, X2) -> active# X1, active# and(X1, X2) -> and#(active X1, X2)) (active# cons(X1, X2) -> active# X1, active# and(X1, X2) -> and#(X1, active X2)) (active# cons(X1, X2) -> active# X1, active# and(X1, X2) -> active# X2) (active# cons(X1, X2) -> active# X1, active# and(X1, X2) -> active# X1) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# uLength(tt(), L) -> length# L) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# uLength(tt(), L) -> s# length L) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# uLength(X1, X2) -> uLength#(active X1, X2)) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# uLength(X1, X2) -> active# X1) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# uTake2(tt(), M, N, IL) -> take#(M, IL)) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# uTake2(tt(), M, N, IL) -> cons#(N, take(M, IL))) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# uTake2(X1, X2, X3, X4) -> uTake2#(active X1, X2, X3, X4)) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# uTake2(X1, X2, X3, X4) -> active# X1) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# uTake1 X -> uTake1# active X) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# uTake1 X -> active# X) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# take(s M, cons(N, IL)) -> uTake2#(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL)) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# take(s M, cons(N, IL)) -> isNat# M) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# take(s M, cons(N, IL)) -> isNat# N) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# take(s M, cons(N, IL)) -> isNatIList# IL) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# take(s M, cons(N, IL)) -> and#(isNat M, and(isNat N, isNatIList IL))) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# take(s M, cons(N, IL)) -> and#(isNat N, isNatIList IL)) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# take(0(), IL) -> uTake1# isNatIList IL) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# take(0(), IL) -> isNatIList# IL) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# take(X1, X2) -> take#(active X1, X2)) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# take(X1, X2) -> take#(X1, active X2)) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# take(X1, X2) -> active# X2) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# take(X1, X2) -> active# X1) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# cons(X1, X2) -> cons#(active X1, X2)) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# cons(X1, X2) -> active# X1) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# zeros() -> cons#(0(), zeros())) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# length cons(N, L) -> uLength#(and(isNat N, isNatList L), L)) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# length cons(N, L) -> isNat# N) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# length cons(N, L) -> isNatList# L) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# length cons(N, L) -> and#(isNat N, isNatList L)) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# length X -> length# active X) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# length X -> active# X) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# s X -> s# active X) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# s X -> active# X) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# isNat length L -> isNatList# L) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# isNat s N -> isNat# N) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# isNatIList cons(N, IL) -> isNat# N) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# isNatIList cons(N, IL) -> isNatIList# IL) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# isNatIList cons(N, IL) -> and#(isNat N, isNatIList IL)) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# isNatIList IL -> isNatList# IL) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# isNatList take(N, IL) -> isNat# N) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# isNatList take(N, IL) -> isNatIList# IL) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# isNatList take(N, IL) -> and#(isNat N, isNatIList IL)) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# isNatList cons(N, L) -> isNat# N) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# isNatList cons(N, L) -> isNatList# L) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# isNatList cons(N, L) -> and#(isNat N, isNatList L)) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# and(X1, X2) -> and#(active X1, X2)) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# and(X1, X2) -> and#(X1, active X2)) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# and(X1, X2) -> active# X2) (active# uTake2(X1, X2, X3, X4) -> active# X1, active# and(X1, X2) -> active# X1) (proper# and(X1, X2) -> proper# X1, proper# uLength(X1, X2) -> proper# X2) (proper# and(X1, X2) -> proper# X1, proper# uLength(X1, X2) -> proper# X1) (proper# and(X1, X2) -> proper# X1, proper# uLength(X1, X2) -> uLength#(proper X1, proper X2)) (proper# and(X1, X2) -> proper# X1, proper# uTake2(X1, X2, X3, X4) -> proper# X4) (proper# and(X1, X2) -> proper# X1, proper# uTake2(X1, X2, X3, X4) -> proper# X3) (proper# and(X1, X2) -> proper# X1, proper# uTake2(X1, X2, X3, X4) -> proper# X2) (proper# and(X1, X2) -> proper# X1, proper# uTake2(X1, X2, X3, X4) -> proper# X1) (proper# and(X1, X2) -> proper# X1, proper# uTake2(X1, X2, X3, X4) -> uTake2#(proper X1, proper X2, proper X3, proper X4)) (proper# and(X1, X2) -> proper# X1, proper# uTake1 X -> proper# X) (proper# and(X1, X2) -> proper# X1, proper# uTake1 X -> uTake1# proper X) (proper# and(X1, X2) -> proper# X1, proper# take(X1, X2) -> proper# X2) (proper# and(X1, X2) -> proper# X1, proper# take(X1, X2) -> proper# X1) (proper# and(X1, X2) -> proper# X1, proper# take(X1, X2) -> take#(proper X1, proper X2)) (proper# and(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2) (proper# and(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X1) (proper# and(X1, X2) -> proper# X1, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (proper# and(X1, X2) -> proper# X1, proper# length X -> proper# X) (proper# and(X1, X2) -> proper# X1, proper# length X -> length# proper X) (proper# and(X1, X2) -> proper# X1, proper# s X -> proper# X) (proper# and(X1, X2) -> proper# X1, proper# s X -> s# proper X) (proper# and(X1, X2) -> proper# X1, proper# isNat X -> proper# X) (proper# and(X1, X2) -> proper# X1, proper# 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uLength(X1, X2) -> proper# X1, proper# uLength(X1, X2) -> uLength#(proper X1, proper X2)) (proper# uLength(X1, X2) -> proper# X1, proper# uTake2(X1, X2, X3, X4) -> proper# X4) (proper# uLength(X1, X2) -> proper# X1, proper# uTake2(X1, X2, X3, X4) -> proper# X3) (proper# uLength(X1, X2) -> proper# X1, proper# uTake2(X1, X2, X3, X4) -> proper# X2) (proper# uLength(X1, X2) -> proper# X1, proper# uTake2(X1, X2, X3, X4) -> proper# X1) (proper# uLength(X1, X2) -> proper# X1, proper# uTake2(X1, X2, X3, X4) -> uTake2#(proper X1, proper X2, proper X3, proper X4)) (proper# uLength(X1, X2) -> proper# X1, proper# uTake1 X -> proper# X) (proper# uLength(X1, X2) -> proper# X1, proper# uTake1 X -> uTake1# proper X) (proper# uLength(X1, X2) -> proper# X1, proper# take(X1, X2) -> proper# X2) (proper# uLength(X1, X2) -> proper# X1, proper# take(X1, X2) -> proper# X1) (proper# uLength(X1, X2) -> proper# X1, proper# take(X1, X2) -> take#(proper X1, proper X2)) (proper# uLength(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2) (proper# uLength(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X1) (proper# uLength(X1, X2) -> proper# X1, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (proper# uLength(X1, X2) -> proper# X1, proper# length X -> proper# X) (proper# uLength(X1, X2) -> proper# X1, proper# length X -> length# proper X) (proper# uLength(X1, X2) -> proper# X1, proper# s X -> proper# X) (proper# uLength(X1, X2) -> proper# X1, proper# s X -> s# proper X) (proper# uLength(X1, X2) -> proper# X1, proper# isNat X -> proper# X) (proper# uLength(X1, X2) -> proper# X1, proper# isNat X -> isNat# proper X) (proper# uLength(X1, X2) -> proper# X1, proper# isNatIList X -> proper# X) (proper# uLength(X1, X2) -> proper# X1, proper# isNatIList X -> isNatIList# proper X) (proper# uLength(X1, X2) -> proper# X1, proper# isNatList X -> proper# X) (proper# uLength(X1, X2) -> proper# X1, proper# isNatList X -> isNatList# proper X) (proper# uLength(X1, X2) -> proper# X1, proper# and(X1, X2) -> proper# X2) (proper# uLength(X1, X2) -> proper# X1, proper# and(X1, X2) -> proper# X1) (proper# uLength(X1, X2) -> proper# X1, proper# and(X1, X2) -> and#(proper X1, proper X2)) (active# isNatList cons(N, L) -> and#(isNat N, isNatList L), and#(ok X1, ok X2) -> and#(X1, X2)) (active# isNatList cons(N, L) -> and#(isNat N, isNatList L), and#(mark X1, X2) -> and#(X1, X2)) (active# isNatList cons(N, L) -> and#(isNat N, isNatList L), and#(X1, mark X2) -> and#(X1, X2)) (active# isNatIList cons(N, IL) -> and#(isNat N, isNatIList IL), and#(ok X1, ok X2) -> and#(X1, X2)) (active# isNatIList cons(N, IL) -> and#(isNat N, isNatIList IL), and#(mark X1, X2) -> and#(X1, X2)) (active# isNatIList cons(N, IL) -> and#(isNat N, isNatIList IL), and#(X1, mark X2) -> and#(X1, X2)) (active# take(X1, X2) -> take#(X1, active X2), take#(ok X1, ok X2) -> take#(X1, X2)) (active# take(X1, X2) -> take#(X1, active X2), take#(X1, mark X2) -> take#(X1, X2)) (proper# and(X1, X2) -> and#(proper X1, proper X2), and#(ok X1, ok X2) -> and#(X1, X2)) (proper# and(X1, X2) -> and#(proper X1, proper X2), and#(mark X1, X2) -> and#(X1, X2)) (proper# and(X1, X2) -> and#(proper X1, proper X2), and#(X1, mark X2) -> and#(X1, X2)) (proper# take(X1, X2) -> take#(proper X1, proper X2), take#(ok X1, ok X2) -> take#(X1, X2)) (proper# take(X1, X2) -> take#(proper X1, proper X2), take#(mark X1, X2) -> take#(X1, X2)) (proper# take(X1, X2) -> take#(proper X1, proper X2), take#(X1, mark X2) -> take#(X1, X2)) (active# uTake2(tt(), M, N, IL) -> take#(M, IL), take#(ok X1, ok X2) -> take#(X1, X2)) (active# uTake2(tt(), M, N, IL) -> take#(M, IL), take#(mark X1, X2) -> take#(X1, X2)) (active# uTake2(tt(), M, N, IL) -> take#(M, IL), take#(X1, mark X2) -> take#(X1, X2)) (active# length X -> active# X, active# uLength(tt(), L) -> length# L) (active# length X -> active# X, active# uLength(tt(), L) -> s# length L) (active# length X -> active# X, active# uLength(X1, X2) -> uLength#(active X1, X2)) (active# length X -> 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(active# length X -> active# X, active# take(s M, cons(N, IL)) -> and#(isNat N, isNatIList IL)) (active# length X -> active# X, active# take(0(), IL) -> uTake1# isNatIList IL) (active# length X -> active# X, active# take(0(), IL) -> isNatIList# IL) (active# length X -> active# X, active# take(X1, X2) -> take#(active X1, X2)) (active# length X -> active# X, active# take(X1, X2) -> take#(X1, active X2)) (active# length X -> active# X, active# take(X1, X2) -> active# X2) (active# length X -> active# X, active# take(X1, X2) -> active# X1) (active# length X -> active# X, active# cons(X1, X2) -> cons#(active X1, X2)) (active# length X -> active# X, active# cons(X1, X2) -> active# X1) (active# length X -> active# X, active# zeros() -> cons#(0(), zeros())) (active# length X -> active# X, active# length cons(N, L) -> uLength#(and(isNat N, isNatList L), L)) (active# length X -> active# X, active# length cons(N, L) -> isNat# N) (active# length X -> active# X, active# length cons(N, L) -> isNatList# L) (active# length X -> active# X, active# length cons(N, L) -> and#(isNat N, isNatList L)) (active# length X -> active# X, active# length X -> length# active X) (active# length X -> active# X, active# length X -> active# X) (active# length X -> active# X, active# s X -> s# active X) (active# length X -> active# X, active# s X -> active# X) (active# length X -> active# X, active# isNat length L -> isNatList# L) (active# length X -> active# X, active# isNat s N -> isNat# N) (active# length X -> active# X, active# isNatIList cons(N, IL) -> isNat# N) (active# length X -> active# X, active# isNatIList cons(N, IL) -> isNatIList# IL) (active# length X -> active# X, active# isNatIList cons(N, IL) -> and#(isNat N, isNatIList IL)) (active# length X -> active# X, active# isNatIList IL -> isNatList# IL) (active# length X -> active# X, active# isNatList take(N, IL) -> isNat# N) (active# length X -> active# X, active# isNatList take(N, IL) -> isNatIList# IL) (active# length X -> active# 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-> uTake1# active X) (top# ok X -> active# X, active# uTake1 X -> active# X) (top# ok X -> active# X, active# take(s M, cons(N, IL)) -> uTake2#(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL)) (top# ok X -> active# X, active# take(s M, cons(N, IL)) -> isNat# M) (top# ok X -> active# X, active# take(s M, cons(N, IL)) -> isNat# N) (top# ok X -> active# X, active# take(s M, cons(N, IL)) -> isNatIList# IL) (top# ok X -> active# X, active# take(s M, cons(N, IL)) -> and#(isNat M, and(isNat N, isNatIList IL))) (top# ok X -> active# X, active# take(s M, cons(N, IL)) -> and#(isNat N, isNatIList IL)) (top# ok X -> active# X, active# take(0(), IL) -> uTake1# isNatIList IL) (top# ok X -> active# X, active# take(0(), IL) -> isNatIList# IL) (top# ok X -> active# X, active# take(X1, X2) -> take#(active X1, X2)) (top# ok X -> active# X, active# take(X1, X2) -> take#(X1, active X2)) (top# ok X -> active# X, active# take(X1, X2) -> active# X2) (top# ok X -> active# X, active# take(X1, X2) -> 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-> proper# X4, proper# and(X1, X2) -> proper# X1) (proper# uTake2(X1, X2, X3, X4) -> proper# X4, proper# and(X1, X2) -> proper# X2) (proper# uTake2(X1, X2, X3, X4) -> proper# X4, proper# isNatList X -> isNatList# proper X) (proper# uTake2(X1, X2, X3, X4) -> proper# X4, proper# isNatList X -> proper# X) (proper# uTake2(X1, X2, X3, X4) -> proper# X4, proper# isNatIList X -> isNatIList# proper X) (proper# uTake2(X1, X2, X3, X4) -> proper# X4, proper# isNatIList X -> proper# X) (proper# uTake2(X1, X2, X3, X4) -> proper# X4, proper# isNat X -> isNat# proper X) (proper# uTake2(X1, X2, X3, X4) -> proper# X4, proper# isNat X -> proper# X) (proper# uTake2(X1, X2, X3, X4) -> proper# X4, proper# s X -> s# proper X) (proper# uTake2(X1, X2, X3, X4) -> proper# X4, proper# s X -> proper# X) (proper# uTake2(X1, X2, X3, X4) -> proper# X4, proper# length X -> length# proper X) (proper# uTake2(X1, X2, X3, X4) -> proper# X4, proper# length X -> proper# X) (proper# uTake2(X1, X2, X3, X4) -> proper# X4, 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-> proper# X4, proper# uTake2(X1, X2, X3, X4) -> proper# X3) (proper# uTake2(X1, X2, X3, X4) -> proper# X4, proper# uTake2(X1, X2, X3, X4) -> proper# X4) (proper# uTake2(X1, X2, X3, X4) -> proper# X4, proper# uLength(X1, X2) -> uLength#(proper X1, proper X2)) (proper# uTake2(X1, X2, X3, X4) -> proper# X4, proper# uLength(X1, X2) -> proper# X1) (proper# uTake2(X1, X2, X3, X4) -> proper# X4, proper# uLength(X1, X2) -> proper# X2) (top# mark X -> proper# X, proper# and(X1, X2) -> and#(proper X1, proper X2)) (top# mark X -> proper# X, proper# and(X1, X2) -> proper# X1) (top# mark X -> proper# X, proper# and(X1, X2) -> proper# X2) (top# mark X -> proper# X, proper# isNatList X -> isNatList# proper X) (top# mark X -> proper# X, proper# isNatList X -> proper# X) (top# mark X -> proper# X, proper# isNatIList X -> isNatIList# proper X) (top# mark X -> proper# X, proper# isNatIList X -> proper# X) (top# mark X -> proper# X, proper# isNat X -> isNat# proper X) (top# mark X -> proper# X, proper# 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length X -> proper# X, proper# uTake2(X1, X2, X3, X4) -> proper# X1) (proper# length X -> proper# X, proper# uTake2(X1, X2, X3, X4) -> proper# X2) (proper# length X -> proper# X, proper# uTake2(X1, X2, X3, X4) -> proper# X3) (proper# length X -> proper# X, proper# uTake2(X1, X2, X3, X4) -> proper# X4) (proper# length X -> proper# X, proper# uLength(X1, X2) -> uLength#(proper X1, proper X2)) (proper# length X -> proper# X, proper# uLength(X1, X2) -> proper# X1) (proper# length X -> proper# X, proper# uLength(X1, X2) -> proper# X2) (proper# isNat X -> proper# X, proper# and(X1, X2) -> and#(proper X1, proper X2)) (proper# isNat X -> proper# X, proper# and(X1, X2) -> proper# X1) (proper# isNat X -> proper# X, proper# and(X1, X2) -> proper# X2) (proper# isNat X -> proper# X, proper# isNatList X -> isNatList# proper X) (proper# isNat X -> proper# X, proper# isNatList X -> proper# X) (proper# isNat X -> proper# X, proper# isNatIList X -> isNatIList# proper X) (proper# isNat X -> proper# X, 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-> active# X1, active# zeros() -> cons#(0(), zeros())) (active# uLength(X1, X2) -> active# X1, active# cons(X1, X2) -> active# X1) (active# uLength(X1, X2) -> active# X1, active# cons(X1, X2) -> cons#(active X1, X2)) (active# uLength(X1, X2) -> active# X1, active# take(X1, X2) -> active# X1) (active# uLength(X1, X2) -> active# X1, active# take(X1, X2) -> active# X2) (active# uLength(X1, X2) -> active# X1, active# take(X1, X2) -> take#(X1, active X2)) (active# uLength(X1, X2) -> active# X1, active# take(X1, X2) -> take#(active X1, X2)) (active# uLength(X1, X2) -> active# X1, active# take(0(), IL) -> isNatIList# IL) (active# uLength(X1, X2) -> active# X1, active# take(0(), IL) -> uTake1# isNatIList IL) (active# uLength(X1, X2) -> active# X1, active# take(s M, cons(N, IL)) -> and#(isNat N, isNatIList IL)) (active# uLength(X1, X2) -> active# X1, active# take(s M, cons(N, IL)) -> and#(isNat M, and(isNat N, isNatIList IL))) (active# uLength(X1, X2) -> active# X1, active# take(s M, cons(N, 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cons(N, IL)) -> and#(isNat M, and(isNat N, isNatIList IL))) (active# take(X1, X2) -> active# X1, active# take(s M, cons(N, IL)) -> isNatIList# IL) (active# take(X1, X2) -> active# X1, active# take(s M, cons(N, IL)) -> isNat# N) (active# take(X1, X2) -> active# X1, active# take(s M, cons(N, IL)) -> isNat# M) (active# take(X1, X2) -> active# X1, active# take(s M, cons(N, IL)) -> uTake2#(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL)) (active# take(X1, X2) -> active# X1, active# uTake1 X -> active# X) (active# take(X1, X2) -> active# X1, active# uTake1 X -> uTake1# active X) (active# take(X1, X2) -> active# X1, active# uTake2(X1, X2, X3, X4) -> active# X1) (active# take(X1, X2) -> active# X1, active# uTake2(X1, X2, X3, X4) -> uTake2#(active X1, X2, X3, X4)) (active# take(X1, X2) -> active# X1, active# uTake2(tt(), M, N, IL) -> cons#(N, take(M, IL))) (active# take(X1, X2) -> active# X1, active# uTake2(tt(), M, N, IL) -> take#(M, IL)) (active# take(X1, X2) -> active# X1, active# 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proper# X2, proper# isNat X -> isNat# proper X) (proper# cons(X1, X2) -> proper# X2, proper# isNat X -> proper# X) (proper# cons(X1, X2) -> proper# X2, proper# s X -> s# proper X) (proper# cons(X1, X2) -> proper# X2, proper# s X -> proper# X) (proper# cons(X1, X2) -> proper# X2, proper# length X -> length# proper X) (proper# cons(X1, X2) -> proper# X2, proper# length X -> proper# X) (proper# cons(X1, X2) -> proper# X2, proper# cons(X1, X2) -> cons#(proper X1, proper X2)) (proper# cons(X1, X2) -> proper# X2, proper# cons(X1, X2) -> proper# X1) (proper# cons(X1, X2) -> proper# X2, proper# cons(X1, X2) -> proper# X2) (proper# cons(X1, X2) -> proper# X2, proper# take(X1, X2) -> take#(proper X1, proper X2)) (proper# cons(X1, X2) -> proper# X2, proper# take(X1, X2) -> proper# X1) (proper# cons(X1, X2) -> proper# X2, proper# take(X1, X2) -> proper# X2) (proper# cons(X1, X2) -> proper# X2, proper# uTake1 X -> uTake1# proper X) (proper# cons(X1, X2) -> proper# X2, proper# uTake1 X -> proper# X) (proper# cons(X1, X2) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> uTake2#(proper X1, proper X2, proper X3, proper X4)) (proper# cons(X1, X2) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> proper# X1) (proper# cons(X1, X2) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> proper# X2) (proper# cons(X1, X2) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> proper# X3) (proper# cons(X1, X2) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> proper# X4) (proper# cons(X1, X2) -> proper# X2, proper# uLength(X1, X2) -> uLength#(proper X1, proper