NO
proof of Transformed_CSR_04_Ex4_DLMMU04_FR.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished


Termination w.r.t. Q of the given QTRS could be disproven:

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 0 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) QDP
(5) QDPOrderProof [EQUIVALENT, 146 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 0 ms]
(8) AND
    (9) QDP
        (10) QDPOrderProof [EQUIVALENT, 101 ms]
        (11) QDP
        (12) DependencyGraphProof [EQUIVALENT, 0 ms]
        (13) AND
            (14) QDP
                (15) UsableRulesProof [EQUIVALENT, 2 ms]
                (16) QDP
                (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (18) YES
            (19) QDP
                (20) QDPOrderProof [EQUIVALENT, 134 ms]
                (21) QDP
                (22) PisEmptyProof [EQUIVALENT, 0 ms]
                (23) YES
            (24) QDP
                (25) TransformationProof [EQUIVALENT, 0 ms]
                (26) QDP
                (27) TransformationProof [EQUIVALENT, 0 ms]
                (28) QDP
                (29) DependencyGraphProof [EQUIVALENT, 0 ms]
                (30) QDP
                (31) TransformationProof [EQUIVALENT, 0 ms]
                (32) QDP
                (33) DependencyGraphProof [EQUIVALENT, 0 ms]
                (34) QDP
                (35) TransformationProof [EQUIVALENT, 9 ms]
                (36) QDP
                (37) DependencyGraphProof [EQUIVALENT, 0 ms]
                (38) QDP
                (39) TransformationProof [EQUIVALENT, 0 ms]
                (40) QDP
                (41) TransformationProof [EQUIVALENT, 0 ms]
                (42) QDP
                (43) QDPOrderProof [EQUIVALENT, 66 ms]
                (44) QDP
                (45) DependencyGraphProof [EQUIVALENT, 0 ms]
                (46) AND
                    (47) QDP
                        (48) UsableRulesProof [EQUIVALENT, 0 ms]
                        (49) QDP
                        (50) MNOCProof [EQUIVALENT, 1 ms]
                        (51) QDP
                        (52) TransformationProof [EQUIVALENT, 0 ms]
                        (53) QDP
                        (54) UsableRulesProof [EQUIVALENT, 0 ms]
                        (55) QDP
                        (56) QReductionProof [EQUIVALENT, 0 ms]
                        (57) QDP
                        (58) TransformationProof [EQUIVALENT, 0 ms]
                        (59) QDP
                        (60) NonTerminationLoopProof [COMPLETE, 5 ms]
                        (61) NO
                    (62) QDP
                        (63) QDPOrderProof [EQUIVALENT, 61 ms]
                        (64) QDP
            (65) QDP
                (66) TransformationProof [EQUIVALENT, 0 ms]
                (67) QDP
                (68) TransformationProof [EQUIVALENT, 0 ms]
                (69) QDP
                (70) DependencyGraphProof [EQUIVALENT, 0 ms]
                (71) QDP
                (72) TransformationProof [EQUIVALENT, 0 ms]
                (73) QDP
                (74) DependencyGraphProof [EQUIVALENT, 0 ms]
                (75) QDP
                (76) TransformationProof [EQUIVALENT, 0 ms]
                (77) QDP
                (78) DependencyGraphProof [EQUIVALENT, 0 ms]
                (79) QDP
                (80) TransformationProof [EQUIVALENT, 0 ms]
                (81) QDP
                (82) TransformationProof [EQUIVALENT, 2 ms]
                (83) QDP
                (84) QDPOrderProof [EQUIVALENT, 104 ms]
                (85) QDP
                (86) DependencyGraphProof [EQUIVALENT, 0 ms]
                (87) AND
                    (88) QDP
                        (89) UsableRulesProof [EQUIVALENT, 0 ms]
                        (90) QDP
                        (91) MNOCProof [EQUIVALENT, 0 ms]
                        (92) QDP
                        (93) TransformationProof [EQUIVALENT, 0 ms]
                        (94) QDP
                        (95) UsableRulesProof [EQUIVALENT, 0 ms]
                        (96) QDP
                        (97) QReductionProof [EQUIVALENT, 0 ms]
                        (98) QDP
                        (99) TransformationProof [EQUIVALENT, 0 ms]
                        (100) QDP
                    (101) QDP
                        (102) QDPOrderProof [EQUIVALENT, 40 ms]
                        (103) QDP
    (104) QDP
        (105) QDPOrderProof [EQUIVALENT, 192 ms]
        (106) QDP
        (107) DependencyGraphProof [EQUIVALENT, 0 ms]
        (108) TRUE


----------------------------------------

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.