X2)) (proper# cons(X1, X2) -> proper# X2, proper# uLength(X1, X2) -> proper# X1) (proper# cons(X1, X2) -> proper# X2, proper# uLength(X1, X2) -> proper# X2) (active# take(X1, X2) -> active# X2, active# and(X1, X2) -> active# X1) (active# take(X1, X2) -> active# X2, active# and(X1, X2) -> active# X2) (active# take(X1, X2) -> active# X2, active# and(X1, X2) -> and#(X1, active X2)) (active# take(X1, X2) -> active# X2, active# and(X1, X2) -> and#(active X1, X2)) (active# take(X1, X2) -> active# X2, active# isNatList cons(N, L) -> and#(isNat N, isNatList L)) (active# take(X1, X2) -> active# X2, active# isNatList cons(N, L) -> isNatList# L) (active# take(X1, X2) -> active# X2, active# isNatList cons(N, L) -> isNat# N) (active# take(X1, X2) -> active# X2, active# isNatList take(N, IL) -> and#(isNat N, isNatIList IL)) (active# take(X1, X2) -> active# X2, active# isNatList take(N, IL) -> isNatIList# IL) (active# take(X1, X2) -> active# X2, active# isNatList take(N, IL) -> isNat# N) (active# take(X1, X2) -> active# X2, active# isNatIList IL -> isNatList# IL) (active# take(X1, X2) -> active# X2, active# isNatIList cons(N, IL) -> and#(isNat N, isNatIList IL)) (active# take(X1, X2) -> active# X2, active# isNatIList cons(N, IL) -> isNatIList# IL) (active# take(X1, X2) -> active# X2, active# isNatIList cons(N, IL) -> isNat# N) (active# take(X1, X2) -> active# X2, active# isNat s N -> isNat# N) (active# take(X1, X2) -> active# X2, active# isNat length L -> isNatList# L) (active# take(X1, X2) -> active# X2, active# s X -> active# X) (active# take(X1, X2) -> active# X2, active# s X -> s# active X) (active# take(X1, X2) -> active# X2, active# length X -> active# X) (active# take(X1, X2) -> active# X2, active# length X -> length# active X) (active# take(X1, X2) -> active# X2, active# length cons(N, L) -> and#(isNat N, isNatList L)) (active# take(X1, X2) -> active# X2, active# length cons(N, L) -> isNatList# L) (active# take(X1, X2) -> active# X2, active# length cons(N, L) -> isNat# N) (active# take(X1, X2) -> active# X2, active# length cons(N, L) -> uLength#(and(isNat N, isNatList L), L)) (active# take(X1, X2) -> active# X2, active# zeros() -> cons#(0(), zeros())) (active# take(X1, X2) -> active# X2, active# cons(X1, X2) -> active# X1) (active# take(X1, X2) -> active# X2, active# cons(X1, X2) -> cons#(active X1, X2)) (active# take(X1, X2) -> active# X2, active# take(X1, X2) -> active# X1) (active# take(X1, X2) -> active# X2, active# take(X1, X2) -> active# X2) (active# take(X1, X2) -> active# X2, active# take(X1, X2) -> take#(X1, active X2)) (active# take(X1, X2) -> active# X2, active# take(X1, X2) -> take#(active X1, X2)) (active# take(X1, X2) -> active# X2, active# take(0(), IL) -> isNatIList# IL) (active# take(X1, X2) -> active# X2, active# take(0(), IL) -> uTake1# isNatIList IL) (active# take(X1, X2) -> active# X2, active# take(s M, cons(N, IL)) -> and#(isNat N, isNatIList IL)) (active# take(X1, X2) -> active# X2, active# take(s M, cons(N, IL)) -> and#(isNat M, and(isNat N, isNatIList IL))) (active# take(X1, X2) -> active# X2, active# take(s M, cons(N, IL)) -> isNatIList# IL) (active# take(X1, X2) -> active# X2, active# take(s M, cons(N, IL)) -> isNat# N) (active# take(X1, X2) -> active# X2, active# take(s M, cons(N, IL)) -> isNat# M) (active# take(X1, X2) -> active# X2, active# take(s M, cons(N, IL)) -> uTake2#(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL)) (active# take(X1, X2) -> active# X2, active# uTake1 X -> active# X) (active# take(X1, X2) -> active# X2, active# uTake1 X -> uTake1# active X) (active# take(X1, X2) -> active# X2, active# uTake2(X1, X2, X3, X4) -> active# X1) (active# take(X1, X2) -> active# X2, active# uTake2(X1, X2, X3, X4) -> uTake2#(active X1, X2, X3, X4)) (active# take(X1, X2) -> active# X2, active# uTake2(tt(), M, N, IL) -> cons#(N, take(M, IL))) (active# take(X1, X2) -> active# X2, active# uTake2(tt(), M, N, IL) -> take#(M, IL)) (active# take(X1, X2) -> active# X2, active# uLength(X1, X2) -> active# X1) (active# take(X1, X2) -> active# X2, active# uLength(X1, X2) -> uLength#(active X1, X2)) (active# take(X1, X2) -> active# X2, active# uLength(tt(), L) -> s# length L) (active# take(X1, X2) -> active# X2, active# uLength(tt(), L) -> length# L) (active# take(s M, cons(N, IL)) -> uTake2#(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), uTake2#(mark X1, X2, X3, X4) -> uTake2#(X1, X2, X3, X4)) (uTake2#(mark X1, X2, X3, X4) -> uTake2#(X1, X2, X3, X4), uTake2#(mark X1, X2, X3, X4) -> uTake2#(X1, X2, X3, X4)) (uTake2#(mark X1, X2, X3, X4) -> uTake2#(X1, X2, X3, X4), uTake2#(ok X1, ok X2, ok X3, ok X4) -> uTake2#(X1, X2, X3, X4)) (active# length cons(N, L) -> uLength#(and(isNat N, isNatList L), L), uLength#(mark X1, X2) -> uLength#(X1, X2)) (active# length cons(N, L) -> uLength#(and(isNat N, isNatList L), L), uLength#(ok X1, ok X2) -> uLength#(X1, X2)) (active# length cons(N, L) -> isNat# N, isNat# ok X -> isNat# X) (active# isNatIList cons(N, IL) -> isNat# N, isNat# ok X -> isNat# X) (active# isNatList cons(N, L) -> isNat# N, isNat# ok X -> isNat# X) (proper# uTake2(X1, X2, X3, X4) -> uTake2#(proper X1, proper X2, proper X3, proper X4), uTake2#(mark X1, X2, X3, X4) -> uTake2#(X1, X2, X3, X4)) (proper# uTake2(X1, X2, X3, X4) -> uTake2#(proper X1, proper X2, proper X3, proper X4), uTake2#(ok X1, ok X2, ok X3, ok X4) -> uTake2#(X1, X2, X3, X4)) (uLength#(mark X1, X2) -> uLength#(X1, X2), uLength#(mark X1, X2) -> uLength#(X1, X2)) (uLength#(mark X1, X2) -> uLength#(X1, X2), uLength#(ok X1, ok X2) -> uLength#(X1, X2)) (take#(mark X1, X2) -> take#(X1, X2), take#(X1, mark X2) -> take#(X1, X2)) (take#(mark X1, X2) -> take#(X1, X2), take#(mark X1, X2) -> take#(X1, X2)) (take#(mark X1, X2) -> take#(X1, X2), take#(ok X1, ok X2) -> take#(X1, X2)) (cons#(ok X1, ok X2) -> cons#(X1, X2), cons#(mark X1, X2) -> cons#(X1, X2)) (cons#(ok X1, ok X2) -> cons#(X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2)) (and#(ok X1, ok X2) -> and#(X1, X2), and#(X1, mark X2) -> and#(X1, X2)) (and#(ok X1, ok X2) -> and#(X1, X2), and#(mark X1, X2) -> and#(X1, X2)) (and#(ok X1, ok X2) -> and#(X1, X2), and#(ok X1, ok X2) -> and#(X1, X2)) (and#(X1, mark X2) -> and#(X1, X2), and#(X1, mark X2) -> and#(X1, X2)) (and#(X1, mark X2) -> and#(X1, X2), and#(mark X1, X2) -> and#(X1, X2)) (and#(X1, mark X2) -> and#(X1, X2), and#(ok X1, ok X2) -> and#(X1, X2)) (active# take(X1, X2) -> take#(active X1, X2), take#(mark X1, X2) -> take#(X1, X2)) (active# take(X1, X2) -> take#(active X1, X2), take#(ok X1, ok X2) -> take#(X1, X2)) (active# and(X1, X2) -> and#(active X1, X2), and#(mark X1, X2) -> and#(X1, X2)) (active# and(X1, X2) -> and#(active X1, X2), and#(ok X1, ok X2) -> and#(X1, X2)) } STATUS: arrows: 0.884343 SCCS (14): Scc: {top# mark X -> top# proper X, top# ok X -> top# active X} Scc: { proper# and(X1, X2) -> proper# X1, proper# and(X1, X2) -> proper# X2, proper# isNatList X -> proper# X, proper# isNatIList X -> proper# X, proper# isNat X -> proper# X, proper# s X -> proper# X, proper# length X -> proper# X, proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# take(X1, X2) -> proper# X1, proper# take(X1, X2) -> proper# X2, proper# uTake1 X -> proper# X, proper# uTake2(X1, X2, X3, X4) -> proper# X1, proper# uTake2(X1, X2, X3, X4) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> proper# X3, proper# uTake2(X1, X2, X3, X4) -> proper# X4, proper# uLength(X1, X2) -> proper# X1, proper# uLength(X1, X2) -> proper# X2} Scc: { active# and(X1, X2) -> active# X1, active# and(X1, X2) -> active# X2, active# s X -> active# X, active# length X -> active# X, active# cons(X1, X2) -> active# X1, active# take(X1, X2) -> active# X1, active# take(X1, X2) -> active# X2, active# uTake1 X -> active# X, active# uTake2(X1, X2, X3, X4) -> active# X1, active# uLength(X1, X2) -> active# X1} Scc: {uTake1# mark X -> uTake1# X, uTake1# ok X -> uTake1# X} Scc: {length# mark X -> length# X, length# ok X -> length# X} Scc: {s# mark X -> s# X, s# ok X -> s# X} Scc: {isNatIList# ok X -> isNatIList# X} Scc: {isNatList# ok X -> isNatList# X} Scc: {isNat# ok X -> isNat# X} Scc: { uTake2#(mark X1, X2, X3, X4) -> uTake2#(X1, X2, X3, X4), uTake2#(ok X1, ok X2, ok X3, ok X4) -> uTake2#(X1, X2, X3, X4)} Scc: { uLength#(mark X1, X2) -> uLength#(X1, X2), uLength#(ok X1, ok X2) -> uLength#(X1, X2)} Scc: { take#(X1, mark X2) -> take#(X1, X2), take#(mark X1, X2) -> take#(X1, X2), take#(ok X1, ok X2) -> take#(X1, X2)} Scc: { cons#(mark X1, X2) -> cons#(X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2)} Scc: { and#(X1, mark X2) -> and#(X1, X2), and#(mark X1, X2) -> and#(X1, X2), and#(ok X1, ok X2) -> and#(X1, X2)} SCC (2): Strict: {top# mark X -> top# proper X, top# ok X -> top# active X} Weak: { active and(X1, X2) -> and(X1, active X2), active and(X1, X2) -> and(active X1, X2), active and(tt(), T) -> mark T, active isNatList cons(N, L) -> mark and(isNat N, isNatList L), active isNatList nil() -> mark tt(), active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL), active isNatIList IL -> mark isNatList IL, active isNatIList zeros() -> mark tt(), active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL), active isNat 0() -> mark tt(), active isNat s N -> mark isNat N, active isNat length L -> mark isNatList L, active s X -> s active X, active length X -> length active X, active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L), active zeros() -> mark cons(0(), zeros()), active cons(X1, X2) -> cons(active X1, X2), active take(X1, X2) -> take(X1, active X2), active take(X1, X2) -> take(active X1, X2), active take(0(), IL) -> mark uTake1 isNatIList IL, active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), active uTake1 X -> uTake1 active X, active uTake1 tt() -> mark nil(), active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4), active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)), active uLength(X1, X2) -> uLength(active X1, X2), active uLength(tt(), L) -> mark s length L, and(X1, mark X2) -> mark and(X1, X2), and(mark X1, X2) -> mark and(X1, X2), and(ok X1, ok X2) -> ok and(X1, X2), isNatList ok X -> ok isNatList X, isNatIList ok X -> ok isNatIList X, isNat ok X -> ok isNat X, s mark X -> mark s X, s ok X -> ok s X, length mark X -> mark length X, length ok X -> ok length X, cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), take(X1, mark X2) -> mark take(X1, X2), take(mark X1, X2) -> mark take(X1, X2), take(ok X1, ok X2) -> ok take(X1, X2), uTake1 mark X -> mark uTake1 X, uTake1 ok X -> ok uTake1 X, uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4), uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4), uLength(mark X1, X2) -> mark uLength(X1, X2), uLength(ok X1, ok X2) -> ok uLength(X1, X2), proper and(X1, X2) -> and(proper X1, proper X2), proper tt() -> ok tt(), proper isNatList X -> isNatList proper X, proper isNatIList X -> isNatIList proper X, proper isNat X -> isNat proper X, proper 0() -> ok 0(), proper s X -> s proper X, proper length X -> length proper X, proper zeros() -> ok zeros(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper nil() -> ok nil(), proper take(X1, X2) -> take(proper X1, proper X2), proper uTake1 X -> uTake1 proper X, proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4), proper uLength(X1, X2) -> uLength(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} Fail SCC (18): Strict: { proper# and(X1, X2) -> proper# X1, proper# and(X1, X2) -> proper# X2, proper# isNatList X -> proper# X, proper# isNatIList X -> proper# X, proper# isNat X -> proper# X, proper# s X -> proper# X, proper# length X -> proper# X, proper# cons(X1, X2) -> proper# X1, proper# cons(X1, X2) -> proper# X2, proper# take(X1, X2) -> proper# X1, proper# take(X1, X2) -> proper# X2, proper# uTake1 X -> proper# X, proper# uTake2(X1, X2, X3, X4) -> proper# X1, proper# uTake2(X1, X2, X3, X4) -> proper# X2, proper# uTake2(X1, X2, X3, X4) -> proper# X3, proper# uTake2(X1, X2, X3, X4) -> proper# X4, proper# uLength(X1, X2) -> proper# X1, proper# uLength(X1, X2) -> proper# X2} Weak: { active and(X1, X2) -> and(X1, active X2), active and(X1, X2) -> and(active X1, X2), active and(tt(), T) -> mark T, active isNatList cons(N, L) -> mark and(isNat N, isNatList L), active isNatList nil() -> mark tt(), active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL), active isNatIList IL -> mark isNatList IL, active isNatIList zeros() -> mark tt(), active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL), active isNat 0() -> mark tt(), active isNat s N -> mark isNat N, active isNat length L -> mark isNatList L, active s X -> s active X, active length X -> length active X, active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L), active zeros() -> mark cons(0(), zeros()), active cons(X1, X2) -> cons(active X1, X2), active take(X1, X2) -> take(X1, active X2), active take(X1, X2) -> take(active X1, X2), active take(0(), IL) -> mark uTake1 isNatIList IL, active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), active uTake1 X -> uTake1 active X, active uTake1 tt() -> mark nil(), active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4), active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)), active uLength(X1, X2) -> uLength(active X1, X2), active uLength(tt(), L) -> mark s length L, and(X1, mark X2) -> mark and(X1, X2), and(mark X1, X2) -> mark and(X1, X2), and(ok X1, ok X2) -> ok and(X1, X2), isNatList ok X -> ok isNatList X, isNatIList ok X -> ok isNatIList X, isNat ok X -> ok isNat X, s mark X -> mark s X, s ok X -> ok s X, length mark X -> mark length X, length ok X -> ok length X, cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), take(X1, mark X2) -> mark take(X1, X2), take(mark X1, X2) -> mark take(X1, X2), take(ok X1, ok X2) -> ok take(X1, X2), uTake1 mark X -> mark uTake1 X, uTake1 ok X -> ok uTake1 X, uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4), uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4), uLength(mark X1, X2) -> mark uLength(X1, X2), uLength(ok X1, ok X2) -> ok uLength(X1, X2), proper and(X1, X2) -> and(proper X1, proper X2), proper tt() -> ok tt(), proper isNatList X -> isNatList proper X, proper isNatIList X -> isNatIList proper X, proper isNat X -> isNat proper X, proper 0() -> ok 0(), proper s X -> s proper X, proper length X -> length proper X, proper zeros() -> ok zeros(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper nil() -> ok nil(), proper take(X1, X2) -> take(proper X1, proper X2), proper uTake1 X -> uTake1 proper X, proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4), proper uLength(X1, X2) -> uLength(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + x3 + 1, [and](x0, x1) = x0 + x1 + 1, [cons](x0, x1) = x0 + x1 + 1, [take](x0, x1) = x0 + x1 + 1, [uLength](x0, x1) = x0 + x1 + 1, [mark](x0) = 1, [active](x0) = x0 + 1, [isNatList](x0) = x0 + 1, [isNatIList](x0) = x0 + 1, [isNat](x0) = x0 + 1, [s](x0) = x0 + 1, [length](x0) = x0 + 1, [uTake1](x0) = x0 + 1, [proper](x0) = x0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 0, [zeros] = 0, [nil] = 0, [proper#](x0) = x0 + 1 Strict: proper# uLength(X1, X2) -> proper# X2 2 + 1X1 + 1X2 >= 1 + 1X2 proper# uLength(X1, X2) -> proper# X1 2 + 1X1 + 1X2 >= 1 + 1X1 proper# uTake2(X1, X2, X3, X4) -> proper# X4 2 + 1X1 + 1X2 + 1X3 + 1X4 >= 1 + 1X4 proper# uTake2(X1, X2, X3, X4) -> proper# X3 2 + 1X1 + 1X2 + 1X3 + 1X4 >= 1 + 1X3 proper# uTake2(X1, X2, X3, X4) -> proper# X2 2 + 1X1 + 1X2 + 1X3 + 1X4 >= 1 + 1X2 proper# uTake2(X1, X2, X3, X4) -> proper# X1 2 + 1X1 + 1X2 + 1X3 + 1X4 >= 1 + 1X1 proper# uTake1 X -> proper# X 2 + 1X >= 1 + 1X proper# take(X1, X2) -> proper# X2 2 + 1X1 + 1X2 >= 1 + 1X2 proper# take(X1, X2) -> proper# X1 2 + 1X1 + 1X2 >= 1 + 1X1 proper# cons(X1, X2) -> proper# X2 2 + 1X1 + 1X2 >= 1 + 1X2 proper# cons(X1, X2) -> proper# X1 2 + 1X1 + 1X2 >= 1 + 1X1 proper# length X -> proper# X 2 + 1X >= 1 + 1X proper# s X -> proper# X 2 + 1X >= 1 + 1X proper# isNat X -> proper# X 2 + 1X >= 1 + 1X proper# isNatIList X -> proper# X 2 + 1X >= 1 + 1X proper# isNatList X -> proper# X 2 + 1X >= 1 + 1X proper# and(X1, X2) -> proper# X2 2 + 1X1 + 1X2 >= 1 + 1X2 proper# and(X1, X2) -> proper# X1 2 + 1X1 + 1X2 >= 1 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper uLength(X1, X2) -> uLength(proper X1, proper X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4) 1 + 1X1 + 1X2 + 1X3 + 1X4 >= 1 + 1X1 + 1X2 + 1X3 + 1X4 proper uTake1 X -> uTake1 proper X 1 + 1X >= 1 + 1X proper take(X1, X2) -> take(proper X1, proper X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 proper nil() -> ok nil() 0 >= 1 proper cons(X1, X2) -> cons(proper X1, proper X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 proper zeros() -> ok zeros() 0 >= 1 proper length X -> length proper X 1 + 1X >= 1 + 1X proper s X -> s proper X 1 + 1X >= 1 + 1X proper 0() -> ok 0() 0 >= 1 proper isNat X -> isNat proper X 1 + 1X >= 1 + 1X proper isNatIList X -> isNatIList proper X 1 + 1X >= 1 + 1X proper isNatList X -> isNatList proper X 1 + 1X >= 1 + 1X proper tt() -> ok tt() 0 >= 1 proper and(X1, X2) -> and(proper X1, proper X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 uLength(ok X1, ok X2) -> ok uLength(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 uLength(mark X1, X2) -> mark uLength(X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 0X2 uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4) 5 + 1X1 + 1X2 + 1X3 + 1X4 >= 2 + 1X1 + 1X2 + 1X3 + 1X4 uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4) 2 + 0X1 + 1X2 + 1X3 + 1X4 >= 1 + 0X1 + 0X2 + 0X3 + 0X4 uTake1 ok X -> ok uTake1 X 2 + 1X >= 2 + 1X uTake1 mark X -> mark uTake1 X 2 + 0X >= 1 + 0X take(ok X1, ok X2) -> ok take(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 take(mark X1, X2) -> mark take(X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 0X2 take(X1, mark X2) -> mark take(X1, X2) 2 + 1X1 + 0X2 >= 1 + 0X1 + 0X2 cons(ok X1, ok X2) -> ok cons(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 0X2 length ok X -> ok length X 2 + 1X >= 2 + 1X length mark X -> mark length X 2 + 0X >= 1 + 0X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 2 + 0X >= 1 + 0X isNat ok X -> ok isNat X 2 + 1X >= 2 + 1X isNatIList ok X -> ok isNatIList X 2 + 1X >= 2 + 1X isNatList ok X -> ok isNatList X 2 + 1X >= 2 + 1X and(ok X1, ok X2) -> ok and(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 and(mark X1, X2) -> mark and(X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 0X2 and(X1, mark X2) -> mark and(X1, X2) 2 + 1X1 + 0X2 >= 1 + 0X1 + 0X2 active uLength(tt(), L) -> mark s length L 2 + 1L >= 1 + 0L active uLength(X1, X2) -> uLength(active X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)) 2 + 1IL + 1N + 1M >= 1 + 0IL + 0N + 0M active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4) 2 + 1X1 + 1X2 + 1X3 + 1X4 >= 2 + 1X1 + 1X2 + 1X3 + 1X4 active uTake1 tt() -> mark nil() 2 >= 1 active uTake1 X -> uTake1 active X 2 + 1X >= 2 + 1X active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL) 4 + 1IL + 1N + 1M >= 1 + 0IL + 0N + 0M active take(0(), IL) -> mark uTake1 isNatIList IL 2 + 1IL >= 1 + 0IL active take(X1, X2) -> take(active X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active take(X1, X2) -> take(X1, active X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active zeros() -> mark cons(0(), zeros()) 1 >= 1 active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L) 3 + 1N + 1L >= 1 + 0N + 0L active length X -> length active X 2 + 1X >= 2 + 1X active s X -> s active X 2 + 1X >= 2 + 1X active isNat length L -> mark isNatList L 3 + 1L >= 1 + 0L active isNat s N -> mark isNat N 3 + 1N >= 1 + 0N active isNat 0() -> mark tt() 2 >= 1 active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL) 3 + 1IL + 1N >= 1 + 0IL + 0N active isNatIList zeros() -> mark tt() 2 >= 1 active isNatIList IL -> mark isNatList IL 2 + 1IL >= 1 + 0IL active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL) 3 + 1IL + 1N >= 1 + 0IL + 0N active isNatList nil() -> mark tt() 2 >= 1 active isNatList cons(N, L) -> mark and(isNat N, isNatList L) 3 + 1N + 1L >= 1 + 0N + 0L active and(tt(), T) -> mark T 2 + 1T >= 1 + 0T active and(X1, X2) -> and(active X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active and(X1, X2) -> and(X1, active X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 Qed SCC (10): Strict: { active# and(X1, X2) -> active# X1, active# and(X1, X2) -> active# X2, active# s X -> active# X, active# length X -> active# X, active# cons(X1, X2) -> active# X1, active# take(X1, X2) -> active# X1, active# take(X1, X2) -> active# X2, active# uTake1 X -> active# X, active# uTake2(X1, X2, X3, X4) -> active# X1, active# uLength(X1, X2) -> active# X1} Weak: { active and(X1, X2) -> and(X1, active X2), active and(X1, X2) -> and(active X1, X2), active and(tt(), T) -> mark T, active isNatList cons(N, L) -> mark and(isNat N, isNatList L), active isNatList nil() -> mark tt(), active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL), active isNatIList IL -> mark isNatList IL, active isNatIList zeros() -> mark tt(), active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL), active isNat 0() -> mark tt(), active isNat s N -> mark isNat N, active isNat length L -> mark isNatList L, active s X -> s active X, active length X -> length active X, active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L), active zeros() -> mark cons(0(), zeros()), active cons(X1, X2) -> cons(active X1, X2), active take(X1, X2) -> take(X1, active X2), active take(X1, X2) -> take(active X1, X2), active take(0(), IL) -> mark uTake1 isNatIList IL, active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), active uTake1 X -> uTake1 active X, active uTake1 tt() -> mark nil(), active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4), active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)), active uLength(X1, X2) -> uLength(active X1, X2), active uLength(tt(), L) -> mark s length L, and(X1, mark X2) -> mark and(X1, X2), and(mark X1, X2) -> mark and(X1, X2), and(ok X1, ok X2) -> ok and(X1, X2), isNatList ok X -> ok isNatList X, isNatIList ok X -> ok isNatIList X, isNat ok X -> ok isNat X, s mark X -> mark s X, s ok X -> ok s X, length mark X -> mark length X, length ok X -> ok length X, cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), take(X1, mark X2) -> mark take(X1, X2), take(mark X1, X2) -> mark take(X1, X2), take(ok X1, ok X2) -> ok take(X1, X2), uTake1 mark X -> mark uTake1 X, uTake1 ok X -> ok uTake1 X, uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4), uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4), uLength(mark X1, X2) -> mark uLength(X1, X2), uLength(ok X1, ok X2) -> ok uLength(X1, X2), proper and(X1, X2) -> and(proper X1, proper X2), proper tt() -> ok tt(), proper isNatList X -> isNatList proper X, proper isNatIList X -> isNatIList proper X, proper isNat X -> isNat proper X, proper 0() -> ok 0(), proper s X -> s proper X, proper length X -> length proper X, proper zeros() -> ok zeros(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper nil() -> ok nil(), proper take(X1, X2) -> take(proper X1, proper X2), proper uTake1 X -> uTake1 proper X, proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4), proper uLength(X1, X2) -> uLength(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [uTake2](x0, x1, x2, x3) = x0 + 1, [and](x0, x1) = x0 + x1 + 1, [cons](x0, x1) = x0 + 1, [take](x0, x1) = x0 + x1 + 1, [uLength](x0, x1) = x0 + 1, [mark](x0) = x0, [active](x0) = x0 + 1, [isNatList](x0) = 1, [isNatIList](x0) = 0, [isNat](x0) = 1, [s](x0) = x0 + 1, [length](x0) = x0 + 1, [uTake1](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0, [top](x0) = 0, [tt] = 0, [0] = 1, [zeros] = 1, [nil] = 1, [active#](x0) = x0 + 1 Strict: active# uLength(X1, X2) -> active# X1 2 + 1X1 + 0X2 >= 1 + 1X1 active# uTake2(X1, X2, X3, X4) -> active# X1 2 + 1X1 + 0X2 + 0X3 + 0X4 >= 1 + 1X1 active# uTake1 X -> active# X 2 + 1X >= 1 + 1X active# take(X1, X2) -> active# X2 2 + 1X1 + 1X2 >= 1 + 1X2 active# take(X1, X2) -> active# X1 2 + 1X1 + 1X2 >= 1 + 1X1 active# cons(X1, X2) -> active# X1 2 + 1X1 + 0X2 >= 1 + 1X1 active# length X -> active# X 2 + 1X >= 1 + 1X active# s X -> active# X 2 + 1X >= 1 + 1X active# and(X1, X2) -> active# X2 2 + 1X1 + 1X2 >= 1 + 1X2 active# and(X1, X2) -> active# X1 2 + 1X1 + 1X2 >= 1 + 1X1 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper uLength(X1, X2) -> uLength(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4) 0 + 0X1 + 0X2 + 0X3 + 0X4 >= 1 + 0X1 + 0X2 + 0X3 + 0X4 proper uTake1 X -> uTake1 proper X 0 + 0X >= 1 + 0X proper take(X1, X2) -> take(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 1 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper zeros() -> ok zeros() 0 >= 1 proper length X -> length proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 1 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper isNatIList X -> isNatIList proper X 0 + 0X >= 0 + 0X proper isNatList X -> isNatList proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 0 proper and(X1, X2) -> and(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 uLength(ok X1, ok X2) -> ok uLength(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 uLength(mark X1, X2) -> mark uLength(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4) 1 + 1X1 + 0X2 + 0X3 + 0X4 >= 1 + 1X1 + 0X2 + 0X3 + 0X4 uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4) 1 + 1X1 + 0X2 + 0X3 + 0X4 >= 1 + 1X1 + 0X2 + 0X3 + 0X4 uTake1 ok X -> ok uTake1 X 1 + 1X >= 1 + 1X uTake1 mark X -> mark uTake1 X 1 + 1X >= 1 + 1X take(ok X1, ok X2) -> ok take(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 take(mark X1, X2) -> mark take(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 take(X1, mark X2) -> mark take(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 cons(ok X1, ok X2) -> ok cons(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 length ok X -> ok length X 1 + 1X >= 1 + 1X length mark X -> mark length X 1 + 1X >= 1 + 1X s ok X -> ok s X 1 + 1X >= 1 + 1X s mark X -> mark s X 1 + 1X >= 1 + 1X isNat ok X -> ok isNat X 1 + 0X >= 1 + 0X isNatIList ok X -> ok isNatIList X 0 + 0X >= 0 + 0X isNatList ok X -> ok isNatList X 1 + 0X >= 1 + 0X and(ok X1, ok X2) -> ok and(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 and(mark X1, X2) -> mark and(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 and(X1, mark X2) -> mark and(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 active uLength(tt(), L) -> mark s length L 2 + 0L >= 2 + 1L active uLength(X1, X2) -> uLength(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)) 2 + 0IL + 0N + 0M >= 1 + 0IL + 1N + 0M active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 0X4 >= 2 + 1X1 + 0X2 + 0X3 + 0X4 active uTake1 tt() -> mark nil() 2 >= 1 active uTake1 X -> uTake1 active X 2 + 1X >= 2 + 1X active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL) 4 + 0IL + 1N + 1M >= 5 + 0IL + 0N + 0M active take(0(), IL) -> mark uTake1 isNatIList IL 3 + 1IL >= 1 + 0IL active take(X1, X2) -> take(active X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active take(X1, X2) -> take(X1, active X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 0X2 >= 2 + 1X1 + 0X2 active zeros() -> mark cons(0(), zeros()) 2 >= 2 active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L) 3 + 1N + 0L >= 4 + 0N + 0L active length X -> length active X 2 + 1X >= 2 + 1X active s X -> s active X 2 + 1X >= 2 + 1X active isNat length L -> mark isNatList L 2 + 0L >= 1 + 0L active isNat s N -> mark isNat N 2 + 0N >= 1 + 0N active isNat 0() -> mark tt() 2 >= 0 active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL) 1 + 0IL + 0N >= 2 + 0IL + 0N active isNatIList zeros() -> mark tt() 1 >= 0 active isNatIList IL -> mark isNatList IL 1 + 0IL >= 1 + 0IL active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 2 + 0IL + 0N active isNatList nil() -> mark tt() 2 >= 0 active isNatList cons(N, L) -> mark and(isNat N, isNatList L) 2 + 0N + 0L >= 3 + 0N + 0L active and(tt(), T) -> mark T 2 + 1T >= 0 + 1T active and(X1, X2) -> and(active X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active and(X1, X2) -> and(X1, active X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 Qed SCC (2): Strict: {uTake1# mark X -> uTake1# X, uTake1# ok X -> uTake1# X} Weak: { active and(X1, X2) -> and(X1, active X2), active and(X1, X2) -> and(active X1, X2), active and(tt(), T) -> mark T, active isNatList cons(N, L) -> mark and(isNat N, isNatList L), active isNatList nil() -> mark tt(), active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL), active isNatIList IL -> mark isNatList IL, active isNatIList zeros() -> mark tt(), active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL), active isNat 0() -> mark tt(), active isNat s N -> mark isNat N, active isNat length L -> mark isNatList L, active s X -> s active X, active length X -> length active X, active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L), active zeros() -> mark cons(0(), zeros()), active cons(X1, X2) -> cons(active X1, X2), active take(X1, X2) -> take(X1, active X2), active take(X1, X2) -> take(active X1, X2), active take(0(), IL) -> mark uTake1 isNatIList IL, active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), active uTake1 X -> uTake1 active X, active uTake1 tt() -> mark nil(), active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4), active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)), active uLength(X1, X2) -> uLength(active X1, X2), active uLength(tt(), L) -> mark s length L, and(X1, mark X2) -> mark and(X1, X2), and(mark X1, X2) -> mark and(X1, X2), and(ok X1, ok X2) -> ok and(X1, X2), isNatList ok X -> ok isNatList X, isNatIList ok X -> ok isNatIList X, isNat ok X -> ok isNat X, s mark X -> mark s X, s ok X -> ok s X, length mark X -> mark length X, length ok X -> ok length X, cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), take(X1, mark X2) -> mark take(X1, X2), take(mark X1, X2) -> mark take(X1, X2), take(ok X1, ok X2) -> ok take(X1, X2), uTake1 mark X -> mark uTake1 X, uTake1 ok X -> ok uTake1 X, uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4), uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4), uLength(mark X1, X2) -> mark uLength(X1, X2), uLength(ok X1, ok X2) -> ok uLength(X1, X2), proper and(X1, X2) -> and(proper X1, proper X2), proper tt() -> ok tt(), proper isNatList X -> isNatList proper X, proper isNatIList X -> isNatIList proper X, proper isNat X -> isNat proper X, proper 0() -> ok 0(), proper s X -> s proper X, proper length X -> length proper X, proper zeros() -> ok zeros(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper nil() -> ok nil(), proper take(X1, X2) -> take(proper X1, proper X2), proper uTake1 X -> uTake1 proper X, proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4), proper uLength(X1, X2) -> uLength(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [uTake2](x0, x1, x2, x3) = x0 + x1 + 1, [and](x0, x1) = x0 + 1, [cons](x0, x1) = 0, [take](x0, x1) = x0 + 1, [uLength](x0, x1) = x0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [isNatList](x0) = 1, [isNatIList](x0) = 1, [isNat](x0) = 1, [s](x0) = x0 + 1, [length](x0) = x0 + 1, [uTake1](x0) = x0 + 1, [proper](x0) = x0 + 1, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 1, [0] = 1, [zeros] = 1, [nil] = 1, [uTake1#](x0) = x0 + 1 Strict: uTake1# ok X -> uTake1# X 2 + 1X >= 1 + 1X uTake1# mark X -> uTake1# X 2 + 1X >= 1 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper uLength(X1, X2) -> uLength(proper X1, proper X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4) 2 + 1X1 + 0X2 + 0X3 + 1X4 >= 3 + 1X1 + 0X2 + 0X3 + 1X4 proper uTake1 X -> uTake1 proper X 2 + 1X >= 2 + 1X proper take(X1, X2) -> take(proper X1, proper X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 proper nil() -> ok nil() 2 >= 2 proper cons(X1, X2) -> cons(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper zeros() -> ok zeros() 2 >= 2 proper length X -> length proper X 2 + 1X >= 2 + 1X proper s X -> s proper X 2 + 1X >= 2 + 1X proper 0() -> ok 0() 2 >= 2 proper isNat X -> isNat proper X 2 + 0X >= 1 + 0X proper isNatIList X -> isNatIList proper X 2 + 0X >= 1 + 0X proper isNatList X -> isNatList proper X 2 + 0X >= 1 + 0X proper tt() -> ok tt() 2 >= 2 proper and(X1, X2) -> and(proper X1, proper X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 uLength(ok X1, ok X2) -> ok uLength(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 uLength(mark X1, X2) -> mark uLength(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4) 3 + 1X1 + 0X2 + 0X3 + 1X4 >= 2 + 1X1 + 0X2 + 0X3 + 1X4 uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 1X4 >= 2 + 1X1 + 0X2 + 0X3 + 1X4 uTake1 ok X -> ok uTake1 X 2 + 1X >= 2 + 1X uTake1 mark X -> mark uTake1 X 2 + 1X >= 2 + 1X take(ok X1, ok X2) -> ok take(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 take(mark X1, X2) -> mark take(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 take(X1, mark X2) -> mark take(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 cons(ok X1, ok X2) -> ok cons(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 length ok X -> ok length X 2 + 1X >= 2 + 1X length mark X -> mark length X 2 + 1X >= 2 + 1X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 1 + 0X >= 2 + 0X isNatIList ok X -> ok isNatIList X 1 + 0X >= 2 + 0X isNatList ok X -> ok isNatList X 1 + 0X >= 2 + 0X and(ok X1, ok X2) -> ok and(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 and(mark X1, X2) -> mark and(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 and(X1, mark X2) -> mark and(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active uLength(tt(), L) -> mark s length L 2 + 0L >= 3 + 1L active uLength(X1, X2) -> uLength(active X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)) 3 + 1IL + 0N + 0M >= 1 + 0IL + 0N + 0M active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 1X4 >= 2 + 1X1 + 0X2 + 0X3 + 1X4 active uTake1 tt() -> mark nil() 3 >= 2 active uTake1 X -> uTake1 active X 2 + 1X >= 2 + 1X active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL) 2 + 0IL + 0N + 0M >= 5 + 1IL + 0N + 0M active take(0(), IL) -> mark uTake1 isNatIList IL 2 + 1IL >= 3 + 0IL active take(X1, X2) -> take(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active take(X1, X2) -> take(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active cons(X1, X2) -> cons(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active zeros() -> mark cons(0(), zeros()) 2 >= 1 active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L) 2 + 0N + 0L >= 3 + 0N + 0L active length X -> length active X 2 + 1X >= 2 + 1X active s X -> s active X 2 + 1X >= 2 + 1X active isNat length L -> mark isNatList L 2 + 0L >= 2 + 0L active isNat s N -> mark isNat N 2 + 0N >= 2 + 0N active isNat 0() -> mark tt() 2 >= 2 active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 3 + 0IL + 0N active isNatIList zeros() -> mark tt() 2 >= 2 active isNatIList IL -> mark isNatList IL 2 + 0IL >= 2 + 0IL active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 3 + 0IL + 0N active isNatList nil() -> mark tt() 2 >= 2 active isNatList cons(N, L) -> mark and(isNat N, isNatList L) 2 + 0N + 0L >= 3 + 0N + 0L active and(tt(), T) -> mark T 2 + 1T >= 1 + 1T active and(X1, X2) -> and(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active and(X1, X2) -> and(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 Qed SCC (2): Strict: {length# mark X -> length# X, length# ok X -> length# X} Weak: { active and(X1, X2) -> and(X1, active X2), active and(X1, X2) -> and(active X1, X2), active and(tt(), T) -> mark T, active isNatList cons(N, L) -> mark and(isNat N, isNatList L), active isNatList nil() -> mark tt(), active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL), active isNatIList IL -> mark isNatList IL, active isNatIList zeros() -> mark tt(), active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL), active isNat 0() -> mark tt(), active isNat s N -> mark isNat N, active isNat length L -> mark isNatList L, active s X -> s active X, active length X -> length active X, active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L), active zeros() -> mark cons(0(), zeros()), active cons(X1, X2) -> cons(active X1, X2), active take(X1, X2) -> take(X1, active X2), active take(X1, X2) -> take(active X1, X2), active take(0(), IL) -> mark uTake1 isNatIList IL, active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), active uTake1 X -> uTake1 active X, active uTake1 tt() -> mark nil(), active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4), active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)), active uLength(X1, X2) -> uLength(active X1, X2), active uLength(tt(), L) -> mark s length L, and(X1, mark X2) -> mark and(X1, X2), and(mark X1, X2) -> mark and(X1, X2), and(ok X1, ok X2) -> ok and(X1, X2), isNatList ok X -> ok isNatList X, isNatIList ok X -> ok isNatIList X, isNat ok X -> ok isNat X, s mark X -> mark s X, s ok X -> ok s X, length mark X -> mark length X, length ok X -> ok length X, cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), take(X1, mark X2) -> mark take(X1, X2), take(mark X1, X2) -> mark take(X1, X2), take(ok X1, ok X2) -> ok take(X1, X2), uTake1 mark X -> mark uTake1 X, uTake1 ok X -> ok uTake1 X, uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4), uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4), uLength(mark X1, X2) -> mark uLength(X1, X2), uLength(ok X1, ok X2) -> ok uLength(X1, X2), proper and(X1, X2) -> and(proper X1, proper X2), proper tt() -> ok tt(), proper isNatList X -> isNatList proper X, proper isNatIList X -> isNatIList proper X, proper isNat X -> isNat proper X, proper 0() -> ok 0(), proper s X -> s proper X, proper length X -> length proper X, proper zeros() -> ok zeros(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper nil() -> ok nil(), proper take(X1, X2) -> take(proper X1, proper X2), proper uTake1 X -> uTake1 proper X, proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4), proper uLength(X1, X2) -> uLength(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [uTake2](x0, x1, x2, x3) = x0 + x1 + 1, [and](x0, x1) = x0 + 1, [cons](x0, x1) = 0, [take](x0, x1) = x0 + 1, [uLength](x0, x1) = x0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [isNatList](x0) = 1, [isNatIList](x0) = 1, [isNat](x0) = 1, [s](x0) = x0 + 1, [length](x0) = x0 + 1, [uTake1](x0) = x0 + 1, [proper](x0) = x0 + 1, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 1, [0] = 1, [zeros] = 1, [nil] = 1, [length#](x0) = x0 + 1 Strict: length# ok X -> length# X 2 + 1X >= 1 + 1X length# mark X -> length# X 2 + 1X >= 1 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper uLength(X1, X2) -> uLength(proper X1, proper X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4) 2 + 1X1 + 0X2 + 0X3 + 1X4 >= 3 + 1X1 + 0X2 + 0X3 + 1X4 proper uTake1 X -> uTake1 proper X 2 + 1X >= 2 + 1X proper take(X1, X2) -> take(proper X1, proper X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 proper nil() -> ok nil() 2 >= 2 proper cons(X1, X2) -> cons(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper zeros() -> ok zeros() 2 >= 2 proper length X -> length proper X 2 + 1X >= 2 + 1X proper s X -> s proper X 2 + 1X >= 2 + 1X proper 0() -> ok 0() 2 >= 2 proper isNat X -> isNat proper X 2 + 0X >= 1 + 0X proper isNatIList X -> isNatIList proper X 2 + 0X >= 1 + 0X proper isNatList X -> isNatList proper X 2 + 0X >= 1 + 0X proper tt() -> ok tt() 2 >= 2 proper and(X1, X2) -> and(proper X1, proper X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 uLength(ok X1, ok X2) -> ok uLength(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 uLength(mark X1, X2) -> mark uLength(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4) 3 + 1X1 + 0X2 + 0X3 + 1X4 >= 2 + 1X1 + 0X2 + 0X3 + 1X4 uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 1X4 >= 2 + 1X1 + 0X2 + 0X3 + 1X4 uTake1 ok X -> ok uTake1 X 2 + 1X >= 2 + 1X uTake1 mark X -> mark uTake1 X 2 + 1X >= 2 + 1X take(ok X1, ok X2) -> ok take(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 take(mark X1, X2) -> mark take(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 take(X1, mark X2) -> mark take(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 cons(ok X1, ok X2) -> ok cons(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 length ok X -> ok length X 2 + 1X >= 2 + 1X length mark X -> mark length X 2 + 1X >= 2 + 1X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 1 + 0X >= 2 + 0X isNatIList ok X -> ok isNatIList X 1 + 0X >= 2 + 0X isNatList ok X -> ok isNatList X 1 + 0X >= 2 + 0X and(ok X1, ok X2) -> ok and(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 and(mark X1, X2) -> mark and(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 and(X1, mark X2) -> mark and(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active uLength(tt(), L) -> mark s length L 2 + 0L >= 3 + 1L active uLength(X1, X2) -> uLength(active X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)) 3 + 1IL + 0N + 0M >= 1 + 0IL + 0N + 0M active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 1X4 >= 2 + 1X1 + 0X2 + 0X3 + 1X4 active uTake1 tt() -> mark nil() 3 >= 2 active uTake1 X -> uTake1 active X 2 + 1X >= 2 + 1X active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL) 2 + 0IL + 0N + 0M >= 5 + 1IL + 0N + 0M active take(0(), IL) -> mark uTake1 isNatIList IL 2 + 1IL >= 3 + 0IL active take(X1, X2) -> take(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active take(X1, X2) -> take(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active cons(X1, X2) -> cons(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active zeros() -> mark cons(0(), zeros()) 2 >= 1 active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L) 2 + 0N + 0L >= 3 + 0N + 0L active length X -> length active X 2 + 1X >= 2 + 1X active s X -> s active X 2 + 1X >= 2 + 1X active isNat length L -> mark isNatList L 2 + 0L >= 2 + 0L active isNat s N -> mark isNat N 2 + 0N >= 2 + 0N active isNat 0() -> mark tt() 2 >= 2 active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 3 + 0IL + 0N active isNatIList zeros() -> mark tt() 2 >= 2 active isNatIList IL -> mark isNatList IL 2 + 0IL >= 2 + 0IL active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 3 + 0IL + 0N active isNatList nil() -> mark tt() 2 >= 2 active isNatList cons(N, L) -> mark and(isNat N, isNatList L) 2 + 0N + 0L >= 3 + 0N + 0L active and(tt(), T) -> mark T 2 + 1T >= 1 + 1T active and(X1, X2) -> and(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active and(X1, X2) -> and(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 Qed SCC (2): Strict: {s# mark X -> s# X, s# ok X -> s# X} Weak: { active and(X1, X2) -> and(X1, active X2), active and(X1, X2) -> and(active X1, X2), active and(tt(), T) -> mark T, active isNatList cons(N, L) -> mark and(isNat N, isNatList L), active isNatList nil() -> mark tt(), active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL), active isNatIList IL -> mark isNatList IL, active isNatIList zeros() -> mark tt(), active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL), active isNat 0() -> mark tt(), active isNat s N -> mark isNat N, active isNat length L -> mark isNatList L, active s X -> s active X, active length X -> length active X, active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L), active zeros() -> mark cons(0(), zeros()), active cons(X1, X2) -> cons(active X1, X2), active take(X1, X2) -> take(X1, active X2), active take(X1, X2) -> take(active X1, X2), active take(0(), IL) -> mark uTake1 isNatIList IL, active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), active uTake1 X -> uTake1 active X, active uTake1 tt() -> mark nil(), active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4), active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)), active uLength(X1, X2) -> uLength(active X1, X2), active uLength(tt(), L) -> mark s length L, and(X1, mark X2) -> mark and(X1, X2), and(mark X1, X2) -> mark and(X1, X2), and(ok X1, ok X2) -> ok and(X1, X2), isNatList ok X -> ok isNatList X, isNatIList ok X -> ok isNatIList X, isNat ok X -> ok isNat X, s mark X -> mark s X, s ok X -> ok s X, length mark X -> mark length X, length ok X -> ok length X, cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), take(X1, mark X2) -> mark take(X1, X2), take(mark X1, X2) -> mark take(X1, X2), take(ok X1, ok X2) -> ok take(X1, X2), uTake1 mark X -> mark uTake1 X, uTake1 ok X -> ok uTake1 X, uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4), uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4), uLength(mark X1, X2) -> mark uLength(X1, X2), uLength(ok X1, ok X2) -> ok uLength(X1, X2), proper and(X1, X2) -> and(proper X1, proper X2), proper tt() -> ok tt(), proper isNatList X -> isNatList proper X, proper isNatIList X -> isNatIList proper X, proper isNat X -> isNat proper X, proper 0() -> ok 0(), proper s X -> s proper X, proper length X -> length proper X, proper zeros() -> ok zeros(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper nil() -> ok nil(), proper take(X1, X2) -> take(proper X1, proper X2), proper uTake1 X -> uTake1 proper X, proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4), proper uLength(X1, X2) -> uLength(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [uTake2](x0, x1, x2, x3) = x0 + x1 + 1, [and](x0, x1) = x0 + 1, [cons](x0, x1) = 0, [take](x0, x1) = x0 + 1, [uLength](x0, x1) = x0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [isNatList](x0) = 1, [isNatIList](x0) = 1, [isNat](x0) = 1, [s](x0) = x0 + 1, [length](x0) = x0 + 1, [uTake1](x0) = x0 + 1, [proper](x0) = x0 + 1, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 1, [0] = 1, [zeros] = 1, [nil] = 1, [s#](x0) = x0 + 1 Strict: s# ok X -> s# X 2 + 1X >= 1 + 1X s# mark X -> s# X 2 + 1X >= 1 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper uLength(X1, X2) -> uLength(proper X1, proper X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4) 2 + 1X1 + 0X2 + 0X3 + 1X4 >= 3 + 1X1 + 0X2 + 0X3 + 1X4 proper uTake1 X -> uTake1 proper X 2 + 1X >= 2 + 1X proper take(X1, X2) -> take(proper X1, proper X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 proper nil() -> ok nil() 2 >= 2 proper cons(X1, X2) -> cons(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper zeros() -> ok zeros() 2 >= 2 proper length X -> length proper X 2 + 1X >= 2 + 1X proper s X -> s proper X 2 + 1X >= 2 + 1X proper 0() -> ok 0() 2 >= 2 proper isNat X -> isNat proper X 2 + 0X >= 1 + 0X proper isNatIList X -> isNatIList proper X 2 + 0X >= 1 + 0X proper isNatList X -> isNatList proper X 2 + 0X >= 1 + 0X proper tt() -> ok tt() 2 >= 2 proper and(X1, X2) -> and(proper X1, proper X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 uLength(ok X1, ok X2) -> ok uLength(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 uLength(mark X1, X2) -> mark uLength(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4) 3 + 1X1 + 0X2 + 0X3 + 1X4 >= 2 + 1X1 + 0X2 + 0X3 + 1X4 uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 1X4 >= 2 + 1X1 + 0X2 + 0X3 + 1X4 uTake1 ok X -> ok uTake1 X 2 + 1X >= 2 + 1X uTake1 mark X -> mark uTake1 X 2 + 1X >= 2 + 1X take(ok X1, ok X2) -> ok take(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 take(mark X1, X2) -> mark take(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 take(X1, mark X2) -> mark take(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 cons(ok X1, ok X2) -> ok cons(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 length ok X -> ok length X 2 + 1X >= 2 + 1X length mark X -> mark length X 2 + 1X >= 2 + 1X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 1 + 0X >= 2 + 0X isNatIList ok X -> ok isNatIList X 1 + 0X >= 2 + 0X isNatList ok X -> ok isNatList X 1 + 0X >= 2 + 0X and(ok X1, ok X2) -> ok and(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 and(mark X1, X2) -> mark and(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 and(X1, mark X2) -> mark and(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active uLength(tt(), L) -> mark s length L 2 + 0L >= 3 + 1L active uLength(X1, X2) -> uLength(active X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)) 3 + 1IL + 0N + 0M >= 1 + 0IL + 0N + 0M active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 1X4 >= 2 + 1X1 + 0X2 + 0X3 + 1X4 active uTake1 tt() -> mark nil() 3 >= 2 active uTake1 X -> uTake1 active X 2 + 1X >= 2 + 1X active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL) 2 + 0IL + 0N + 0M >= 5 + 1IL + 0N + 0M active take(0(), IL) -> mark uTake1 isNatIList IL 2 + 1IL >= 3 + 0IL active take(X1, X2) -> take(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active take(X1, X2) -> take(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active cons(X1, X2) -> cons(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active zeros() -> mark cons(0(), zeros()) 2 >= 1 active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L) 2 + 0N + 0L >= 3 + 0N + 0L active length X -> length active X 2 + 1X >= 2 + 1X active s X -> s active X 2 + 1X >= 2 + 1X active isNat length L -> mark isNatList L 2 + 0L >= 2 + 0L active isNat s N -> mark isNat N 2 + 0N >= 2 + 0N active isNat 0() -> mark tt() 2 >= 2 active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 3 + 0IL + 0N active isNatIList zeros() -> mark tt() 2 >= 2 active isNatIList IL -> mark isNatList IL 2 + 0IL >= 2 + 0IL active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 3 + 0IL + 0N active isNatList nil() -> mark tt() 2 >= 2 active isNatList cons(N, L) -> mark and(isNat N, isNatList L) 2 + 0N + 0L >= 3 + 0N + 0L active and(tt(), T) -> mark T 2 + 1T >= 1 + 1T active and(X1, X2) -> and(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active and(X1, X2) -> and(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 Qed SCC (1): Strict: {isNatIList# ok X -> isNatIList# X} Weak: { active and(X1, X2) -> and(X1, active X2), active and(X1, X2) -> and(active X1, X2), active and(tt(), T) -> mark T, active isNatList cons(N, L) -> mark and(isNat N, isNatList L), active isNatList nil() -> mark tt(), active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL), active isNatIList IL -> mark isNatList IL, active isNatIList zeros() -> mark tt(), active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL), active isNat 0() -> mark tt(), active isNat s N -> mark isNat N, active isNat length L -> mark isNatList L, active s X -> s active X, active length X -> length active X, active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L), active zeros() -> mark cons(0(), zeros()), active cons(X1, X2) -> cons(active X1, X2), active take(X1, X2) -> take(X1, active X2), active take(X1, X2) -> take(active X1, X2), active take(0(), IL) -> mark uTake1 isNatIList IL, active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), active uTake1 X -> uTake1 active X, active uTake1 tt() -> mark nil(), active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4), active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)), active uLength(X1, X2) -> uLength(active X1, X2), active uLength(tt(), L) -> mark s length L, and(X1, mark X2) -> mark and(X1, X2), and(mark X1, X2) -> mark and(X1, X2), and(ok X1, ok X2) -> ok and(X1, X2), isNatList ok X -> ok isNatList X, isNatIList ok X -> ok isNatIList X, isNat ok X -> ok isNat X, s mark X -> mark s X, s ok X -> ok s X, length mark X -> mark length X, length ok X -> ok length X, cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), take(X1, mark X2) -> mark take(X1, X2), take(mark X1, X2) -> mark take(X1, X2), take(ok X1, ok X2) -> ok take(X1, X2), uTake1 mark X -> mark uTake1 X, uTake1 ok X -> ok uTake1 X, uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4), uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4), uLength(mark X1, X2) -> mark uLength(X1, X2), uLength(ok X1, ok X2) -> ok uLength(X1, X2), proper and(X1, X2) -> and(proper X1, proper X2), proper tt() -> ok tt(), proper isNatList X -> isNatList proper X, proper isNatIList X -> isNatIList proper X, proper isNat X -> isNat proper X, proper 0() -> ok 0(), proper s X -> s proper X, proper length X -> length proper X, proper zeros() -> ok zeros(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper nil() -> ok nil(), proper take(X1, X2) -> take(proper X1, proper X2), proper uTake1 X -> uTake1 proper X, proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4), proper uLength(X1, X2) -> uLength(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [uTake2](x0, x1, x2, x3) = x0 + 1, [and](x0, x1) = x0 + 1, [cons](x0, x1) = 0, [take](x0, x1) = x0 + 1, [uLength](x0, x1) = x0, [mark](x0) = x0, [active](x0) = x0 + 1, [isNatList](x0) = 1, [isNatIList](x0) = 1, [isNat](x0) = 1, [s](x0) = x0 + 1, [length](x0) = x0 + 1, [uTake1](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [zeros] = 1, [nil] = 0, [isNatIList#](x0) = x0 + 1 Strict: isNatIList# ok X -> isNatIList# X 2 + 1X >= 1 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper uLength(X1, X2) -> uLength(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4) 0 + 0X1 + 0X2 + 0X3 + 0X4 >= 1 + 0X1 + 0X2 + 0X3 + 0X4 proper uTake1 X -> uTake1 proper X 0 + 0X >= 1 + 0X proper take(X1, X2) -> take(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 1 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper zeros() -> ok zeros() 0 >= 2 proper length X -> length proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper isNatIList X -> isNatIList proper X 0 + 0X >= 1 + 0X proper isNatList X -> isNatList proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper and(X1, X2) -> and(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 uLength(ok X1, ok X2) -> ok uLength(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 uLength(mark X1, X2) -> mark uLength(X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 0X4 >= 2 + 1X1 + 0X2 + 0X3 + 0X4 uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4) 1 + 1X1 + 0X2 + 0X3 + 0X4 >= 1 + 1X1 + 0X2 + 0X3 + 0X4 uTake1 ok X -> ok uTake1 X 2 + 1X >= 2 + 1X uTake1 mark X -> mark uTake1 X 1 + 1X >= 1 + 1X take(ok X1, ok X2) -> ok take(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 take(mark X1, X2) -> mark take(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 take(X1, mark X2) -> mark take(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 cons(ok X1, ok X2) -> ok cons(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 length ok X -> ok length X 2 + 1X >= 2 + 1X length mark X -> mark length X 1 + 1X >= 1 + 1X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 1 + 1X >= 1 + 1X isNat ok X -> ok isNat X 1 + 0X >= 2 + 0X isNatIList ok X -> ok isNatIList X 1 + 0X >= 2 + 0X isNatList ok X -> ok isNatList X 1 + 0X >= 2 + 0X and(ok X1, ok X2) -> ok and(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 and(mark X1, X2) -> mark and(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 and(X1, mark X2) -> mark and(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active uLength(tt(), L) -> mark s length L 1 + 0L >= 2 + 1L active uLength(X1, X2) -> uLength(active X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)) 2 + 0IL + 0N + 0M >= 0 + 0IL + 0N + 0M active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 0X4 >= 2 + 1X1 + 0X2 + 0X3 + 0X4 active uTake1 tt() -> mark nil() 2 >= 0 active uTake1 X -> uTake1 active X 2 + 1X >= 2 + 1X active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL) 2 + 0IL + 0N + 0M >= 4 + 0IL + 0N + 0M active take(0(), IL) -> mark uTake1 isNatIList IL 2 + 1IL >= 2 + 0IL active take(X1, X2) -> take(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active take(X1, X2) -> take(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active cons(X1, X2) -> cons(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active zeros() -> mark cons(0(), zeros()) 2 >= 0 active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L) 2 + 0N + 0L >= 2 + 0N + 0L active length X -> length active X 2 + 1X >= 2 + 1X active s X -> s active X 2 + 1X >= 2 + 1X active isNat length L -> mark isNatList L 2 + 0L >= 1 + 0L active isNat s N -> mark isNat N 2 + 0N >= 1 + 0N active isNat 0() -> mark tt() 2 >= 0 active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 2 + 0IL + 0N active isNatIList zeros() -> mark tt() 2 >= 0 active isNatIList IL -> mark isNatList IL 2 + 0IL >= 1 + 0IL active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 2 + 0IL + 0N active isNatList nil() -> mark tt() 2 >= 0 active isNatList cons(N, L) -> mark and(isNat N, isNatList L) 2 + 0N + 0L >= 2 + 0N + 0L active and(tt(), T) -> mark T 2 + 1T >= 0 + 1T active and(X1, X2) -> and(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active and(X1, X2) -> and(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 Qed SCC (1): Strict: {isNatList# ok X -> isNatList# X} Weak: { active and(X1, X2) -> and(X1, active X2), active and(X1, X2) -> and(active X1, X2), active and(tt(), T) -> mark T, active isNatList cons(N, L) -> mark and(isNat N, isNatList L), active isNatList nil() -> mark tt(), active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL), active isNatIList IL -> mark isNatList IL, active isNatIList zeros() -> mark tt(), active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL), active isNat 0() -> mark tt(), active isNat s N -> mark isNat N, active isNat length L -> mark isNatList L, active s X -> s active X, active length X -> length active X, active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L), active zeros() -> mark cons(0(), zeros()), active cons(X1, X2) -> cons(active X1, X2), active take(X1, X2) -> take(X1, active X2), active take(X1, X2) -> take(active X1, X2), active take(0(), IL) -> mark uTake1 isNatIList IL, active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), active uTake1 X -> uTake1 active X, active uTake1 tt() -> mark nil(), active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4), active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)), active uLength(X1, X2) -> uLength(active X1, X2), active uLength(tt(), L) -> mark s length L, and(X1, mark X2) -> mark and(X1, X2), and(mark X1, X2) -> mark and(X1, X2), and(ok X1, ok X2) -> ok and(X1, X2), isNatList ok X -> ok isNatList X, isNatIList ok X -> ok isNatIList X, isNat ok X -> ok isNat X, s mark X -> mark s X, s ok X -> ok s X, length mark X -> mark length X, length ok X -> ok length X, cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), take(X1, mark X2) -> mark take(X1, X2), take(mark X1, X2) -> mark take(X1, X2), take(ok X1, ok X2) -> ok take(X1, X2), uTake1 mark X -> mark uTake1 X, uTake1 ok X -> ok uTake1 X, uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4), uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4), uLength(mark X1, X2) -> mark uLength(X1, X2), uLength(ok X1, ok X2) -> ok uLength(X1, X2), proper and(X1, X2) -> and(proper X1, proper X2), proper tt() -> ok tt(), proper isNatList X -> isNatList proper X, proper isNatIList X -> isNatIList proper X, proper isNat X -> isNat proper X, proper 0() -> ok 0(), proper s X -> s proper X, proper length X -> length proper X, proper zeros() -> ok zeros(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper nil() -> ok nil(), proper take(X1, X2) -> take(proper X1, proper X2), proper uTake1 X -> uTake1 proper X, proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4), proper uLength(X1, X2) -> uLength(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [uTake2](x0, x1, x2, x3) = x0 + 1, [and](x0, x1) = x0 + 1, [cons](x0, x1) = 0, [take](x0, x1) = x0 + 1, [uLength](x0, x1) = x0, [mark](x0) = x0, [active](x0) = x0 + 1, [isNatList](x0) = 1, [isNatIList](x0) = 1, [isNat](x0) = 1, [s](x0) = x0 + 1, [length](x0) = x0 + 1, [uTake1](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [zeros] = 1, [nil] = 0, [isNatList#](x0) = x0 + 1 Strict: isNatList# ok X -> isNatList# X 2 + 1X >= 1 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper uLength(X1, X2) -> uLength(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4) 0 + 0X1 + 0X2 + 0X3 + 0X4 >= 1 + 0X1 + 0X2 + 0X3 + 0X4 proper uTake1 X -> uTake1 proper X 0 + 0X >= 1 + 0X proper take(X1, X2) -> take(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 1 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper zeros() -> ok zeros() 0 >= 2 proper length X -> length proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper isNatIList X -> isNatIList proper X 0 + 0X >= 1 + 0X proper isNatList X -> isNatList proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper and(X1, X2) -> and(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 uLength(ok X1, ok X2) -> ok uLength(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 uLength(mark X1, X2) -> mark uLength(X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 0X4 >= 2 + 1X1 + 0X2 + 0X3 + 0X4 uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4) 1 + 1X1 + 0X2 + 0X3 + 0X4 >= 1 + 1X1 + 0X2 + 0X3 + 0X4 uTake1 ok X -> ok uTake1 X 2 + 1X >= 2 + 1X uTake1 mark X -> mark uTake1 X 1 + 1X >= 1 + 1X take(ok X1, ok X2) -> ok take(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 take(mark X1, X2) -> mark take(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 take(X1, mark X2) -> mark take(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 cons(ok X1, ok X2) -> ok cons(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 length ok X -> ok length X 2 + 1X >= 2 + 1X length mark X -> mark length X 1 + 1X >= 1 + 1X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 1 + 1X >= 1 + 1X isNat ok X -> ok isNat X 1 + 0X >= 2 + 0X isNatIList ok X -> ok isNatIList X 1 + 0X >= 2 + 0X isNatList ok X -> ok isNatList X 1 + 0X >= 2 + 0X and(ok X1, ok X2) -> ok and(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 and(mark X1, X2) -> mark and(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 and(X1, mark X2) -> mark and(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active