----------------------------------------

(1) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(2)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(IL) -> ISNATLIST(activate(IL))
   ISNATILIST(IL) -> ACTIVATE(IL)
   ISNAT(n__s(N)) -> ISNAT(activate(N))
   ISNAT(n__s(N)) -> ACTIVATE(N)
   ISNAT(n__length(L)) -> ISNATLIST(activate(L))
   ISNAT(n__length(L)) -> ACTIVATE(L)
   ISNATILIST(n__cons(N, IL)) -> AND(isNat(activate(N)), isNatIList(activate(IL)))
   ISNATILIST(n__cons(N, IL)) -> ISNAT(activate(N))
   ISNATILIST(n__cons(N, IL)) -> ACTIVATE(N)
   ISNATILIST(n__cons(N, IL)) -> ISNATILIST(activate(IL))
   ISNATILIST(n__cons(N, IL)) -> ACTIVATE(IL)
   ISNATLIST(n__cons(N, L)) -> AND(isNat(activate(N)), isNatList(activate(L)))
   ISNATLIST(n__cons(N, L)) -> ISNAT(activate(N))
   ISNATLIST(n__cons(N, L)) -> ACTIVATE(N)
   ISNATLIST(n__cons(N, L)) -> ISNATLIST(activate(L))
   ISNATLIST(n__cons(N, L)) -> ACTIVATE(L)
   ISNATLIST(n__take(N, IL)) -> AND(isNat(activate(N)), isNatIList(activate(IL)))
   ISNATLIST(n__take(N, IL)) -> ISNAT(activate(N))
   ISNATLIST(n__take(N, IL)) -> ACTIVATE(N)
   ISNATLIST(n__take(N, IL)) -> ISNATILIST(activate(IL))
   ISNATLIST(n__take(N, IL)) -> ACTIVATE(IL)
   ZEROS -> CONS(0, n__zeros)
   ZEROS -> 0^1
   TAKE(0, IL) -> UTAKE1(isNatIList(IL))
   TAKE(0, IL) -> ISNATILIST(IL)
   UTAKE1(tt) -> NIL
   TAKE(s(M), cons(N, IL)) -> UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   TAKE(s(M), cons(N, IL)) -> AND(isNat(M), and(isNat(N), isNatIList(activate(IL))))
   TAKE(s(M), cons(N, IL)) -> ISNAT(M)
   TAKE(s(M), cons(N, IL)) -> AND(isNat(N), isNatIList(activate(IL)))
   TAKE(s(M), cons(N, IL)) -> ISNAT(N)
   TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL))
   TAKE(s(M), cons(N, IL)) -> ACTIVATE(IL)
   UTAKE2(tt, M, N, IL) -> CONS(activate(N), n__take(activate(M), activate(IL)))
   UTAKE2(tt, M, N, IL) -> ACTIVATE(N)
   UTAKE2(tt, M, N, IL) -> ACTIVATE(M)
   UTAKE2(tt, M, N, IL) -> ACTIVATE(IL)
   LENGTH(cons(N, L)) -> ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
   LENGTH(cons(N, L)) -> AND(isNat(N), isNatList(activate(L)))
   LENGTH(cons(N, L)) -> ISNAT(N)
   LENGTH(cons(N, L)) -> ISNATLIST(activate(L))
   LENGTH(cons(N, L)) -> ACTIVATE(L)
   ULENGTH(tt, L) -> S(length(activate(L)))
   ULENGTH(tt, L) -> LENGTH(activate(L))
   ULENGTH(tt, L) -> ACTIVATE(L)
   ACTIVATE(n__0) -> 0^1
   ACTIVATE(n__s(X)) -> S(activate(X))
   ACTIVATE(n__s(X)) -> ACTIVATE(X)
   ACTIVATE(n__length(X)) -> LENGTH(activate(X))
   ACTIVATE(n__length(X)) -> ACTIVATE(X)
   ACTIVATE(n__zeros) -> ZEROS
   ACTIVATE(n__cons(X1, X2)) -> CONS(activate(X1), X2)
   ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__nil) -> NIL
   ACTIVATE(n__take(X1, X2)) -> TAKE(activate(X1), activate(X2))
   ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X2)

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 17 less nodes.
----------------------------------------

(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(N, L)) -> ISNAT(activate(N))
   ISNAT(n__s(N)) -> ISNAT(activate(N))
   ISNAT(n__s(N)) -> ACTIVATE(N)
   ACTIVATE(n__s(X)) -> ACTIVATE(X)
   ACTIVATE(n__length(X)) -> LENGTH(activate(X))
   LENGTH(cons(N, L)) -> ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
   ULENGTH(tt, L) -> LENGTH(activate(L))
   LENGTH(cons(N, L)) -> ISNAT(N)
   ISNAT(n__length(L)) -> ISNATLIST(activate(L))
   ISNATLIST(n__cons(N, L)) -> ACTIVATE(N)
   ACTIVATE(n__length(X)) -> ACTIVATE(X)
   ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__take(X1, X2)) -> TAKE(activate(X1), activate(X2))
   TAKE(0, IL) -> ISNATILIST(IL)
   ISNATILIST(IL) -> ISNATLIST(activate(IL))
   ISNATLIST(n__cons(N, L)) -> ISNATLIST(activate(L))
   ISNATLIST(n__cons(N, L)) -> ACTIVATE(L)
   ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X2)
   ISNATLIST(n__take(N, IL)) -> ISNAT(activate(N))
   ISNAT(n__length(L)) -> ACTIVATE(L)
   ISNATLIST(n__take(N, IL)) -> ACTIVATE(N)
   ISNATLIST(n__take(N, IL)) -> ISNATILIST(activate(IL))
   ISNATILIST(IL) -> ACTIVATE(IL)
   ISNATILIST(n__cons(N, IL)) -> ISNAT(activate(N))
   ISNATILIST(n__cons(N, IL)) -> ACTIVATE(N)
   ISNATILIST(n__cons(N, IL)) -> ISNATILIST(activate(IL))
   ISNATILIST(n__cons(N, IL)) -> ACTIVATE(IL)
   ISNATLIST(n__take(N, IL)) -> ACTIVATE(IL)
   TAKE(s(M), cons(N, IL)) -> UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   UTAKE2(tt, M, N, IL) -> ACTIVATE(N)
   UTAKE2(tt, M, N, IL) -> ACTIVATE(M)
   UTAKE2(tt, M, N, IL) -> ACTIVATE(IL)
   TAKE(s(M), cons(N, IL)) -> ISNAT(M)
   TAKE(s(M), cons(N, IL)) -> ISNAT(N)
   TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL))
   TAKE(s(M), cons(N, IL)) -> ACTIVATE(IL)
   LENGTH(cons(N, L)) -> ISNATLIST(activate(L))
   LENGTH(cons(N, L)) -> ACTIVATE(L)
   ULENGTH(tt, L) -> ACTIVATE(L)