uLength(tt(), L) -> mark s length L 1 + 0L >= 2 + 1L active uLength(X1, X2) -> uLength(active X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)) 2 + 0IL + 0N + 0M >= 0 + 0IL + 0N + 0M active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 0X4 >= 2 + 1X1 + 0X2 + 0X3 + 0X4 active uTake1 tt() -> mark nil() 2 >= 0 active uTake1 X -> uTake1 active X 2 + 1X >= 2 + 1X active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL) 2 + 0IL + 0N + 0M >= 4 + 0IL + 0N + 0M active take(0(), IL) -> mark uTake1 isNatIList IL 2 + 1IL >= 2 + 0IL active take(X1, X2) -> take(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active take(X1, X2) -> take(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active cons(X1, X2) -> cons(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active zeros() -> mark cons(0(), zeros()) 2 >= 0 active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L) 2 + 0N + 0L >= 2 + 0N + 0L active length X -> length active X 2 + 1X >= 2 + 1X active s X -> s active X 2 + 1X >= 2 + 1X active isNat length L -> mark isNatList L 2 + 0L >= 1 + 0L active isNat s N -> mark isNat N 2 + 0N >= 1 + 0N active isNat 0() -> mark tt() 2 >= 0 active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 2 + 0IL + 0N active isNatIList zeros() -> mark tt() 2 >= 0 active isNatIList IL -> mark isNatList IL 2 + 0IL >= 1 + 0IL active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 2 + 0IL + 0N active isNatList nil() -> mark tt() 2 >= 0 active isNatList cons(N, L) -> mark and(isNat N, isNatList L) 2 + 0N + 0L >= 2 + 0N + 0L active and(tt(), T) -> mark T 2 + 1T >= 0 + 1T active and(X1, X2) -> and(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active and(X1, X2) -> and(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 Qed SCC (1): Strict: {isNat# ok X -> isNat# X} Weak: { active and(X1, X2) -> and(X1, active X2), active and(X1, X2) -> and(active X1, X2), active and(tt(), T) -> mark T, active isNatList cons(N, L) -> mark and(isNat N, isNatList L), active isNatList nil() -> mark tt(), active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL), active isNatIList IL -> mark isNatList IL, active isNatIList zeros() -> mark tt(), active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL), active isNat 0() -> mark tt(), active isNat s N -> mark isNat N, active isNat length L -> mark isNatList L, active s X -> s active X, active length X -> length active X, active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L), active zeros() -> mark cons(0(), zeros()), active cons(X1, X2) -> cons(active X1, X2), active take(X1, X2) -> take(X1, active X2), active take(X1, X2) -> take(active X1, X2), active take(0(), IL) -> mark uTake1 isNatIList IL, active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), active uTake1 X -> uTake1 active X, active uTake1 tt() -> mark nil(), active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4), active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)), active uLength(X1, X2) -> uLength(active X1, X2), active uLength(tt(), L) -> mark s length L, and(X1, mark X2) -> mark and(X1, X2), and(mark X1, X2) -> mark and(X1, X2), and(ok X1, ok X2) -> ok and(X1, X2), isNatList ok X -> ok isNatList X, isNatIList ok X -> ok isNatIList X, isNat ok X -> ok isNat X, s mark X -> mark s X, s ok X -> ok s X, length mark X -> mark length X, length ok X -> ok length X, cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), take(X1, mark X2) -> mark take(X1, X2), take(mark X1, X2) -> mark take(X1, X2), take(ok X1, ok X2) -> ok take(X1, X2), uTake1 mark X -> mark uTake1 X, uTake1 ok X -> ok uTake1 X, uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4), uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4), uLength(mark X1, X2) -> mark uLength(X1, X2), uLength(ok X1, ok X2) -> ok uLength(X1, X2), proper and(X1, X2) -> and(proper X1, proper X2), proper tt() -> ok tt(), proper isNatList X -> isNatList proper X, proper isNatIList X -> isNatIList proper X, proper isNat X -> isNat proper X, proper 0() -> ok 0(), proper s X -> s proper X, proper length X -> length proper X, proper zeros() -> ok zeros(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper nil() -> ok nil(), proper take(X1, X2) -> take(proper X1, proper X2), proper uTake1 X -> uTake1 proper X, proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4), proper uLength(X1, X2) -> uLength(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [uTake2](x0, x1, x2, x3) = x0 + 1, [and](x0, x1) = x0 + 1, [cons](x0, x1) = 0, [take](x0, x1) = x0 + 1, [uLength](x0, x1) = x0, [mark](x0) = x0, [active](x0) = x0 + 1, [isNatList](x0) = 1, [isNatIList](x0) = 1, [isNat](x0) = 1, [s](x0) = x0 + 1, [length](x0) = x0 + 1, [uTake1](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [zeros] = 1, [nil] = 0, [isNat#](x0) = x0 + 1 Strict: isNat# ok X -> isNat# X 2 + 1X >= 1 + 1X Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper uLength(X1, X2) -> uLength(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4) 0 + 0X1 + 0X2 + 0X3 + 0X4 >= 1 + 0X1 + 0X2 + 0X3 + 0X4 proper uTake1 X -> uTake1 proper X 0 + 0X >= 1 + 0X proper take(X1, X2) -> take(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 1 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper zeros() -> ok zeros() 0 >= 2 proper length X -> length proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper isNatIList X -> isNatIList proper X 0 + 0X >= 1 + 0X proper isNatList X -> isNatList proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper and(X1, X2) -> and(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 uLength(ok X1, ok X2) -> ok uLength(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 uLength(mark X1, X2) -> mark uLength(X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 0X4 >= 2 + 1X1 + 0X2 + 0X3 + 0X4 uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4) 1 + 1X1 + 0X2 + 0X3 + 0X4 >= 1 + 1X1 + 0X2 + 0X3 + 0X4 uTake1 ok X -> ok uTake1 X 2 + 1X >= 2 + 1X uTake1 mark X -> mark uTake1 X 1 + 1X >= 1 + 1X take(ok X1, ok X2) -> ok take(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 take(mark X1, X2) -> mark take(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 take(X1, mark X2) -> mark take(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 cons(ok X1, ok X2) -> ok cons(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 length ok X -> ok length X 2 + 1X >= 2 + 1X length mark X -> mark length X 1 + 1X >= 1 + 1X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 1 + 1X >= 1 + 1X isNat ok X -> ok isNat X 1 + 0X >= 2 + 0X isNatIList ok X -> ok isNatIList X 1 + 0X >= 2 + 0X isNatList ok X -> ok isNatList X 1 + 0X >= 2 + 0X and(ok X1, ok X2) -> ok and(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 and(mark X1, X2) -> mark and(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 and(X1, mark X2) -> mark and(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active uLength(tt(), L) -> mark s length L 1 + 0L >= 2 + 1L active uLength(X1, X2) -> uLength(active X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)) 2 + 0IL + 0N + 0M >= 0 + 0IL + 0N + 0M active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 0X4 >= 2 + 1X1 + 0X2 + 0X3 + 0X4 active uTake1 tt() -> mark nil() 2 >= 0 active uTake1 X -> uTake1 active X 2 + 1X >= 2 + 1X active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL) 2 + 0IL + 0N + 0M >= 4 + 0IL + 0N + 0M active take(0(), IL) -> mark uTake1 isNatIList IL 2 + 1IL >= 2 + 0IL active take(X1, X2) -> take(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active take(X1, X2) -> take(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active cons(X1, X2) -> cons(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active zeros() -> mark cons(0(), zeros()) 2 >= 0 active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L) 2 + 0N + 0L >= 2 + 0N + 0L active length X -> length active X 2 + 1X >= 2 + 1X active s X -> s active X 2 + 1X >= 2 + 1X active isNat length L -> mark isNatList L 2 + 0L >= 1 + 0L active isNat s N -> mark isNat N 2 + 0N >= 1 + 0N active isNat 0() -> mark tt() 2 >= 0 active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 2 + 0IL + 0N active isNatIList zeros() -> mark tt() 2 >= 0 active isNatIList IL -> mark isNatList IL 2 + 0IL >= 1 + 0IL active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 2 + 0IL + 0N active isNatList nil() -> mark tt() 2 >= 0 active isNatList cons(N, L) -> mark and(isNat N, isNatList L) 2 + 0N + 0L >= 2 + 0N + 0L active and(tt(), T) -> mark T 2 + 1T >= 0 + 1T active and(X1, X2) -> and(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active and(X1, X2) -> and(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 Qed SCC (2): Strict: { uTake2#(mark X1, X2, X3, X4) -> uTake2#(X1, X2, X3, X4), uTake2#(ok X1, ok X2, ok X3, ok X4) -> uTake2#(X1, X2, X3, X4)} Weak: { active and(X1, X2) -> and(X1, active X2), active and(X1, X2) -> and(active X1, X2), active and(tt(), T) -> mark T, active isNatList cons(N, L) -> mark and(isNat N, isNatList L), active isNatList nil() -> mark tt(), active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL), active isNatIList IL -> mark isNatList IL, active isNatIList zeros() -> mark tt(), active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL), active isNat 0() -> mark tt(), active isNat s N -> mark isNat N, active isNat length L -> mark isNatList L, active s X -> s active X, active length X -> length active X, active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L), active zeros() -> mark cons(0(), zeros()), active cons(X1, X2) -> cons(active X1, X2), active take(X1, X2) -> take(X1, active X2), active take(X1, X2) -> take(active X1, X2), active take(0(), IL) -> mark uTake1 isNatIList IL, active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), active uTake1 X -> uTake1 active X, active uTake1 tt() -> mark nil(), active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4), active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)), active uLength(X1, X2) -> uLength(active X1, X2), active uLength(tt(), L) -> mark s length L, and(X1, mark X2) -> mark and(X1, X2), and(mark X1, X2) -> mark and(X1, X2), and(ok X1, ok X2) -> ok and(X1, X2), isNatList ok X -> ok isNatList X, isNatIList ok X -> ok isNatIList X, isNat ok X -> ok isNat X, s mark X -> mark s X, s ok X -> ok s X, length mark X -> mark length X, length ok X -> ok length X, cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), take(X1, mark X2) -> mark take(X1, X2), take(mark X1, X2) -> mark take(X1, X2), take(ok X1, ok X2) -> ok take(X1, X2), uTake1 mark X -> mark uTake1 X, uTake1 ok X -> ok uTake1 X, uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4), uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4), uLength(mark X1, X2) -> mark uLength(X1, X2), uLength(ok X1, ok X2) -> ok uLength(X1, X2), proper and(X1, X2) -> and(proper X1, proper X2), proper tt() -> ok tt(), proper isNatList X -> isNatList proper X, proper isNatIList X -> isNatIList proper X, proper isNat X -> isNat proper X, proper 0() -> ok 0(), proper s X -> s proper X, proper length X -> length proper X, proper zeros() -> ok zeros(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper nil() -> ok nil(), proper take(X1, X2) -> take(proper X1, proper X2), proper uTake1 X -> uTake1 proper X, proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4), proper uLength(X1, X2) -> uLength(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [uTake2](x0, x1, x2, x3) = x0 + 1, [and](x0, x1) = x0 + 1, [cons](x0, x1) = 0, [take](x0, x1) = x0 + 1, [uLength](x0, x1) = x0 + 1, [mark](x0) = x0, [active](x0) = x0 + 1, [isNatList](x0) = 1, [isNatIList](x0) = 1, [isNat](x0) = 1, [s](x0) = x0 + 1, [length](x0) = x0 + 1, [uTake1](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [zeros] = 1, [nil] = 0, [uTake2#](x0, x1, x2, x3) = x0 Strict: uTake2#(ok X1, ok X2, ok X3, ok X4) -> uTake2#(X1, X2, X3, X4) 1 + 0X1 + 0X2 + 0X3 + 1X4 >= 0 + 0X1 + 0X2 + 0X3 + 1X4 uTake2#(mark X1, X2, X3, X4) -> uTake2#(X1, X2, X3, X4) 0 + 0X1 + 0X2 + 0X3 + 1X4 >= 0 + 0X1 + 0X2 + 0X3 + 1X4 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper uLength(X1, X2) -> uLength(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4) 0 + 0X1 + 0X2 + 0X3 + 0X4 >= 1 + 0X1 + 0X2 + 0X3 + 0X4 proper uTake1 X -> uTake1 proper X 0 + 0X >= 1 + 0X proper take(X1, X2) -> take(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 1 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper zeros() -> ok zeros() 0 >= 2 proper length X -> length proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper isNatIList X -> isNatIList proper X 0 + 0X >= 1 + 0X proper isNatList X -> isNatList proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper and(X1, X2) -> and(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 uLength(ok X1, ok X2) -> ok uLength(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 uLength(mark X1, X2) -> mark uLength(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 0X4 >= 2 + 1X1 + 0X2 + 0X3 + 0X4 uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4) 1 + 1X1 + 0X2 + 0X3 + 0X4 >= 1 + 1X1 + 0X2 + 0X3 + 0X4 uTake1 ok X -> ok uTake1 X 2 + 1X >= 2 + 1X uTake1 mark X -> mark uTake1 X 1 + 1X >= 1 + 1X take(ok X1, ok X2) -> ok take(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 take(mark X1, X2) -> mark take(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 take(X1, mark X2) -> mark take(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 cons(ok X1, ok X2) -> ok cons(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 length ok X -> ok length X 2 + 1X >= 2 + 1X length mark X -> mark length X 1 + 1X >= 1 + 1X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 1 + 1X >= 1 + 1X isNat ok X -> ok isNat X 1 + 0X >= 2 + 0X isNatIList ok X -> ok isNatIList X 1 + 0X >= 2 + 0X isNatList ok X -> ok isNatList X 1 + 0X >= 2 + 0X and(ok X1, ok X2) -> ok and(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 and(mark X1, X2) -> mark and(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 and(X1, mark X2) -> mark and(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active uLength(tt(), L) -> mark s length L 2 + 1L >= 2 + 1L active uLength(X1, X2) -> uLength(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)) 2 + 0IL + 0N + 0M >= 0 + 0IL + 0N + 0M active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 0X4 >= 2 + 1X1 + 0X2 + 0X3 + 0X4 active uTake1 tt() -> mark nil() 2 >= 0 active uTake1 X -> uTake1 active X 2 + 1X >= 2 + 1X active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL) 2 + 0IL + 0N + 0M >= 4 + 0IL + 0N + 0M active take(0(), IL) -> mark uTake1 isNatIList IL 2 + 1IL >= 2 + 0IL active take(X1, X2) -> take(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active take(X1, X2) -> take(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active cons(X1, X2) -> cons(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active zeros() -> mark cons(0(), zeros()) 2 >= 0 active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L) 2 + 0N + 0L >= 1 + 0N + 1L active length X -> length active X 2 + 1X >= 2 + 1X active s X -> s active X 2 + 1X >= 2 + 1X active isNat length L -> mark isNatList L 2 + 0L >= 1 + 0L active isNat s N -> mark isNat N 2 + 0N >= 1 + 0N active isNat 0() -> mark tt() 2 >= 0 active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 2 + 0IL + 0N active isNatIList zeros() -> mark tt() 2 >= 0 active isNatIList IL -> mark isNatList IL 2 + 0IL >= 1 + 0IL active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 2 + 0IL + 0N active isNatList nil() -> mark tt() 2 >= 0 active isNatList cons(N, L) -> mark and(isNat N, isNatList L) 2 + 0N + 0L >= 2 + 0N + 0L active and(tt(), T) -> mark T 2 + 1T >= 0 + 1T active and(X1, X2) -> and(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active and(X1, X2) -> and(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 SCCS (1): Scc: {uTake2#(mark X1, X2, X3, X4) -> uTake2#(X1, X2, X3, X4)} SCC (1): Strict: {uTake2#(mark X1, X2, X3, X4) -> uTake2#(X1, X2, X3, X4)} Weak: { active and(X1, X2) -> and(X1, active X2), active and(X1, X2) -> and(active X1, X2), active and(tt(), T) -> mark T, active isNatList cons(N, L) -> mark and(isNat N, isNatList L), active isNatList nil() -> mark tt(), active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL), active isNatIList IL -> mark isNatList IL, active isNatIList zeros() -> mark tt(), active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL), active isNat 0() -> mark tt(), active isNat s N -> mark isNat N, active isNat length L -> mark isNatList L, active s X -> s active X, active length X -> length active X, active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L), active zeros() -> mark cons(0(), zeros()), active cons(X1, X2) -> cons(active X1, X2), active take(X1, X2) -> take(X1, active X2), active take(X1, X2) -> take(active X1, X2), active take(0(), IL) -> mark uTake1 isNatIList IL, active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), active uTake1 X -> uTake1 active X, active uTake1 tt() -> mark nil(), active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4), active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)), active uLength(X1, X2) -> uLength(active X1, X2), active uLength(tt(), L) -> mark s length L, and(X1, mark X2) -> mark and(X1, X2), and(mark X1, X2) -> mark and(X1, X2), and(ok X1, ok X2) -> ok and(X1, X2), isNatList ok X -> ok isNatList X, isNatIList ok X -> ok isNatIList X, isNat ok X -> ok isNat X, s mark X -> mark s X, s ok X -> ok s X, length mark X -> mark length X, length ok X -> ok length X, cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), take(X1, mark X2) -> mark take(X1, X2), take(mark X1, X2) -> mark take(X1, X2), take(ok X1, ok X2) -> ok take(X1, X2), uTake1 mark X -> mark uTake1 X, uTake1 ok X -> ok uTake1 X, uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4), uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4), uLength(mark X1, X2) -> mark uLength(X1, X2), uLength(ok X1, ok X2) -> ok uLength(X1, X2), proper and(X1, X2) -> and(proper X1, proper X2), proper tt() -> ok tt(), proper isNatList X -> isNatList proper X, proper isNatIList X -> isNatIList proper X, proper isNat X -> isNat proper X, proper 0() -> ok 0(), proper s X -> s proper X, proper length X -> length proper X, proper zeros() -> ok zeros(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper nil() -> ok nil(), proper take(X1, X2) -> take(proper X1, proper X2), proper uTake1 X -> uTake1 proper X, proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4), proper uLength(X1, X2) -> uLength(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [uTake2](x0, x1, x2, x3) = x0 + x1 + 1, [and](x0, x1) = x0 + 1, [cons](x0, x1) = 0, [take](x0, x1) = x0 + 1, [uLength](x0, x1) = x0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [isNatList](x0) = 1, [isNatIList](x0) = 1, [isNat](x0) = 1, [s](x0) = x0 + 1, [length](x0) = x0 + 1, [uTake1](x0) = x0 + 1, [proper](x0) = x0 + 1, [ok](x0) = 0, [top](x0) = x0 + 1, [tt] = 1, [0] = 1, [zeros] = 1, [nil] = 1, [uTake2#](x0, x1, x2, x3) = x0 + 1 Strict: uTake2#(mark X1, X2, X3, X4) -> uTake2#(X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 0X4 >= 1 + 1X1 + 0X2 + 0X3 + 0X4 Weak: top ok X -> top active X 1 + 0X >= 2 + 1X top mark X -> top proper X 2 + 1X >= 2 + 1X proper uLength(X1, X2) -> uLength(proper X1, proper X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4) 2 + 1X1 + 0X2 + 1X3 + 0X4 >= 3 + 1X1 + 0X2 + 1X3 + 0X4 proper uTake1 X -> uTake1 proper X 2 + 1X >= 2 + 1X proper take(X1, X2) -> take(proper X1, proper X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 proper nil() -> ok nil() 2 >= 0 proper cons(X1, X2) -> cons(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper zeros() -> ok zeros() 2 >= 0 proper length X -> length proper X 2 + 1X >= 2 + 1X proper s X -> s proper X 2 + 1X >= 2 + 1X proper 0() -> ok 0() 2 >= 0 proper isNat X -> isNat proper X 2 + 0X >= 1 + 0X proper isNatIList X -> isNatIList proper X 2 + 0X >= 1 + 0X proper isNatList X -> isNatList proper X 2 + 0X >= 1 + 0X proper tt() -> ok tt() 2 >= 0 proper and(X1, X2) -> and(proper X1, proper X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 uLength(ok X1, ok X2) -> ok uLength(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 uLength(mark X1, X2) -> mark uLength(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4) 1 + 0X1 + 0X2 + 0X3 + 0X4 >= 0 + 0X1 + 0X2 + 0X3 + 0X4 uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4) 2 + 1X1 + 0X2 + 1X3 + 0X4 >= 2 + 1X1 + 0X2 + 1X3 + 0X4 uTake1 ok X -> ok uTake1 X 1 + 0X >= 0 + 0X uTake1 mark X -> mark uTake1 X 2 + 1X >= 2 + 1X take(ok X1, ok X2) -> ok take(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 take(mark X1, X2) -> mark take(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 take(X1, mark X2) -> mark take(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 cons(ok X1, ok X2) -> ok cons(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 length ok X -> ok length X 1 + 0X >= 0 + 0X length mark X -> mark length X 2 + 1X >= 2 + 1X s ok X -> ok s X 1 + 0X >= 0 + 0X s mark X -> mark s X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 1 + 0X >= 0 + 0X isNatIList ok X -> ok isNatIList X 1 + 0X >= 0 + 0X isNatList ok X -> ok isNatList X 1 + 0X >= 0 + 0X and(ok X1, ok X2) -> ok and(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 and(mark X1, X2) -> mark and(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 and(X1, mark X2) -> mark and(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active uLength(tt(), L) -> mark s length L 2 + 0L >= 3 + 1L active uLength(X1, X2) -> uLength(active X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)) 3 + 0IL + 1N + 0M >= 1 + 0IL + 0N + 0M active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4) 2 + 1X1 + 0X2 + 1X3 + 0X4 >= 2 + 1X1 + 0X2 + 1X3 + 0X4 active uTake1 tt() -> mark nil() 3 >= 2 active uTake1 X -> uTake1 active X 2 + 1X >= 2 + 1X active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL) 2 + 0IL + 0N + 0M >= 5 + 0IL + 1N + 0M active take(0(), IL) -> mark uTake1 isNatIList IL 2 + 1IL >= 3 + 0IL active take(X1, X2) -> take(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active take(X1, X2) -> take(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active cons(X1, X2) -> cons(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active zeros() -> mark cons(0(), zeros()) 2 >= 1 active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L) 2 + 0N + 0L >= 3 + 0N + 0L active length X -> length active X 2 + 1X >= 2 + 1X active s X -> s active X 2 + 1X >= 2 + 1X active isNat length L -> mark isNatList L 2 + 0L >= 2 + 0L active isNat s N -> mark isNat N 2 + 0N >= 2 + 0N active isNat 0() -> mark tt() 2 >= 2 active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 3 + 0IL + 0N active isNatIList zeros() -> mark tt() 2 >= 2 active isNatIList IL -> mark isNatList IL 2 + 0IL >= 2 + 0IL active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 3 + 0IL + 0N active isNatList nil() -> mark tt() 2 >= 2 active isNatList cons(N, L) -> mark and(isNat N, isNatList L) 2 + 0N + 0L >= 3 + 0N + 0L active and(tt(), T) -> mark T 2 + 1T >= 1 + 1T active and(X1, X2) -> and(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active and(X1, X2) -> and(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 Qed SCC (2): Strict: { uLength#(mark X1, X2) -> uLength#(X1, X2), uLength#(ok X1, ok X2) -> uLength#(X1, X2)} Weak: { active and(X1, X2) -> and(X1, active X2), active and(X1, X2) -> and(active X1, X2), active and(tt(), T) -> mark T, active isNatList cons(N, L) -> mark and(isNat N, isNatList L), active isNatList nil() -> mark tt(), active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL), active isNatIList IL -> mark isNatList IL, active isNatIList zeros() -> mark tt(), active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL), active isNat 0() -> mark tt(), active isNat s N -> mark isNat N, active isNat length L -> mark isNatList L, active s X -> s active X, active length X -> length active X, active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L), active zeros() -> mark cons(0(), zeros()), active cons(X1, X2) -> cons(active X1, X2), active take(X1, X2) -> take(X1, active X2), active take(X1, X2) -> take(active X1, X2), active take(0(), IL) -> mark uTake1 isNatIList IL, active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), active uTake1 X -> uTake1 active X, active uTake1 tt() -> mark nil(), active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4), active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)), active uLength(X1, X2) -> uLength(active X1, X2), active uLength(tt(), L) -> mark s length L, and(X1, mark X2) -> mark and(X1, X2), and(mark X1, X2) -> mark and(X1, X2), and(ok X1, ok X2) -> ok and(X1, X2), isNatList ok X -> ok isNatList X, isNatIList ok X -> ok isNatIList X, isNat ok X -> ok isNat X, s mark X -> mark s X, s ok X -> ok s X, length mark X -> mark length X, length ok X -> ok length X, cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), take(X1, mark X2) -> mark take(X1, X2), take(mark X1, X2) -> mark take(X1, X2), take(ok X1, ok X2) -> ok take(X1, X2), uTake1 mark X -> mark uTake1 X, uTake1 ok X -> ok uTake1 X, uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4), uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4), uLength(mark X1, X2) -> mark uLength(X1, X2), uLength(ok X1, ok X2) -> ok uLength(X1, X2), proper and(X1, X2) -> and(proper X1, proper X2), proper tt() -> ok tt(), proper isNatList X -> isNatList proper X, proper isNatIList X -> isNatIList proper X, proper isNat X -> isNat proper X, proper 0() -> ok 0(), proper s X -> s proper X, proper length X -> length proper X, proper zeros() -> ok zeros(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper nil() -> ok nil(), proper take(X1, X2) -> take(proper X1, proper X2), proper uTake1 X -> uTake1 proper X, proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4), proper uLength(X1, X2) -> uLength(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [uTake2](x0, x1, x2, x3) = x0 + 1, [and](x0, x1) = x0 + 1, [cons](x0, x1) = x0 + x1 + 1, [take](x0, x1) = x0 + 1, [uLength](x0, x1) = x0, [mark](x0) = x0, [active](x0) = x0 + 1, [isNatList](x0) = 1, [isNatIList](x0) = 1, [isNat](x0) = 1, [s](x0) = x0 + 1, [length](x0) = 1, [uTake1](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 0, [zeros] = 1, [nil] = 0, [uLength#](x0, x1) = x0 Strict: uLength#(ok X1, ok X2) -> uLength#(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 1X2 uLength#(mark X1, X2) -> uLength#(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 1X2 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper uLength(X1, X2) -> uLength(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4) 0 + 0X1 + 0X2 + 0X3 + 0X4 >= 1 + 0X1 + 0X2 + 0X3 + 0X4 proper uTake1 X -> uTake1 proper X 0 + 0X >= 1 + 0X proper take(X1, X2) -> take(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 1 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper zeros() -> ok zeros() 0 >= 2 proper length X -> length proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 1 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper isNatIList X -> isNatIList proper X 0 + 0X >= 1 + 0X proper isNatList X -> isNatList proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper and(X1, X2) -> and(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 uLength(ok X1, ok X2) -> ok uLength(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 uLength(mark X1, X2) -> mark uLength(X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 0X4 >= 2 + 1X1 + 0X2 + 0X3 + 0X4 uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4) 1 + 1X1 + 0X2 + 0X3 + 0X4 >= 1 + 1X1 + 0X2 + 0X3 + 0X4 uTake1 ok X -> ok uTake1 X 2 + 1X >= 2 + 1X uTake1 mark X -> mark uTake1 X 1 + 1X >= 1 + 1X take(ok X1, ok X2) -> ok take(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 take(mark X1, X2) -> mark take(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 take(X1, mark X2) -> mark take(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 cons(ok X1, ok X2) -> ok cons(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 length ok X -> ok length X 1 + 0X >= 2 + 0X length mark X -> mark length X 1 + 0X >= 1 + 0X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 1 + 1X >= 1 + 1X isNat ok X -> ok isNat X 1 + 0X >= 2 + 0X isNatIList ok X -> ok isNatIList X 1 + 0X >= 2 + 0X isNatList ok X -> ok isNatList X 1 + 0X >= 2 + 0X and(ok X1, ok X2) -> ok and(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 and(mark X1, X2) -> mark and(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 and(X1, mark X2) -> mark and(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active uLength(tt(), L) -> mark s length L 1 + 0L >= 2 + 0L active uLength(X1, X2) -> uLength(active X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)) 2 + 0IL + 0N + 0M >= 2 + 1IL + 1N + 0M active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 0X4 >= 2 + 1X1 + 0X2 + 0X3 + 0X4 active uTake1 tt() -> mark nil() 2 >= 0 active uTake1 X -> uTake1 active X 2 + 1X >= 2 + 1X active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL) 3 + 1IL + 1N + 0M >= 4 + 0IL + 0N + 0M active take(0(), IL) -> mark uTake1 isNatIList IL 2 + 1IL >= 2 + 0IL active take(X1, X2) -> take(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active take(X1, X2) -> take(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active zeros() -> mark cons(0(), zeros()) 2 >= 2 active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L) 2 + 0N + 0L >= 2 + 0N + 0L active length X -> length active X 2 + 0X >= 1 + 0X active s X -> s active X 2 + 1X >= 2 + 1X active isNat length L -> mark isNatList L 2 + 0L >= 1 + 0L active isNat s N -> mark isNat N 2 + 0N >= 1 + 0N active isNat 0() -> mark tt() 2 >= 0 active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 2 + 0IL + 0N active isNatIList zeros() -> mark tt() 2 >= 0 active isNatIList IL -> mark isNatList IL 2 + 0IL >= 1 + 0IL active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 2 + 0IL + 0N active isNatList nil() -> mark tt() 2 >= 0 active isNatList cons(N, L) -> mark and(isNat N, isNatList L) 2 + 0N + 0L >= 2 + 0N + 0L active and(tt(), T) -> mark T 2 + 1T >= 0 + 1T active and(X1, X2) -> and(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active and(X1, X2) -> and(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 SCCS (1): Scc: {uLength#(mark X1, X2) -> uLength#(X1, X2)} SCC (1): Strict: {uLength#(mark X1, X2) -> uLength#(X1, X2)} Weak: { active and(X1, X2) -> and(X1, active X2), active and(X1, X2) -> and(active X1, X2), active and(tt(), T) -> mark T, active isNatList cons(N, L) -> mark and(isNat N, isNatList L), active isNatList nil() -> mark tt(), active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL), active isNatIList IL -> mark isNatList IL, active isNatIList zeros() -> mark tt(), active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL), active isNat 0() -> mark tt(), active isNat s N -> mark isNat N, active isNat length L -> mark isNatList L, active s X -> s active X, active length X -> length active X, active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L), active zeros() -> mark cons(0(), zeros()), active cons(X1, X2) -> cons(active X1, X2), active take(X1, X2) -> take(X1, active X2), active take(X1, X2) -> take(active X1, X2), active take(0(), IL) -> mark uTake1 isNatIList IL, active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), active uTake1 X -> uTake1 active X, active uTake1 tt() -> mark nil(), active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4), active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)), active uLength(X1, X2) -> uLength(active X1, X2), active uLength(tt(), L) -> mark s length L, and(X1, mark X2) -> mark and(X1, X2), and(mark X1, X2) -> mark and(X1, X2), and(ok X1, ok X2) -> ok and(X1, X2), isNatList ok X -> ok isNatList X, isNatIList ok X -> ok isNatIList X, isNat ok X -> ok isNat X, s mark X -> mark s X, s ok X -> ok s X, length mark X -> mark length X, length ok X -> ok length X, cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), take(X1, mark X2) -> mark take(X1, X2), take(mark X1, X2) -> mark take(X1, X2), take(ok X1, ok X2) -> ok take(X1, X2), uTake1 mark X -> mark uTake1 X, uTake1 ok X -> ok uTake1 X, uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4), uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4), uLength(mark X1, X2) -> mark uLength(X1, X2), uLength(ok X1, ok X2) -> ok uLength(X1, X2), proper and(X1, X2) -> and(proper X1, proper X2), proper tt() -> ok tt(), proper isNatList X -> isNatList proper X, proper isNatIList X -> isNatIList proper X, proper isNat X -> isNat proper X, proper 0() -> ok 0(), proper s X -> s proper X, proper length X -> length proper X, proper zeros() -> ok zeros(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper nil() -> ok nil(), proper take(X1, X2) -> take(proper X1, proper X2), proper uTake1 X -> uTake1 proper X, proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4), proper uLength(X1, X2) -> uLength(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [uTake2](x0, x1, x2, x3) = x0 + x1 + 1, [and](x0, x1) = x0 + 1, [cons](x0, x1) = 0, [take](x0, x1) = x0 + 1, [uLength](x0, x1) = x0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [isNatList](x0) = 1, [isNatIList](x0) = 1, [isNat](x0) = 1, [s](x0) = x0 + 1, [length](x0) = x0 + 1, [uTake1](x0) = x0 + 1, [proper](x0) = x0 + 1, [ok](x0) = 0, [top](x0) = x0 + 1, [tt] = 1, [0] = 1, [zeros] = 1, [nil] = 1, [uLength#](x0, x1) = x0 + 1 Strict: uLength#(mark X1, X2) -> uLength#(X1, X2) 2 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 Weak: top ok X -> top active X 1 + 0X >= 2 + 1X top mark X -> top proper X 2 + 1X >= 2 + 1X proper uLength(X1, X2) -> uLength(proper X1, proper X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4) 2 + 1X1 + 0X2 + 1X3 + 0X4 >= 3 + 1X1 + 0X2 + 1X3 + 0X4 proper uTake1 X -> uTake1 proper X 2 + 1X >= 2 + 1X proper take(X1, X2) -> take(proper X1, proper X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 proper nil() -> ok nil() 2 >= 0 proper cons(X1, X2) -> cons(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper zeros() -> ok zeros() 2 >= 0 proper length X -> length proper X 2 + 1X >= 2 + 1X proper s X -> s proper X 2 + 1X >= 2 + 1X proper 0() -> ok 0() 2 >= 0 proper isNat X -> isNat proper X 2 + 0X >= 1 + 0X proper isNatIList X -> isNatIList proper X 2 + 0X >= 1 + 0X proper isNatList X -> isNatList proper X 2 + 0X >= 1 + 0X proper tt() -> ok tt() 2 >= 0 proper and(X1, X2) -> and(proper X1, proper X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 uLength(ok X1, ok X2) -> ok uLength(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 uLength(mark X1, X2) -> mark uLength(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4) 1 + 0X1 + 0X2 + 0X3 + 0X4 >= 0 + 0X1 + 0X2 + 0X3 + 0X4 uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4) 2 + 1X1 + 0X2 + 1X3 + 0X4 >= 2 + 1X1 + 0X2 + 1X3 + 0X4 uTake1 ok X -> ok uTake1 X 1 + 0X >= 0 + 0X uTake1 mark X -> mark uTake1 X 2 + 1X >= 2 + 1X take(ok X1, ok X2) -> ok take(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 take(mark X1, X2) -> mark take(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 take(X1, mark X2) -> mark take(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 cons(ok X1, ok X2) -> ok cons(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 length ok X -> ok length X 1 + 0X >= 0 + 0X length mark X -> mark length X 2 + 1X >= 2 + 1X s ok X -> ok s X 1 + 0X >= 0 + 0X s mark X -> mark s X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 1 + 0X >= 0 + 0X isNatIList ok X -> ok isNatIList X 1 + 0X >= 0 + 0X isNatList ok X -> ok isNatList X 1 + 0X >= 0 + 0X and(ok X1, ok X2) -> ok and(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 and(mark X1, X2) -> mark and(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 and(X1, mark X2) -> mark and(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active uLength(tt(), L) -> mark s length L 2 + 0L >= 3 + 1L active uLength(X1, X2) -> uLength(active X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)) 3 + 0IL + 1N + 0M >= 1 + 0IL + 0N + 0M active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4) 2 + 1X1 + 0X2 + 1X3 + 0X4 >= 2 + 1X1 + 0X2 + 1X3 + 0X4 active uTake1 tt() -> mark nil() 3 >= 2 active uTake1 X -> uTake1 active X 2 + 1X >= 2 + 1X active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL) 2 + 0IL + 0N + 0M >= 5 + 0IL + 1N + 0M active take(0(), IL) -> mark uTake1 isNatIList IL 2 + 1IL >= 3 + 0IL active take(X1, X2) -> take(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active take(X1, X2) -> take(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active cons(X1, X2) -> cons(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active zeros() -> mark cons(0(), zeros()) 2 >= 1 active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L) 2 + 0N + 0L >= 3 + 0N + 0L active length X -> length active X 2 + 1X >= 2 + 1X active s X -> s active X 2 + 1X >= 2 + 1X active isNat length L -> mark isNatList L 2 + 0L >= 2 + 0L active isNat s N -> mark isNat N 2 + 0N >= 2 + 0N active isNat 0() -> mark tt() 2 >= 2 active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 3 + 0IL + 0N active isNatIList zeros() -> mark tt() 2 >= 2 active isNatIList IL -> mark isNatList IL 2 + 0IL >= 2 + 0IL active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 3 + 0IL + 0N active isNatList nil() -> mark tt() 2 >= 2 active isNatList cons(N, L) -> mark and(isNat N, isNatList L) 2 + 0N + 0L >= 3 + 0N + 0L active and(tt(), T) -> mark T 2 + 1T >= 1 + 1T active and(X1, X2) -> and(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active and(X1, X2) -> and(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 Qed SCC (3): Strict: { take#(X1, mark X2) -> take#(X1, X2), take#(mark X1, X2) -> take#(X1, X2), take#(ok X1, ok X2) -> take#(X1, X2)} Weak: { active and(X1, X2) -> and(X1, active X2), active and(X1, X2) -> and(active X1, X2), active and(tt(), T) -> mark T, active isNatList cons(N, L) -> mark and(isNat N, isNatList L), active isNatList nil() -> mark tt(), active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL), active isNatIList IL -> mark isNatList IL, active isNatIList zeros() -> mark tt(), active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL), active isNat 0() -> mark tt(), active isNat s N -> mark isNat N, active isNat length L -> mark isNatList L, active s X -> s active X, active length X -> length active X, active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L), active zeros() -> mark cons(0(), zeros()), active cons(X1, X2) -> cons(active X1, X2), active take(X1, X2) -> take(X1, active X2), active take(X1, X2) -> take(active X1, X2), active take(0(), IL) -> mark uTake1 isNatIList IL, active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), active uTake1 X -> uTake1 active X, active uTake1 tt() -> mark nil(), active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4), active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)), active uLength(X1, X2) -> uLength(active X1, X2), active uLength(tt(), L) -> mark s length L, and(X1, mark X2) -> mark and(X1, X2), and(mark X1, X2) -> mark and(X1, X2), and(ok X1, ok X2) -> ok and(X1, X2), isNatList ok X -> ok isNatList X, isNatIList ok X -> ok isNatIList X, isNat ok X -> ok isNat X, s mark X -> mark s X, s ok X -> ok s X, length mark X -> mark length X, length ok X -> ok length X, cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), take(X1, mark X2) -> mark take(X1, X2), take(mark X1, X2) -> mark take(X1, X2), take(ok X1, ok X2) -> ok take(X1, X2), uTake1 mark X -> mark uTake1 X, uTake1 ok X -> ok uTake1 X, uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4), uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4), uLength(mark X1, X2) -> mark uLength(X1, X2), uLength(ok X1, ok X2) -> ok uLength(X1, X2), proper and(X1, X2) -> and(proper X1, proper X2), proper tt() -> ok tt(), proper isNatList X -> isNatList proper X, proper isNatIList X -> isNatIList proper X, proper isNat X -> isNat proper X, proper 0() -> ok 0(), proper s X -> s proper X, proper length X -> length proper X, proper zeros() -> ok zeros(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper nil() -> ok nil(), proper take(X1, X2) -> take(proper X1, proper X2), proper uTake1 X -> uTake1 proper X, proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4), proper uLength(X1, X2) -> uLength(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [uTake2](x0, x1, x2, x3) = x0 + 1, [and](x0, x1) = x0 + x1, [cons](x0, x1) = 1, [take](x0, x1) = 1, [uLength](x0, x1) = 0, [mark](x0) = x0 + 1, [active](x0) = 0, [isNatList](x0) = x0 + 1, [isNatIList](x0) = x0 + 1, [isNat](x0) = 1, [s](x0) = 0, [length](x0) = 0, [uTake1](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [zeros] = 1, [nil] = 1, [take#](x0, x1) = x0 + 1 Strict: take#(ok X1, ok X2) -> take#(X1, X2) 2 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 take#(mark X1, X2) -> take#(X1, X2) 2 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 take#(X1, mark X2) -> take#(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper uLength(X1, X2) -> uLength(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4) 0 + 0X1 + 0X2 + 0X3 + 0X4 >= 1 + 0X1 + 0X2 + 0X3 + 0X4 proper uTake1 X -> uTake1 proper X 0 + 0X >= 1 + 0X proper take(X1, X2) -> take(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper zeros() -> ok zeros() 0 >= 2 proper length X -> length proper X 0 + 0X >= 0 + 0X proper s X -> s proper X 0 + 0X >= 0 + 0X proper 0() -> ok 0() 0 >= 2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper isNatIList X -> isNatIList proper X 0 + 0X >= 1 + 0X proper isNatList X -> isNatList proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper and(X1, X2) -> and(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 uLength(ok X1, ok X2) -> ok uLength(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 uLength(mark X1, X2) -> mark uLength(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 0X4 >= 2 + 1X1 + 0X2 + 0X3 + 0X4 uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 0X4 >= 2 + 1X1 + 0X2 + 0X3 + 0X4 uTake1 ok X -> ok uTake1 X 2 + 1X >= 2 + 1X uTake1 mark X -> mark uTake1 X 2 + 1X >= 2 + 1X take(ok X1, ok X2) -> ok take(X1, X2) 1 + 0X1 + 0X2 >= 2 + 0X1 + 0X2 take(mark X1, X2) -> mark take(X1, X2) 1 + 0X1 + 0X2 >= 2 + 0X1 + 0X2 take(X1, mark X2) -> mark take(X1, X2) 1 + 0X1 + 0X2 >= 2 + 0X1 + 0X2 cons(ok X1, ok X2) -> ok cons(X1, X2) 1 + 0X1 + 0X2 >= 2 + 0X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 1 + 0X1 + 0X2 >= 2 + 0X1 + 0X2 length ok X -> ok length X 0 + 0X >= 1 + 0X length mark X -> mark length X 0 + 0X >= 1 + 0X s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X isNat ok X -> ok isNat X 1 + 0X >= 2 + 0X isNatIList ok X -> ok isNatIList X 2 + 1X >= 2 + 1X isNatList ok X -> ok isNatList X 2 + 1X >= 2 + 1X and(ok X1, ok X2) -> ok and(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 and(mark X1, X2) -> mark and(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 and(X1, mark X2) -> mark and(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 active uLength(tt(), L) -> mark s length L 0 + 0L >= 1 + 0L active uLength(X1, X2) -> uLength(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)) 0 + 0IL + 0N + 0M >= 2 + 0IL + 0N + 0M active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4) 0 + 0X1 + 0X2 + 0X3 + 0X4 >= 1 + 0X1 + 0X2 + 0X3 + 0X4 active uTake1 tt() -> mark nil() 0 >= 2 active uTake1 X -> uTake1 active X 0 + 0X >= 1 + 0X active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL) 0 + 0IL + 0N + 0M >= 5 + 1IL + 0N + 0M active take(0(), IL) -> mark uTake1 isNatIList IL 0 + 0IL >= 3 + 1IL active take(X1, X2) -> take(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active take(X1, X2) -> take(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active zeros() -> mark cons(0(), zeros()) 0 >= 2 active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L) 0 + 0N + 0L >= 1 + 0N + 0L active length X -> length active X 0 + 0X >= 0 + 0X active s X -> s active X 0 + 0X >= 0 + 0X active isNat length L -> mark isNatList L 0 + 0L >= 2 + 1L active isNat s N -> mark isNat N 0 + 0N >= 2 + 0N active isNat 0() -> mark tt() 0 >= 1 active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL) 0 + 0IL + 0N >= 3 + 1IL + 0N active isNatIList zeros() -> mark tt() 0 >= 1 active isNatIList IL -> mark isNatList IL 0 + 0IL >= 2 + 1IL active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL) 0 + 0IL + 0N >= 3 + 1IL + 0N active isNatList nil() -> mark tt() 0 >= 1 active isNatList cons(N, L) -> mark and(isNat N, isNatList L) 0 + 0N + 0L >= 3 + 0N + 1L active and(tt(), T) -> mark T 0 + 0T >= 1 + 1T active and(X1, X2) -> and(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active and(X1, X2) -> and(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 1X1 + 0X2 SCCS (1): Scc: {take#(X1, mark X2) -> take#(X1, X2)} SCC (1): Strict: {take#(X1, mark X2) -> take#(X1, X2)} Weak: { active and(X1, X2) -> and(X1, active X2), active and(X1, X2) -> and(active X1, X2), active and(tt(), T) -> mark T, active isNatList cons(N, L) -> mark and(isNat N, isNatList L), active isNatList nil() -> mark tt(), active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL), active isNatIList IL -> mark isNatList IL, active isNatIList zeros() -> mark tt(), active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL), active isNat 0() -> mark tt(), active isNat s N -> mark isNat N, active isNat length L -> mark isNatList L, active s X -> s active X, active length X -> length active X, active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L), active zeros() -> mark cons(0(), zeros()), active cons(X1, X2) -> cons(active X1, X2), active take(X1, X2) -> take(X1, active X2), active take(X1, X2) -> take(active X1, X2), active take(0(), IL) -> mark uTake1 isNatIList IL, active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), active uTake1 X -> uTake1 active X, active uTake1 tt() -> mark nil(), active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4), active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)), active uLength(X1, X2) -> uLength(active X1, X2), active uLength(tt(), L) -> mark s length L, and(X1, mark X2) -> mark and(X1, X2), and(mark X1, X2) -> mark and(X1, X2), and(ok X1, ok X2) -> ok and(X1, X2), isNatList ok X -> ok isNatList X, isNatIList ok X -> ok isNatIList X, isNat ok X -> ok isNat X, s mark X -> mark s X, s ok X -> ok s X, length mark X -> mark length X, length ok X -> ok length X, cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), take(X1, mark X2) -> mark take(X1, X2), take(mark X1, X2) -> mark take(X1, X2), take(ok X1, ok X2) -> ok take(X1, X2), uTake1 mark X -> mark uTake1 X, uTake1 ok X -> ok uTake1 X, uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4), uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4), uLength(mark X1, X2) -> mark uLength(X1, X2), uLength(ok X1, ok X2) -> ok uLength(X1, X2), proper and(X1, X2) -> and(proper X1, proper X2), proper tt() -> ok tt(), proper isNatList X -> isNatList proper X, proper isNatIList X -> isNatIList proper X, proper isNat X -> isNat proper X, proper 0() -> ok 0(), proper s X -> s proper X, proper length X -> length proper X, proper zeros() -> ok zeros(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper nil() -> ok nil(), proper take(X1, X2) -> take(proper X1, proper X2), proper uTake1 X -> uTake1 proper X, proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4), proper uLength(X1, X2) -> uLength(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [uTake2](x0, x1, x2, x3) = x0 + x1 + 1, [and](x0, x1) = x0 + 1, [cons](x0, x1) = 0, [take](x0, x1) = x0 + 1, [uLength](x0, x1) = x0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [isNatList](x0) = 1, [isNatIList](x0) = 1, [isNat](x0) = 1, [s](x0) = x0 + 1, [length](x0) = x0 + 1, [uTake1](x0) = x0 + 1, [proper](x0) = x0 + 1, [ok](x0) = 0, [top](x0) = x0 + 1, [tt] = 1, [0] = 1, [zeros] = 1, [nil] = 1, [take#](x0, x1) = x0 + 1 Strict: take#(X1, mark X2) -> take#(X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 Weak: top ok X -> top active X 1 + 0X >= 2 + 1X top mark X -> top proper X 2 + 1X >= 2 + 1X proper uLength(X1, X2) -> uLength(proper X1, proper X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4) 2 + 1X1 + 0X2 + 1X3 + 0X4 >= 3 + 1X1 + 0X2 + 1X3 + 0X4 proper uTake1 X -> uTake1 proper X 2 + 1X >= 2 + 1X proper take(X1, X2) -> take(proper X1, proper X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 proper nil() -> ok nil() 2 >= 0 proper cons(X1, X2) -> cons(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper zeros() -> ok zeros() 2 >= 0 proper length X -> length proper X 2 + 1X >= 2 + 1X proper s X -> s proper X 2 + 1X >= 2 + 1X proper 0() -> ok 0() 2 >= 0 proper isNat X -> isNat proper X 2 + 0X >= 1 + 0X proper isNatIList X -> isNatIList proper X 2 + 0X >= 1 + 0X proper isNatList X -> isNatList proper X 2 + 0X >= 1 + 0X proper tt() -> ok tt() 2 >= 0 proper and(X1, X2) -> and(proper X1, proper X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 uLength(ok X1, ok X2) -> ok uLength(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 uLength(mark X1, X2) -> mark uLength(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4) 1 + 0X1 + 0X2 + 0X3 + 0X4 >= 0 + 0X1 + 0X2 + 0X3 + 0X4 uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4) 2 + 1X1 + 0X2 + 1X3 + 0X4 >= 2 + 1X1 + 0X2 + 1X3 + 0X4 uTake1 ok X -> ok uTake1 X 1 + 0X >= 0 + 0X uTake1 mark X -> mark uTake1 X 2 + 1X >= 2 + 1X take(ok X1, ok X2) -> ok take(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 take(mark X1, X2) -> mark take(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 take(X1, mark X2) -> mark take(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 cons(ok X1, ok X2) -> ok cons(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 length ok X -> ok length X 1 + 0X >= 0 + 0X length mark X -> mark length X 2 + 1X >= 2 + 1X s ok X -> ok s X 1 + 0X >= 0 + 0X s mark X -> mark s X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 1 + 0X >= 0 + 0X isNatIList ok X -> ok isNatIList X 1 + 0X >= 0 + 0X isNatList ok X -> ok isNatList X 1 + 0X >= 0 + 0X and(ok X1, ok X2) -> ok and(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 and(mark X1, X2) -> mark and(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 and(X1, mark X2) -> mark and(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active uLength(tt(), L) -> mark s length L 2 + 0L >= 3 + 1L active uLength(X1, X2) -> uLength(active X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)) 3 + 0IL + 1N + 0M >= 1 + 0IL + 0N + 0M active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4) 2 + 1X1 + 0X2 + 1X3 + 0X4 >= 2 + 1X1 + 0X2 + 1X3 + 0X4 active uTake1 tt() -> mark nil() 3 >= 2 active uTake1 X -> uTake1 active X 2 + 1X >= 2 + 1X active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL) 2 + 0IL + 0N + 0M >= 5 + 0IL + 1N + 0M active take(0(), IL) -> mark uTake1 isNatIList IL 2 + 1IL >= 3 + 0IL active take(X1, X2) -> take(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active take(X1, X2) -> take(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active cons(X1, X2) -> cons(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active zeros() -> mark cons(0(), zeros()) 2 >= 1 active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L) 2 + 0N + 0L >= 3 + 0N + 0L active length X -> length active X 2 + 1X >= 2 + 1X active s X -> s active X 2 + 1X >= 2 + 1X active isNat length L -> mark isNatList L 2 + 0L >= 2 + 0L active isNat s N -> mark isNat N 2 + 0N >= 2 + 0N active isNat 0() -> mark tt() 2 >= 2 active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 3 + 0IL + 0N active isNatIList zeros() -> mark tt() 2 >= 2 active isNatIList IL -> mark isNatList IL 2 + 0IL >= 2 + 0IL active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 3 + 0IL + 0N active isNatList nil() -> mark tt() 2 >= 2 active isNatList cons(N, L) -> mark and(isNat N, isNatList L) 2 + 0N + 0L >= 3 + 0N + 0L active and(tt(), T) -> mark T 2 + 1T >= 1 + 1T active and(X1, X2) -> and(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active and(X1, X2) -> and(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 Qed SCC (2): Strict: { cons#(mark X1, X2) -> cons#(X1, X2), cons#(ok X1, ok X2) -> cons#(X1, X2)} Weak: { active and(X1, X2) -> and(X1, active X2), active and(X1, X2) -> and(active X1, X2), active and(tt(), T) -> mark T, active isNatList cons(N, L) -> mark and(isNat N, isNatList L), active isNatList nil() -> mark tt(), active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL), active isNatIList IL -> mark isNatList IL, active isNatIList zeros() -> mark tt(), active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL), active isNat 0() -> mark tt(), active isNat s N -> mark isNat N, active isNat length L -> mark isNatList L, active s X -> s active X, active length X -> length active X, active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L), active zeros() -> mark cons(0(), zeros()), active cons(X1, X2) -> cons(active X1, X2), active take(X1, X2) -> take(X1, active X2), active take(X1, X2) -> take(active X1, X2), active take(0(), IL) -> mark uTake1 isNatIList IL, active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), active uTake1 X -> uTake1 active X, active uTake1 tt() -> mark nil(), active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4), active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)), active uLength(X1, X2) -> uLength(active X1, X2), active uLength(tt(), L) -> mark s length L, and(X1, mark X2) -> mark and(X1, X2), and(mark X1, X2) -> mark and(X1, X2), and(ok X1, ok X2) -> ok and(X1, X2), isNatList ok X -> ok isNatList X, isNatIList ok X -> ok isNatIList X, isNat ok X -> ok isNat X, s mark X -> mark s X, s ok X -> ok s X, length mark X -> mark length X, length ok X -> ok length X, cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), take(X1, mark X2) -> mark take(X1, X2), take(mark X1, X2) -> mark take(X1, X2), take(ok X1, ok X2) -> ok take(X1, X2), uTake1 mark X -> mark uTake1 X, uTake1 ok X -> ok uTake1 X, uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4), uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4), uLength(mark X1, X2) -> mark uLength(X1, X2), uLength(ok X1, ok X2) -> ok uLength(X1, X2), proper and(X1, X2) -> and(proper X1, proper X2), proper tt() -> ok tt(), proper isNatList X -> isNatList proper X, proper isNatIList X -> isNatIList proper X, proper isNat X -> isNat proper X, proper 0() -> ok 0(), proper s X -> s proper X, proper length X -> length proper X, proper zeros() -> ok zeros(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper nil() -> ok nil(), proper take(X1, X2) -> take(proper X1, proper X2), proper uTake1 X -> uTake1 proper X, proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4), proper uLength(X1, X2) -> uLength(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [uTake2](x0, x1, x2, x3) = x0 + 1, [and](x0, x1) = x0 + 1, [cons](x0, x1) = x0 + x1 + 1, [take](x0, x1) = x0 + 1, [uLength](x0, x1) = x0, [mark](x0) = x0, [active](x0) = x0 + 1, [isNatList](x0) = 1, [isNatIList](x0) = 1, [isNat](x0) = 1, [s](x0) = x0 + 1, [length](x0) = 1, [uTake1](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 0, [zeros] = 1, [nil] = 0, [cons#](x0, x1) = x0 Strict: cons#(ok X1, ok X2) -> cons#(X1, X2) 1 + 0X1 + 1X2 >= 0 + 0X1 + 1X2 cons#(mark X1, X2) -> cons#(X1, X2) 0 + 0X1 + 1X2 >= 0 + 0X1 + 1X2 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper uLength(X1, X2) -> uLength(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4) 0 + 0X1 + 0X2 + 0X3 + 0X4 >= 1 + 0X1 + 0X2 + 0X3 + 0X4 proper uTake1 X -> uTake1 proper X 0 + 0X >= 1 + 0X proper take(X1, X2) -> take(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 1 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper zeros() -> ok zeros() 0 >= 2 proper length X -> length proper X 0 + 0X >= 1 + 0X proper s X -> s proper X 0 + 0X >= 1 + 0X proper 0() -> ok 0() 0 >= 1 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper isNatIList X -> isNatIList proper X 0 + 0X >= 1 + 0X proper isNatList X -> isNatList proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper and(X1, X2) -> and(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 uLength(ok X1, ok X2) -> ok uLength(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 uLength(mark X1, X2) -> mark uLength(X1, X2) 0 + 1X1 + 0X2 >= 0 + 1X1 + 0X2 uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 0X4 >= 2 + 1X1 + 0X2 + 0X3 + 0X4 uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4) 1 + 1X1 + 0X2 + 0X3 + 0X4 >= 1 + 1X1 + 0X2 + 0X3 + 0X4 uTake1 ok X -> ok uTake1 X 2 + 1X >= 2 + 1X uTake1 mark X -> mark uTake1 X 1 + 1X >= 1 + 1X take(ok X1, ok X2) -> ok take(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 take(mark X1, X2) -> mark take(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 take(X1, mark X2) -> mark take(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 cons(ok X1, ok X2) -> ok cons(X1, X2) 3 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 cons(mark X1, X2) -> mark cons(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 length ok X -> ok length X 1 + 0X >= 2 + 0X length mark X -> mark length X 1 + 0X >= 1 + 0X s ok X -> ok s X 2 + 1X >= 2 + 1X s mark X -> mark s X 1 + 1X >= 1 + 1X isNat ok X -> ok isNat X 1 + 0X >= 2 + 0X isNatIList ok X -> ok isNatIList X 1 + 0X >= 2 + 0X isNatList ok X -> ok isNatList X 1 + 0X >= 2 + 0X and(ok X1, ok X2) -> ok and(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 and(mark X1, X2) -> mark and(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 and(X1, mark X2) -> mark and(X1, X2) 1 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active uLength(tt(), L) -> mark s length L 1 + 0L >= 2 + 0L active uLength(X1, X2) -> uLength(active X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)) 2 + 0IL + 0N + 0M >= 2 + 1IL + 1N + 0M active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 0X4 >= 2 + 1X1 + 0X2 + 0X3 + 0X4 active uTake1 tt() -> mark nil() 2 >= 0 active uTake1 X -> uTake1 active X 2 + 1X >= 2 + 1X active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL) 3 + 1IL + 1N + 0M >= 4 + 0IL + 0N + 0M active take(0(), IL) -> mark uTake1 isNatIList IL 2 + 1IL >= 2 + 0IL active take(X1, X2) -> take(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active take(X1, X2) -> take(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active cons(X1, X2) -> cons(active X1, X2) 2 + 1X1 + 1X2 >= 2 + 1X1 + 1X2 active zeros() -> mark cons(0(), zeros()) 2 >= 2 active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L) 2 + 0N + 0L >= 2 + 0N + 0L active length X -> length active X 2 + 0X >= 1 + 0X active s X -> s active X 2 + 1X >= 2 + 1X active isNat length L -> mark isNatList L 2 + 0L >= 1 + 0L active isNat s N -> mark isNat N 2 + 0N >= 1 + 0N active isNat 0() -> mark tt() 2 >= 0 active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 2 + 0IL + 0N active isNatIList zeros() -> mark tt() 2 >= 0 active isNatIList IL -> mark isNatList IL 2 + 0IL >= 1 + 0IL active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 2 + 0IL + 0N active isNatList nil() -> mark tt() 2 >= 0 active isNatList cons(N, L) -> mark and(isNat N, isNatList L) 2 + 0N + 0L >= 2 + 0N + 0L active and(tt(), T) -> mark T 2 + 1T >= 0 + 1T active and(X1, X2) -> and(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active and(X1, X2) -> and(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 SCCS (1): Scc: {cons#(mark X1, X2) -> cons#(X1, X2)} SCC (1): Strict: {cons#(mark X1, X2) -> cons#(X1, X2)} Weak: { active and(X1, X2) -> and(X1, active X2), active and(X1, X2) -> and(active X1, X2), active and(tt(), T) -> mark T, active isNatList cons(N, L) -> mark and(isNat N, isNatList L), active isNatList nil() -> mark tt(), active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL), active isNatIList IL -> mark isNatList IL, active isNatIList zeros() -> mark tt(), active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL), active isNat 0() -> mark tt(), active isNat s N -> mark isNat N, active isNat length L -> mark isNatList L, active s X -> s active X, active length X -> length active X, active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L), active zeros() -> mark cons(0(), zeros()), active cons(X1, X2) -> cons(active X1, X2), active take(X1, X2) -> take(X1, active X2), active take(X1, X2) -> take(active X1, X2), active take(0(), IL) -> mark uTake1 isNatIList IL, active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), active uTake1 X -> uTake1 active X, active uTake1 tt() -> mark nil(), active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4), active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)), active uLength(X1, X2) -> uLength(active X1, X2), active uLength(tt(), L) -> mark s length L, and(X1, mark X2) -> mark and(X1, X2), and(mark X1, X2) -> mark and(X1, X2), and(ok X1, ok X2) -> ok and(X1, X2), isNatList ok X -> ok isNatList X, isNatIList ok X -> ok isNatIList X, isNat ok X -> ok isNat X, s mark X -> mark s X, s ok X -> ok s X, length mark X -> mark length X, length ok X -> ok length X, cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), take(X1, mark X2) -> mark take(X1, X2), take(mark X1, X2) -> mark take(X1, X2), take(ok X1, ok X2) -> ok take(X1, X2), uTake1 mark X -> mark uTake1 X, uTake1 ok X -> ok uTake1 X, uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4), uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4), uLength(mark X1, X2) -> mark uLength(X1, X2), uLength(ok X1, ok X2) -> ok uLength(X1, X2), proper and(X1, X2) -> and(proper X1, proper X2), proper