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(5) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVATE(n__length(X)) -> LENGTH(activate(X))
   ISNAT(n__length(L)) -> ISNATLIST(activate(L))
   ACTIVATE(n__length(X)) -> ACTIVATE(X)
   ISNAT(n__length(L)) -> ACTIVATE(L)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ISNAT_1(x_1) ) = 2x_1
POL( ISNATILIST_1(x_1) ) = 2x_1
POL( ISNATLIST_1(x_1) ) = 2x_1
POL( LENGTH_1(x_1) ) = 2x_1
POL( TAKE_2(x_1, x_2) ) = 2x_1 + 2x_2
POL( ULENGTH_2(x_1, x_2) ) = 2x_2
POL( UTAKE2_4(x_1, ..., x_4) ) = 2x_1 + 2x_2 + 2x_3 + 2x_4
POL( and_2(x_1, x_2) ) = x_2
POL( cons_2(x_1, x_2) ) = 2x_1 + x_2
POL( isNat_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = max{0, -2}
POL( length_1(x_1) ) = x_1 + 2
POL( s_1(x_1) ) = x_1
POL( take_2(x_1, x_2) ) = 2x_1 + x_2
POL( uLength_2(x_1, x_2) ) = 2x_1 + x_2 + 2
POL( uTake2_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + x_4
POL( isNatIList_1(x_1) ) = 0
POL( n__take_2(x_1, x_2) ) = 2x_1 + x_2
POL( activate_1(x_1) ) = x_1
POL( n__0 ) = 0
POL( 0 ) = 0
POL( n__s_1(x_1) ) = x_1
POL( n__length_1(x_1) ) = x_1 + 2
POL( n__zeros ) = 0
POL( zeros ) = 0
POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2
POL( n__nil ) = 0
POL( nil ) = 0
POL( tt ) = 0
POL( uTake1_1(x_1) ) = max{0, -2}
POL( ACTIVATE_1(x_1) ) = 2x_1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   cons(X1, X2) -> n__cons(X1, X2)
   take(X1, X2) -> n__take(X1, X2)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   0 -> n__0
   nil -> n__nil


----------------------------------------

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(N, L)) -> ISNAT(activate(N))
   ISNAT(n__s(N)) -> ISNAT(activate(N))
   ISNAT(n__s(N)) -> ACTIVATE(N)
   ACTIVATE(n__s(X)) -> ACTIVATE(X)
   LENGTH(cons(N, L)) -> ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
   ULENGTH(tt, L) -> LENGTH(activate(L))
   LENGTH(cons(N, L)) -> ISNAT(N)
   ISNATLIST(n__cons(N, L)) -> ACTIVATE(N)
   ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__take(X1, X2)) -> TAKE(activate(X1), activate(X2))
   TAKE(0, IL) -> ISNATILIST(IL)
   ISNATILIST(IL) -> ISNATLIST(activate(IL))
   ISNATLIST(n__cons(N, L)) -> ISNATLIST(activate(L))
   ISNATLIST(n__cons(N, L)) -> ACTIVATE(L)
   ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X2)
   ISNATLIST(n__take(N, IL)) -> ISNAT(activate(N))
   ISNATLIST(n__take(N, IL)) -> ACTIVATE(N)
   ISNATLIST(n__take(N, IL)) -> ISNATILIST(activate(IL))
   ISNATILIST(IL) -> ACTIVATE(IL)
   ISNATILIST(n__cons(N, IL)) -> ISNAT(activate(N))
   ISNATILIST(n__cons(N, IL)) -> ACTIVATE(N)
   ISNATILIST(n__cons(N, IL)) -> ISNATILIST(activate(IL))
   ISNATILIST(n__cons(N, IL)) -> ACTIVATE(IL)
   ISNATLIST(n__take(N, IL)) -> ACTIVATE(IL)
   TAKE(s(M), cons(N, IL)) -> UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   UTAKE2(tt, M, N, IL) -> ACTIVATE(N)
   UTAKE2(tt, M, N, IL) -> ACTIVATE(M)
   UTAKE2(tt, M, N, IL) -> ACTIVATE(IL)
   TAKE(s(M), cons(N, IL)) -> ISNAT(M)
   TAKE(s(M), cons(N, IL)) -> ISNAT(N)
   TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL))
   TAKE(s(M), cons(N, IL)) -> ACTIVATE(IL)
   LENGTH(cons(N, L)) -> ISNATLIST(activate(L))
   LENGTH(cons(N, L)) -> ACTIVATE(L)
   ULENGTH(tt, L) -> ACTIVATE(L)

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(7) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes.
----------------------------------------

(8)
Complex Obligation (AND)

----------------------------------------

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNAT(n__s(N)) -> ISNAT(activate(N))
   ISNAT(n__s(N)) -> ACTIVATE(N)
   ACTIVATE(n__s(X)) -> ACTIVATE(X)
   ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__take(X1, X2)) -> TAKE(activate(X1), activate(X2))
   TAKE(0, IL) -> ISNATILIST(IL)
   ISNATILIST(IL) -> ISNATLIST(activate(IL))
   ISNATLIST(n__cons(N, L)) -> ISNAT(activate(N))
   ISNATLIST(n__cons(N, L)) -> ACTIVATE(N)
   ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X2)
   ISNATLIST(n__cons(N, L)) -> ISNATLIST(activate(L))
   ISNATLIST(n__cons(N, L)) -> ACTIVATE(L)
   ISNATLIST(n__take(N, IL)) -> ISNAT(activate(N))
   ISNATLIST(n__take(N, IL)) -> ACTIVATE(N)
   ISNATLIST(n__take(N, IL)) -> ISNATILIST(activate(IL))
   ISNATILIST(IL) -> ACTIVATE(IL)
   ISNATILIST(n__cons(N, IL)) -> ISNAT(activate(N))
   ISNATILIST(n__cons(N, IL)) -> ACTIVATE(N)
   ISNATILIST(n__cons(N, IL)) -> ISNATILIST(activate(IL))
   ISNATILIST(n__cons(N, IL)) -> ACTIVATE(IL)
   ISNATLIST(n__take(N, IL)) -> ACTIVATE(IL)
   TAKE(s(M), cons(N, IL)) -> UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   UTAKE2(tt, M, N, IL) -> ACTIVATE(N)
   UTAKE2(tt, M, N, IL) -> ACTIVATE(M)
   UTAKE2(tt, M, N, IL) -> ACTIVATE(IL)
   TAKE(s(M), cons(N, IL)) -> ISNAT(M)
   TAKE(s(M), cons(N, IL)) -> ISNAT(N)
   TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL))
   TAKE(s(M), cons(N, IL)) -> ACTIVATE(IL)

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(10) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   TAKE(0, IL) -> ISNATILIST(IL)
   ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X2)
   ISNATLIST(n__take(N, IL)) -> ISNAT(activate(N))
   ISNATLIST(n__take(N, IL)) -> ACTIVATE(N)
   ISNATLIST(n__take(N, IL)) -> ISNATILIST(activate(IL))
   ISNATLIST(n__take(N, IL)) -> ACTIVATE(IL)
   UTAKE2(tt, M, N, IL) -> ACTIVATE(N)
   UTAKE2(tt, M, N, IL) -> ACTIVATE(M)
   UTAKE2(tt, M, N, IL) -> ACTIVATE(IL)
   TAKE(s(M), cons(N, IL)) -> ISNAT(M)
   TAKE(s(M), cons(N, IL)) -> ISNAT(N)
   TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL))
   TAKE(s(M), cons(N, IL)) -> ACTIVATE(IL)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ISNAT_1(x_1) ) = 2x_1
POL( ISNATILIST_1(x_1) ) = x_1
POL( ISNATLIST_1(x_1) ) = x_1
POL( TAKE_2(x_1, x_2) ) = 2x_1 + 2x_2 + 1
POL( UTAKE2_4(x_1, ..., x_4) ) = 2x_2 + x_3 + x_4 + 1
POL( and_2(x_1, x_2) ) = x_2
POL( cons_2(x_1, x_2) ) = 2x_1 + x_2
POL( isNat_1(x_1) ) = x_1 + 2
POL( isNatList_1(x_1) ) = 1
POL( length_1(x_1) ) = 2x_1 + 1
POL( s_1(x_1) ) = x_1
POL( take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 1
POL( uLength_2(x_1, x_2) ) = 2x_2 + 1
POL( uTake2_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 1
POL( isNatIList_1(x_1) ) = 1
POL( n__take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 1
POL( activate_1(x_1) ) = x_1
POL( n__0 ) = 0
POL( 0 ) = 0
POL( n__s_1(x_1) ) = x_1
POL( n__length_1(x_1) ) = 2x_1 + 1
POL( n__zeros ) = 1
POL( zeros ) = 1
POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2
POL( n__nil ) = 0
POL( nil ) = 0
POL( tt ) = 0
POL( uTake1_1(x_1) ) = 0
POL( ACTIVATE_1(x_1) ) = x_1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(IL) -> isNatList(activate(IL))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   and(tt, T) -> T
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   cons(X1, X2) -> n__cons(X1, X2)
   take(X1, X2) -> n__take(X1, X2)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   0 -> n__0
   nil -> n__nil