tt() -> ok tt(), proper isNatList X -> isNatList proper X, proper isNatIList X -> isNatIList proper X, proper isNat X -> isNat proper X, proper 0() -> ok 0(), proper s X -> s proper X, proper length X -> length proper X, proper zeros() -> ok zeros(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper nil() -> ok nil(), proper take(X1, X2) -> take(proper X1, proper X2), proper uTake1 X -> uTake1 proper X, proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4), proper uLength(X1, X2) -> uLength(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [uTake2](x0, x1, x2, x3) = x0 + x1 + 1, [and](x0, x1) = x0 + 1, [cons](x0, x1) = 0, [take](x0, x1) = x0 + 1, [uLength](x0, x1) = x0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [isNatList](x0) = 1, [isNatIList](x0) = 1, [isNat](x0) = 1, [s](x0) = x0 + 1, [length](x0) = x0 + 1, [uTake1](x0) = x0 + 1, [proper](x0) = x0 + 1, [ok](x0) = 0, [top](x0) = x0 + 1, [tt] = 1, [0] = 1, [zeros] = 1, [nil] = 1, [cons#](x0, x1) = x0 + 1 Strict: cons#(mark X1, X2) -> cons#(X1, X2) 2 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 Weak: top ok X -> top active X 1 + 0X >= 2 + 1X top mark X -> top proper X 2 + 1X >= 2 + 1X proper uLength(X1, X2) -> uLength(proper X1, proper X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4) 2 + 1X1 + 0X2 + 1X3 + 0X4 >= 3 + 1X1 + 0X2 + 1X3 + 0X4 proper uTake1 X -> uTake1 proper X 2 + 1X >= 2 + 1X proper take(X1, X2) -> take(proper X1, proper X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 proper nil() -> ok nil() 2 >= 0 proper cons(X1, X2) -> cons(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper zeros() -> ok zeros() 2 >= 0 proper length X -> length proper X 2 + 1X >= 2 + 1X proper s X -> s proper X 2 + 1X >= 2 + 1X proper 0() -> ok 0() 2 >= 0 proper isNat X -> isNat proper X 2 + 0X >= 1 + 0X proper isNatIList X -> isNatIList proper X 2 + 0X >= 1 + 0X proper isNatList X -> isNatList proper X 2 + 0X >= 1 + 0X proper tt() -> ok tt() 2 >= 0 proper and(X1, X2) -> and(proper X1, proper X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 uLength(ok X1, ok X2) -> ok uLength(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 uLength(mark X1, X2) -> mark uLength(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4) 1 + 0X1 + 0X2 + 0X3 + 0X4 >= 0 + 0X1 + 0X2 + 0X3 + 0X4 uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4) 2 + 1X1 + 0X2 + 1X3 + 0X4 >= 2 + 1X1 + 0X2 + 1X3 + 0X4 uTake1 ok X -> ok uTake1 X 1 + 0X >= 0 + 0X uTake1 mark X -> mark uTake1 X 2 + 1X >= 2 + 1X take(ok X1, ok X2) -> ok take(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 take(mark X1, X2) -> mark take(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 take(X1, mark X2) -> mark take(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 cons(ok X1, ok X2) -> ok cons(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 length ok X -> ok length X 1 + 0X >= 0 + 0X length mark X -> mark length X 2 + 1X >= 2 + 1X s ok X -> ok s X 1 + 0X >= 0 + 0X s mark X -> mark s X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 1 + 0X >= 0 + 0X isNatIList ok X -> ok isNatIList X 1 + 0X >= 0 + 0X isNatList ok X -> ok isNatList X 1 + 0X >= 0 + 0X and(ok X1, ok X2) -> ok and(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 and(mark X1, X2) -> mark and(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 and(X1, mark X2) -> mark and(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active uLength(tt(), L) -> mark s length L 2 + 0L >= 3 + 1L active uLength(X1, X2) -> uLength(active X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)) 3 + 0IL + 1N + 0M >= 1 + 0IL + 0N + 0M active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4) 2 + 1X1 + 0X2 + 1X3 + 0X4 >= 2 + 1X1 + 0X2 + 1X3 + 0X4 active uTake1 tt() -> mark nil() 3 >= 2 active uTake1 X -> uTake1 active X 2 + 1X >= 2 + 1X active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL) 2 + 0IL + 0N + 0M >= 5 + 0IL + 1N + 0M active take(0(), IL) -> mark uTake1 isNatIList IL 2 + 1IL >= 3 + 0IL active take(X1, X2) -> take(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active take(X1, X2) -> take(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active cons(X1, X2) -> cons(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active zeros() -> mark cons(0(), zeros()) 2 >= 1 active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L) 2 + 0N + 0L >= 3 + 0N + 0L active length X -> length active X 2 + 1X >= 2 + 1X active s X -> s active X 2 + 1X >= 2 + 1X active isNat length L -> mark isNatList L 2 + 0L >= 2 + 0L active isNat s N -> mark isNat N 2 + 0N >= 2 + 0N active isNat 0() -> mark tt() 2 >= 2 active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 3 + 0IL + 0N active isNatIList zeros() -> mark tt() 2 >= 2 active isNatIList IL -> mark isNatList IL 2 + 0IL >= 2 + 0IL active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 3 + 0IL + 0N active isNatList nil() -> mark tt() 2 >= 2 active isNatList cons(N, L) -> mark and(isNat N, isNatList L) 2 + 0N + 0L >= 3 + 0N + 0L active and(tt(), T) -> mark T 2 + 1T >= 1 + 1T active and(X1, X2) -> and(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active and(X1, X2) -> and(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 Qed SCC (3): Strict: { and#(X1, mark X2) -> and#(X1, X2), and#(mark X1, X2) -> and#(X1, X2), and#(ok X1, ok X2) -> and#(X1, X2)} Weak: { active and(X1, X2) -> and(X1, active X2), active and(X1, X2) -> and(active X1, X2), active and(tt(), T) -> mark T, active isNatList cons(N, L) -> mark and(isNat N, isNatList L), active isNatList nil() -> mark tt(), active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL), active isNatIList IL -> mark isNatList IL, active isNatIList zeros() -> mark tt(), active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL), active isNat 0() -> mark tt(), active isNat s N -> mark isNat N, active isNat length L -> mark isNatList L, active s X -> s active X, active length X -> length active X, active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L), active zeros() -> mark cons(0(), zeros()), active cons(X1, X2) -> cons(active X1, X2), active take(X1, X2) -> take(X1, active X2), active take(X1, X2) -> take(active X1, X2), active take(0(), IL) -> mark uTake1 isNatIList IL, active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), active uTake1 X -> uTake1 active X, active uTake1 tt() -> mark nil(), active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4), active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)), active uLength(X1, X2) -> uLength(active X1, X2), active uLength(tt(), L) -> mark s length L, and(X1, mark X2) -> mark and(X1, X2), and(mark X1, X2) -> mark and(X1, X2), and(ok X1, ok X2) -> ok and(X1, X2), isNatList ok X -> ok isNatList X, isNatIList ok X -> ok isNatIList X, isNat ok X -> ok isNat X, s mark X -> mark s X, s ok X -> ok s X, length mark X -> mark length X, length ok X -> ok length X, cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), take(X1, mark X2) -> mark take(X1, X2), take(mark X1, X2) -> mark take(X1, X2), take(ok X1, ok X2) -> ok take(X1, X2), uTake1 mark X -> mark uTake1 X, uTake1 ok X -> ok uTake1 X, uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4), uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4), uLength(mark X1, X2) -> mark uLength(X1, X2), uLength(ok X1, ok X2) -> ok uLength(X1, X2), proper and(X1, X2) -> and(proper X1, proper X2), proper tt() -> ok tt(), proper isNatList X -> isNatList proper X, proper isNatIList X -> isNatIList proper X, proper isNat X -> isNat proper X, proper 0() -> ok 0(), proper s X -> s proper X, proper length X -> length proper X, proper zeros() -> ok zeros(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper nil() -> ok nil(), proper take(X1, X2) -> take(proper X1, proper X2), proper uTake1 X -> uTake1 proper X, proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4), proper uLength(X1, X2) -> uLength(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [uTake2](x0, x1, x2, x3) = x0 + 1, [and](x0, x1) = x0 + x1, [cons](x0, x1) = 1, [take](x0, x1) = 1, [uLength](x0, x1) = 0, [mark](x0) = x0 + 1, [active](x0) = 0, [isNatList](x0) = x0 + 1, [isNatIList](x0) = x0 + 1, [isNat](x0) = 1, [s](x0) = 0, [length](x0) = 0, [uTake1](x0) = x0 + 1, [proper](x0) = 0, [ok](x0) = x0 + 1, [top](x0) = 0, [tt] = 0, [0] = 1, [zeros] = 1, [nil] = 1, [and#](x0, x1) = x0 + 1 Strict: and#(ok X1, ok X2) -> and#(X1, X2) 2 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 and#(mark X1, X2) -> and#(X1, X2) 2 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 and#(X1, mark X2) -> and#(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 Weak: top ok X -> top active X 0 + 0X >= 0 + 0X top mark X -> top proper X 0 + 0X >= 0 + 0X proper uLength(X1, X2) -> uLength(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4) 0 + 0X1 + 0X2 + 0X3 + 0X4 >= 1 + 0X1 + 0X2 + 0X3 + 0X4 proper uTake1 X -> uTake1 proper X 0 + 0X >= 1 + 0X proper take(X1, X2) -> take(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper nil() -> ok nil() 0 >= 2 proper cons(X1, X2) -> cons(proper X1, proper X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 proper zeros() -> ok zeros() 0 >= 2 proper length X -> length proper X 0 + 0X >= 0 + 0X proper s X -> s proper X 0 + 0X >= 0 + 0X proper 0() -> ok 0() 0 >= 2 proper isNat X -> isNat proper X 0 + 0X >= 1 + 0X proper isNatIList X -> isNatIList proper X 0 + 0X >= 1 + 0X proper isNatList X -> isNatList proper X 0 + 0X >= 1 + 0X proper tt() -> ok tt() 0 >= 1 proper and(X1, X2) -> and(proper X1, proper X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 uLength(ok X1, ok X2) -> ok uLength(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 uLength(mark X1, X2) -> mark uLength(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 0X4 >= 2 + 1X1 + 0X2 + 0X3 + 0X4 uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4) 2 + 1X1 + 0X2 + 0X3 + 0X4 >= 2 + 1X1 + 0X2 + 0X3 + 0X4 uTake1 ok X -> ok uTake1 X 2 + 1X >= 2 + 1X uTake1 mark X -> mark uTake1 X 2 + 1X >= 2 + 1X take(ok X1, ok X2) -> ok take(X1, X2) 1 + 0X1 + 0X2 >= 2 + 0X1 + 0X2 take(mark X1, X2) -> mark take(X1, X2) 1 + 0X1 + 0X2 >= 2 + 0X1 + 0X2 take(X1, mark X2) -> mark take(X1, X2) 1 + 0X1 + 0X2 >= 2 + 0X1 + 0X2 cons(ok X1, ok X2) -> ok cons(X1, X2) 1 + 0X1 + 0X2 >= 2 + 0X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 1 + 0X1 + 0X2 >= 2 + 0X1 + 0X2 length ok X -> ok length X 0 + 0X >= 1 + 0X length mark X -> mark length X 0 + 0X >= 1 + 0X s ok X -> ok s X 0 + 0X >= 1 + 0X s mark X -> mark s X 0 + 0X >= 1 + 0X isNat ok X -> ok isNat X 1 + 0X >= 2 + 0X isNatIList ok X -> ok isNatIList X 2 + 1X >= 2 + 1X isNatList ok X -> ok isNatList X 2 + 1X >= 2 + 1X and(ok X1, ok X2) -> ok and(X1, X2) 2 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 and(mark X1, X2) -> mark and(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 and(X1, mark X2) -> mark and(X1, X2) 1 + 1X1 + 1X2 >= 1 + 1X1 + 1X2 active uLength(tt(), L) -> mark s length L 0 + 0L >= 1 + 0L active uLength(X1, X2) -> uLength(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)) 0 + 0IL + 0N + 0M >= 2 + 0IL + 0N + 0M active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4) 0 + 0X1 + 0X2 + 0X3 + 0X4 >= 1 + 0X1 + 0X2 + 0X3 + 0X4 active uTake1 tt() -> mark nil() 0 >= 2 active uTake1 X -> uTake1 active X 0 + 0X >= 1 + 0X active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL) 0 + 0IL + 0N + 0M >= 5 + 1IL + 0N + 0M active take(0(), IL) -> mark uTake1 isNatIList IL 0 + 0IL >= 3 + 1IL active take(X1, X2) -> take(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active take(X1, X2) -> take(X1, active X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active cons(X1, X2) -> cons(active X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 active zeros() -> mark cons(0(), zeros()) 0 >= 2 active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L) 0 + 0N + 0L >= 1 + 0N + 0L active length X -> length active X 0 + 0X >= 0 + 0X active s X -> s active X 0 + 0X >= 0 + 0X active isNat length L -> mark isNatList L 0 + 0L >= 2 + 1L active isNat s N -> mark isNat N 0 + 0N >= 2 + 0N active isNat 0() -> mark tt() 0 >= 1 active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL) 0 + 0IL + 0N >= 3 + 1IL + 0N active isNatIList zeros() -> mark tt() 0 >= 1 active isNatIList IL -> mark isNatList IL 0 + 0IL >= 2 + 1IL active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL) 0 + 0IL + 0N >= 3 + 1IL + 0N active isNatList nil() -> mark tt() 0 >= 1 active isNatList cons(N, L) -> mark and(isNat N, isNatList L) 0 + 0N + 0L >= 3 + 0N + 1L active and(tt(), T) -> mark T 0 + 0T >= 1 + 1T active and(X1, X2) -> and(active X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 1X2 active and(X1, X2) -> and(X1, active X2) 0 + 0X1 + 0X2 >= 0 + 1X1 + 0X2 SCCS (1): Scc: {and#(X1, mark X2) -> and#(X1, X2)} SCC (1): Strict: {and#(X1, mark X2) -> and#(X1, X2)} Weak: { active and(X1, X2) -> and(X1, active X2), active and(X1, X2) -> and(active X1, X2), active and(tt(), T) -> mark T, active isNatList cons(N, L) -> mark and(isNat N, isNatList L), active isNatList nil() -> mark tt(), active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL), active isNatIList IL -> mark isNatList IL, active isNatIList zeros() -> mark tt(), active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL), active isNat 0() -> mark tt(), active isNat s N -> mark isNat N, active isNat length L -> mark isNatList L, active s X -> s active X, active length X -> length active X, active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L), active zeros() -> mark cons(0(), zeros()), active cons(X1, X2) -> cons(active X1, X2), active take(X1, X2) -> take(X1, active X2), active take(X1, X2) -> take(active X1, X2), active take(0(), IL) -> mark uTake1 isNatIList IL, active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL), active uTake1 X -> uTake1 active X, active uTake1 tt() -> mark nil(), active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4), active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)), active uLength(X1, X2) -> uLength(active X1, X2), active uLength(tt(), L) -> mark s length L, and(X1, mark X2) -> mark and(X1, X2), and(mark X1, X2) -> mark and(X1, X2), and(ok X1, ok X2) -> ok and(X1, X2), isNatList ok X -> ok isNatList X, isNatIList ok X -> ok isNatIList X, isNat ok X -> ok isNat X, s mark X -> mark s X, s ok X -> ok s X, length mark X -> mark length X, length ok X -> ok length X, cons(mark X1, X2) -> mark cons(X1, X2), cons(ok X1, ok X2) -> ok cons(X1, X2), take(X1, mark X2) -> mark take(X1, X2), take(mark X1, X2) -> mark take(X1, X2), take(ok X1, ok X2) -> ok take(X1, X2), uTake1 mark X -> mark uTake1 X, uTake1 ok X -> ok uTake1 X, uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4), uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4), uLength(mark X1, X2) -> mark uLength(X1, X2), uLength(ok X1, ok X2) -> ok uLength(X1, X2), proper and(X1, X2) -> and(proper X1, proper X2), proper tt() -> ok tt(), proper isNatList X -> isNatList proper X, proper isNatIList X -> isNatIList proper X, proper isNat X -> isNat proper X, proper 0() -> ok 0(), proper s X -> s proper X, proper length X -> length proper X, proper zeros() -> ok zeros(), proper cons(X1, X2) -> cons(proper X1, proper X2), proper nil() -> ok nil(), proper take(X1, X2) -> take(proper X1, proper X2), proper uTake1 X -> uTake1 proper X, proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4), proper uLength(X1, X2) -> uLength(proper X1, proper X2), top mark X -> top proper X, top ok X -> top active X} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [uTake2](x0, x1, x2, x3) = x0 + x1 + 1, [and](x0, x1) = x0 + 1, [cons](x0, x1) = 0, [take](x0, x1) = x0 + 1, [uLength](x0, x1) = x0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [isNatList](x0) = 1, [isNatIList](x0) = 1, [isNat](x0) = 1, [s](x0) = x0 + 1, [length](x0) = x0 + 1, [uTake1](x0) = x0 + 1, [proper](x0) = x0 + 1, [ok](x0) = 0, [top](x0) = x0 + 1, [tt] = 1, [0] = 1, [zeros] = 1, [nil] = 1, [and#](x0, x1) = x0 + 1 Strict: and#(X1, mark X2) -> and#(X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 Weak: top ok X -> top active X 1 + 0X >= 2 + 1X top mark X -> top proper X 2 + 1X >= 2 + 1X proper uLength(X1, X2) -> uLength(proper X1, proper X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 proper uTake2(X1, X2, X3, X4) -> uTake2(proper X1, proper X2, proper X3, proper X4) 2 + 1X1 + 0X2 + 1X3 + 0X4 >= 3 + 1X1 + 0X2 + 1X3 + 0X4 proper uTake1 X -> uTake1 proper X 2 + 1X >= 2 + 1X proper take(X1, X2) -> take(proper X1, proper X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 proper nil() -> ok nil() 2 >= 0 proper cons(X1, X2) -> cons(proper X1, proper X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 proper zeros() -> ok zeros() 2 >= 0 proper length X -> length proper X 2 + 1X >= 2 + 1X proper s X -> s proper X 2 + 1X >= 2 + 1X proper 0() -> ok 0() 2 >= 0 proper isNat X -> isNat proper X 2 + 0X >= 1 + 0X proper isNatIList X -> isNatIList proper X 2 + 0X >= 1 + 0X proper isNatList X -> isNatList proper X 2 + 0X >= 1 + 0X proper tt() -> ok tt() 2 >= 0 proper and(X1, X2) -> and(proper X1, proper X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 uLength(ok X1, ok X2) -> ok uLength(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 uLength(mark X1, X2) -> mark uLength(X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 uTake2(ok X1, ok X2, ok X3, ok X4) -> ok uTake2(X1, X2, X3, X4) 1 + 0X1 + 0X2 + 0X3 + 0X4 >= 0 + 0X1 + 0X2 + 0X3 + 0X4 uTake2(mark X1, X2, X3, X4) -> mark uTake2(X1, X2, X3, X4) 2 + 1X1 + 0X2 + 1X3 + 0X4 >= 2 + 1X1 + 0X2 + 1X3 + 0X4 uTake1 ok X -> ok uTake1 X 1 + 0X >= 0 + 0X uTake1 mark X -> mark uTake1 X 2 + 1X >= 2 + 1X take(ok X1, ok X2) -> ok take(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 take(mark X1, X2) -> mark take(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 take(X1, mark X2) -> mark take(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 cons(ok X1, ok X2) -> ok cons(X1, X2) 0 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 cons(mark X1, X2) -> mark cons(X1, X2) 0 + 0X1 + 0X2 >= 1 + 0X1 + 0X2 length ok X -> ok length X 1 + 0X >= 0 + 0X length mark X -> mark length X 2 + 1X >= 2 + 1X s ok X -> ok s X 1 + 0X >= 0 + 0X s mark X -> mark s X 2 + 1X >= 2 + 1X isNat ok X -> ok isNat X 1 + 0X >= 0 + 0X isNatIList ok X -> ok isNatIList X 1 + 0X >= 0 + 0X isNatList ok X -> ok isNatList X 1 + 0X >= 0 + 0X and(ok X1, ok X2) -> ok and(X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 and(mark X1, X2) -> mark and(X1, X2) 1 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 and(X1, mark X2) -> mark and(X1, X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active uLength(tt(), L) -> mark s length L 2 + 0L >= 3 + 1L active uLength(X1, X2) -> uLength(active X1, X2) 1 + 1X1 + 0X2 >= 1 + 1X1 + 0X2 active uTake2(tt(), M, N, IL) -> mark cons(N, take(M, IL)) 3 + 0IL + 1N + 0M >= 1 + 0IL + 0N + 0M active uTake2(X1, X2, X3, X4) -> uTake2(active X1, X2, X3, X4) 2 + 1X1 + 0X2 + 1X3 + 0X4 >= 2 + 1X1 + 0X2 + 1X3 + 0X4 active uTake1 tt() -> mark nil() 3 >= 2 active uTake1 X -> uTake1 active X 2 + 1X >= 2 + 1X active take(s M, cons(N, IL)) -> mark uTake2(and(isNat M, and(isNat N, isNatIList IL)), M, N, IL) 2 + 0IL + 0N + 0M >= 5 + 0IL + 1N + 0M active take(0(), IL) -> mark uTake1 isNatIList IL 2 + 1IL >= 3 + 0IL active take(X1, X2) -> take(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active take(X1, X2) -> take(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 active cons(X1, X2) -> cons(active X1, X2) 1 + 0X1 + 0X2 >= 0 + 0X1 + 0X2 active zeros() -> mark cons(0(), zeros()) 2 >= 1 active length cons(N, L) -> mark uLength(and(isNat N, isNatList L), L) 2 + 0N + 0L >= 3 + 0N + 0L active length X -> length active X 2 + 1X >= 2 + 1X active s X -> s active X 2 + 1X >= 2 + 1X active isNat length L -> mark isNatList L 2 + 0L >= 2 + 0L active isNat s N -> mark isNat N 2 + 0N >= 2 + 0N active isNat 0() -> mark tt() 2 >= 2 active isNatIList cons(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 3 + 0IL + 0N active isNatIList zeros() -> mark tt() 2 >= 2 active isNatIList IL -> mark isNatList IL 2 + 0IL >= 2 + 0IL active isNatList take(N, IL) -> mark and(isNat N, isNatIList IL) 2 + 0IL + 0N >= 3 + 0IL + 0N active isNatList nil() -> mark tt() 2 >= 2 active isNatList cons(N, L) -> mark and(isNat N, isNatList L) 2 + 0N + 0L >= 3 + 0N + 0L active and(tt(), T) -> mark T 2 + 1T >= 1 + 1T active and(X1, X2) -> and(active X1, X2) 2 + 0X1 + 1X2 >= 1 + 0X1 + 1X2 active and(X1, X2) -> and(X1, active X2) 2 + 0X1 + 1X2 >= 2 + 0X1 + 1X2 Qed