----------------------------------------

(11)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNAT(n__s(N)) -> ISNAT(activate(N))
   ISNAT(n__s(N)) -> ACTIVATE(N)
   ACTIVATE(n__s(X)) -> ACTIVATE(X)
   ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__take(X1, X2)) -> TAKE(activate(X1), activate(X2))
   ISNATILIST(IL) -> ISNATLIST(activate(IL))
   ISNATLIST(n__cons(N, L)) -> ISNAT(activate(N))
   ISNATLIST(n__cons(N, L)) -> ACTIVATE(N)
   ISNATLIST(n__cons(N, L)) -> ISNATLIST(activate(L))
   ISNATLIST(n__cons(N, L)) -> ACTIVATE(L)
   ISNATILIST(IL) -> ACTIVATE(IL)
   ISNATILIST(n__cons(N, IL)) -> ISNAT(activate(N))
   ISNATILIST(n__cons(N, IL)) -> ACTIVATE(N)
   ISNATILIST(n__cons(N, IL)) -> ISNATILIST(activate(IL))
   ISNATILIST(n__cons(N, IL)) -> ACTIVATE(IL)
   TAKE(s(M), cons(N, IL)) -> UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(12) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 11 less nodes.
----------------------------------------

(13)
Complex Obligation (AND)

----------------------------------------

(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__s(X)) -> ACTIVATE(X)

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(15) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__s(X)) -> ACTIVATE(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(17) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1)
The graph contains the following edges 1 > 1


*ACTIVATE(n__s(X)) -> ACTIVATE(X)
The graph contains the following edges 1 > 1


----------------------------------------

(18)
YES

----------------------------------------

(19)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNAT(n__s(N)) -> ISNAT(activate(N))

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(20) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ISNAT(n__s(N)) -> ISNAT(activate(N))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(ISNAT(x_1)) =  	[[4A]] 	 +  	[[5A]] 	* 	x_1
>>>

   <<<
 POL(n__s(x_1)) =  	[[5A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(activate(x_1)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(n__0) =  	[[0A]]
>>>

   <<<
 POL(0) =  	[[0A]]
>>>

   <<<
 POL(s(x_1)) =  	[[5A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(n__length(x_1)) =  	[[5A]] 	 +  	[[2A]] 	* 	x_1
>>>

   <<<
 POL(length(x_1)) =  	[[5A]] 	 +  	[[2A]] 	* 	x_1
>>>

   <<<
 POL(n__zeros) =  	[[1A]]
>>>

   <<<
 POL(zeros) =  	[[2A]]
>>>

   <<<
 POL(n__cons(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(n__nil) =  	[[3A]]
>>>

   <<<
 POL(nil) =  	[[3A]]
>>>

   <<<
 POL(n__take(x_1, x_2)) =  	[[4A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2
>>>

   <<<
 POL(take(x_1, x_2)) =  	[[4A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2
>>>

   <<<
 POL(uTake1(x_1)) =  	[[3A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(isNatIList(x_1)) =  	[[4A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(tt) =  	[[3A]]
>>>

   <<<
 POL(isNatList(x_1)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(and(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(isNat(x_1)) =  	[[3A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(uTake2(x_1, x_2, x_3, x_4)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3 	 +  	[[3A]] 	* 	x_4
>>>

   <<<
 POL(uLength(x_1, x_2)) =  	[[5A]] 	 +  	[[3A]] 	* 	x_1 	 +  	[[3A]] 	* 	x_2
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   cons(X1, X2) -> n__cons(X1, X2)
   take(X1, X2) -> n__take(X1, X2)
   take(0, IL) -> uTake1(isNatIList(IL))
   isNatIList(n__zeros) -> tt
   uTake1(tt) -> nil
   isNatIList(IL) -> isNatList(activate(IL))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNat(n__0) -> tt
   and(tt, T) -> T
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   0 -> n__0
   nil -> n__nil


----------------------------------------

(21)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(22) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(23)
YES

----------------------------------------

(24)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(N, L)) -> ISNATLIST(activate(L))

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(25) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule ISNATLIST(n__cons(N, L)) -> ISNATLIST(activate(L)) at position [0] we obtained the following new rules [LPAR04]:

   (ISNATLIST(n__cons(y0, n__0)) -> ISNATLIST(0),ISNATLIST(n__cons(y0, n__0)) -> ISNATLIST(0))
   (ISNATLIST(n__cons(y0, n__s(x0))) -> ISNATLIST(s(activate(x0))),ISNATLIST(n__cons(y0, n__s(x0))) -> ISNATLIST(s(activate(x0))))
   (ISNATLIST(n__cons(y0, n__length(x0))) -> ISNATLIST(length(activate(x0))),ISNATLIST(n__cons(y0, n__length(x0))) -> ISNATLIST(length(activate(x0))))
   (ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(zeros),ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(zeros))
   (ISNATLIST(n__cons(y0, n__cons(x0, x1))) -> ISNATLIST(cons(activate(x0), x1)),ISNATLIST(n__cons(y0, n__cons(x0, x1))) -> ISNATLIST(cons(activate(x0), x1)))
   (ISNATLIST(n__cons(y0, n__nil)) -> ISNATLIST(nil),ISNATLIST(n__cons(y0, n__nil)) -> ISNATLIST(nil))
   (ISNATLIST(n__cons(y0, n__take(x0, x1))) -> ISNATLIST(take(activate(x0), activate(x1))),ISNATLIST(n__cons(y0, n__take(x0, x1))) -> ISNATLIST(take(activate(x0), activate(x1))))
   (ISNATLIST(n__cons(y0, x0)) -> ISNATLIST(x0),ISNATLIST(n__cons(y0, x0)) -> ISNATLIST(x0))


----------------------------------------

(26)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(y0, n__0)) -> ISNATLIST(0)
   ISNATLIST(n__cons(y0, n__s(x0))) -> ISNATLIST(s(activate(x0)))
   ISNATLIST(n__cons(y0, n__length(x0))) -> ISNATLIST(length(activate(x0)))
   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(zeros)
   ISNATLIST(n__cons(y0, n__cons(x0, x1))) -> ISNATLIST(cons(activate(x0), x1))
   ISNATLIST(n__cons(y0, n__nil)) -> ISNATLIST(nil)
   ISNATLIST(n__cons(y0, n__take(x0, x1))) -> ISNATLIST(take(activate(x0), activate(x1)))
   ISNATLIST(n__cons(y0, x0)) -> ISNATLIST(x0)

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(27) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule ISNATLIST(n__cons(y0, n__0)) -> ISNATLIST(0) at position [0] we obtained the following new rules [LPAR04]:

   (ISNATLIST(n__cons(y0, n__0)) -> ISNATLIST(n__0),ISNATLIST(n__cons(y0, n__0)) -> ISNATLIST(n__0))


----------------------------------------

(28)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(y0, n__s(x0))) -> ISNATLIST(s(activate(x0)))
   ISNATLIST(n__cons(y0, n__length(x0))) -> ISNATLIST(length(activate(x0)))
   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(zeros)
   ISNATLIST(n__cons(y0, n__cons(x0, x1))) -> ISNATLIST(cons(activate(x0), x1))
   ISNATLIST(n__cons(y0, n__nil)) -> ISNATLIST(nil)
   ISNATLIST(n__cons(y0, n__take(x0, x1))) -> ISNATLIST(take(activate(x0), activate(x1)))
   ISNATLIST(n__cons(y0, x0)) -> ISNATLIST(x0)
   ISNATLIST(n__cons(y0, n__0)) -> ISNATLIST(n__0)

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(29) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(30)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(y0, n__s(x0))) -> ISNATLIST(s(activate(x0)))
   ISNATLIST(n__cons(y0, n__length(x0))) -> ISNATLIST(length(activate(x0)))
   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(zeros)
   ISNATLIST(n__cons(y0, n__cons(x0, x1))) -> ISNATLIST(cons(activate(x0), x1))
   ISNATLIST(n__cons(y0, n__nil)) -> ISNATLIST(nil)
   ISNATLIST(n__cons(y0, n__take(x0, x1))) -> ISNATLIST(take(activate(x0), activate(x1)))
   ISNATLIST(n__cons(y0, x0)) -> ISNATLIST(x0)

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(31) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(zeros) at position [0] we obtained the following new rules [LPAR04]:

   (ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(cons(0, n__zeros)),ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(cons(0, n__zeros)))
   (ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__zeros),ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__zeros))


----------------------------------------

(32)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(y0, n__s(x0))) -> ISNATLIST(s(activate(x0)))
   ISNATLIST(n__cons(y0, n__length(x0))) -> ISNATLIST(length(activate(x0)))
   ISNATLIST(n__cons(y0, n__cons(x0, x1))) -> ISNATLIST(cons(activate(x0), x1))
   ISNATLIST(n__cons(y0, n__nil)) -> ISNATLIST(nil)
   ISNATLIST(n__cons(y0, n__take(x0, x1))) -> ISNATLIST(take(activate(x0), activate(x1)))
   ISNATLIST(n__cons(y0, x0)) -> ISNATLIST(x0)
   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(cons(0, n__zeros))
   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__zeros)

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(33) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(34)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(y0, n__s(x0))) -> ISNATLIST(s(activate(x0)))
   ISNATLIST(n__cons(y0, n__length(x0))) -> ISNATLIST(length(activate(x0)))
   ISNATLIST(n__cons(y0, n__cons(x0, x1))) -> ISNATLIST(cons(activate(x0), x1))
   ISNATLIST(n__cons(y0, n__nil)) -> ISNATLIST(nil)
   ISNATLIST(n__cons(y0, n__take(x0, x1))) -> ISNATLIST(take(activate(x0), activate(x1)))
   ISNATLIST(n__cons(y0, x0)) -> ISNATLIST(x0)
   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(cons(0, n__zeros))

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(35) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule ISNATLIST(n__cons(y0, n__nil)) -> ISNATLIST(nil) at position [0] we obtained the following new rules [LPAR04]:

   (ISNATLIST(n__cons(y0, n__nil)) -> ISNATLIST(n__nil),ISNATLIST(n__cons(y0, n__nil)) -> ISNATLIST(n__nil))


----------------------------------------

(36)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(y0, n__s(x0))) -> ISNATLIST(s(activate(x0)))
   ISNATLIST(n__cons(y0, n__length(x0))) -> ISNATLIST(length(activate(x0)))
   ISNATLIST(n__cons(y0, n__cons(x0, x1))) -> ISNATLIST(cons(activate(x0), x1))
   ISNATLIST(n__cons(y0, n__take(x0, x1))) -> ISNATLIST(take(activate(x0), activate(x1)))
   ISNATLIST(n__cons(y0, x0)) -> ISNATLIST(x0)
   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(cons(0, n__zeros))
   ISNATLIST(n__cons(y0, n__nil)) -> ISNATLIST(n__nil)

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(37) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(38)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(y0, n__s(x0))) -> ISNATLIST(s(activate(x0)))
   ISNATLIST(n__cons(y0, n__length(x0))) -> ISNATLIST(length(activate(x0)))
   ISNATLIST(n__cons(y0, n__cons(x0, x1))) -> ISNATLIST(cons(activate(x0), x1))
   ISNATLIST(n__cons(y0, n__take(x0, x1))) -> ISNATLIST(take(activate(x0), activate(x1)))
   ISNATLIST(n__cons(y0, x0)) -> ISNATLIST(x0)
   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(cons(0, n__zeros))

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(39) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(cons(0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

   (ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__cons(0, n__zeros)),ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__cons(0, n__zeros)))
   (ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(cons(n__0, n__zeros)),ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(cons(n__0, n__zeros)))


----------------------------------------

(40)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(y0, n__s(x0))) -> ISNATLIST(s(activate(x0)))
   ISNATLIST(n__cons(y0, n__length(x0))) -> ISNATLIST(length(activate(x0)))
   ISNATLIST(n__cons(y0, n__cons(x0, x1))) -> ISNATLIST(cons(activate(x0), x1))
   ISNATLIST(n__cons(y0, n__take(x0, x1))) -> ISNATLIST(take(activate(x0), activate(x1)))
   ISNATLIST(n__cons(y0, x0)) -> ISNATLIST(x0)
   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__cons(0, n__zeros))
   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(cons(n__0, n__zeros))

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(41) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(cons(n__0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

   (ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__cons(n__0, n__zeros)),ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__cons(n__0, n__zeros)))


----------------------------------------

(42)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(y0, n__s(x0))) -> ISNATLIST(s(activate(x0)))
   ISNATLIST(n__cons(y0, n__length(x0))) -> ISNATLIST(length(activate(x0)))
   ISNATLIST(n__cons(y0, n__cons(x0, x1))) -> ISNATLIST(cons(activate(x0), x1))
   ISNATLIST(n__cons(y0, n__take(x0, x1))) -> ISNATLIST(take(activate(x0), activate(x1)))
   ISNATLIST(n__cons(y0, x0)) -> ISNATLIST(x0)
   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__cons(0, n__zeros))
   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(43) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ISNATLIST(n__cons(y0, n__cons(x0, x1))) -> ISNATLIST(cons(activate(x0), x1))
   ISNATLIST(n__cons(y0, x0)) -> ISNATLIST(x0)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ISNATLIST_1(x_1) ) = x_1 + 2
POL( cons_2(x_1, x_2) ) = x_2 + 1
POL( isNat_1(x_1) ) = 1
POL( isNatList_1(x_1) ) = 1
POL( length_1(x_1) ) = 2
POL( s_1(x_1) ) = 1
POL( take_2(x_1, x_2) ) = 1
POL( uLength_2(x_1, x_2) ) = 2
POL( uTake2_4(x_1, ..., x_4) ) = max{0, 2x_1 - 1}
POL( and_2(x_1, x_2) ) = max{0, x_1 + x_2 - 1}
POL( isNatIList_1(x_1) ) = 1
POL( n__take_2(x_1, x_2) ) = 0
POL( activate_1(x_1) ) = 2x_1 + 2
POL( n__0 ) = 0
POL( 0 ) = 0
POL( n__s_1(x_1) ) = 0
POL( n__length_1(x_1) ) = 1
POL( n__zeros ) = 1
POL( zeros ) = 2
POL( n__cons_2(x_1, x_2) ) = x_2 + 1
POL( n__nil ) = 0
POL( nil ) = 0
POL( uTake1_1(x_1) ) = max{0, x_1 - 2}
POL( tt ) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   s(X) -> n__s(X)
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   length(X) -> n__length(X)
   cons(X1, X2) -> n__cons(X1, X2)
   take(0, IL) -> uTake1(isNatIList(IL))
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   take(X1, X2) -> n__take(X1, X2)
   isNatIList(n__zeros) -> tt
   uTake1(tt) -> nil
   isNatIList(IL) -> isNatList(activate(IL))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNat(n__0) -> tt
   and(tt, T) -> T
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   uLength(tt, L) -> s(length(activate(L)))
   nil -> n__nil


----------------------------------------

(44)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(y0, n__s(x0))) -> ISNATLIST(s(activate(x0)))
   ISNATLIST(n__cons(y0, n__length(x0))) -> ISNATLIST(length(activate(x0)))
   ISNATLIST(n__cons(y0, n__take(x0, x1))) -> ISNATLIST(take(activate(x0), activate(x1)))
   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__cons(0, n__zeros))
   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(45) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
----------------------------------------

(46)
Complex Obligation (AND)

----------------------------------------

(47)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__cons(0, n__zeros))
   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(48) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(49)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__cons(0, n__zeros))
   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

   0 -> n__0

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(50) MNOCProof (EQUIVALENT)
We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
----------------------------------------

(51)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__cons(0, n__zeros))
   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

   0 -> n__0

The set Q consists of the following terms:

   0

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(52) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__cons(0, n__zeros)) at position [0,0] we obtained the following new rules [LPAR04]:

   (ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__cons(n__0, n__zeros)),ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__cons(n__0, n__zeros)))


----------------------------------------

(53)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

   0 -> n__0

The set Q consists of the following terms:

   0

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(54) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(55)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__cons(n__0, n__zeros))

R is empty.
The set Q consists of the following terms:

   0

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(56) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   0


----------------------------------------

(57)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__cons(n__0, n__zeros))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(58) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule ISNATLIST(n__cons(y0, n__zeros)) -> ISNATLIST(n__cons(n__0, n__zeros)) we obtained the following new rules [LPAR04]:

   (ISNATLIST(n__cons(n__0, n__zeros)) -> ISNATLIST(n__cons(n__0, n__zeros)),ISNATLIST(n__cons(n__0, n__zeros)) -> ISNATLIST(n__cons(n__0, n__zeros)))


----------------------------------------

(59)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(n__0, n__zeros)) -> ISNATLIST(n__cons(n__0, n__zeros))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(60) NonTerminationLoopProof (COMPLETE)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = ISNATLIST(n__cons(n__0, n__zeros)) evaluates to  t =ISNATLIST(n__cons(n__0, n__zeros))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
* Matcher: [ ]
* Semiunifier: [ ]

--------------------------------------------------------------------------------
Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from ISNATLIST(n__cons(n__0, n__zeros)) to ISNATLIST(n__cons(n__0, n__zeros)).




----------------------------------------

(61)
NO

----------------------------------------

(62)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(y0, n__s(x0))) -> ISNATLIST(s(activate(x0)))
   ISNATLIST(n__cons(y0, n__length(x0))) -> ISNATLIST(length(activate(x0)))
   ISNATLIST(n__cons(y0, n__take(x0, x1))) -> ISNATLIST(take(activate(x0), activate(x1)))

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(63) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ISNATLIST(n__cons(y0, n__s(x0))) -> ISNATLIST(s(activate(x0)))
   ISNATLIST(n__cons(y0, n__length(x0))) -> ISNATLIST(length(activate(x0)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ISNATLIST_1(x_1) ) = x_1
POL( cons_2(x_1, x_2) ) = 2x_1 + 2x_2
POL( isNat_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( length_1(x_1) ) = 2
POL( s_1(x_1) ) = 1
POL( take_2(x_1, x_2) ) = x_2
POL( uLength_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( uTake2_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_3 + 2x_4 - 2}
POL( and_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( isNatIList_1(x_1) ) = 2
POL( n__take_2(x_1, x_2) ) = x_2
POL( activate_1(x_1) ) = x_1
POL( n__0 ) = 0
POL( 0 ) = 0
POL( n__s_1(x_1) ) = 1
POL( n__length_1(x_1) ) = 2
POL( n__zeros ) = 0
POL( zeros ) = 0
POL( n__cons_2(x_1, x_2) ) = 2x_1 + 2x_2
POL( n__nil ) = 0
POL( nil ) = 0
POL( uTake1_1(x_1) ) = max{0, -2}
POL( tt ) = 2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X
   s(X) -> n__s(X)
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   length(X) -> n__length(X)
   take(0, IL) -> uTake1(isNatIList(IL))
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   take(X1, X2) -> n__take(X1, X2)
   cons(X1, X2) -> n__cons(X1, X2)
   isNatIList(n__zeros) -> tt
   uTake1(tt) -> nil
   isNatIList(IL) -> isNatList(activate(IL))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNat(n__0) -> tt
   and(tt, T) -> T
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   uLength(tt, L) -> s(length(activate(L)))
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   0 -> n__0
   nil -> n__nil


----------------------------------------

(64)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(n__cons(y0, n__take(x0, x1))) -> ISNATLIST(take(activate(x0), activate(x1)))

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(65)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(n__cons(N, IL)) -> ISNATILIST(activate(IL))

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(66) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule ISNATILIST(n__cons(N, IL)) -> ISNATILIST(activate(IL)) at position [0] we obtained the following new rules [LPAR04]:

   (ISNATILIST(n__cons(y0, n__0)) -> ISNATILIST(0),ISNATILIST(n__cons(y0, n__0)) -> ISNATILIST(0))
   (ISNATILIST(n__cons(y0, n__s(x0))) -> ISNATILIST(s(activate(x0))),ISNATILIST(n__cons(y0, n__s(x0))) -> ISNATILIST(s(activate(x0))))
   (ISNATILIST(n__cons(y0, n__length(x0))) -> ISNATILIST(length(activate(x0))),ISNATILIST(n__cons(y0, n__length(x0))) -> ISNATILIST(length(activate(x0))))
   (ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(zeros),ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(zeros))
   (ISNATILIST(n__cons(y0, n__cons(x0, x1))) -> ISNATILIST(cons(activate(x0), x1)),ISNATILIST(n__cons(y0, n__cons(x0, x1))) -> ISNATILIST(cons(activate(x0), x1)))
   (ISNATILIST(n__cons(y0, n__nil)) -> ISNATILIST(nil),ISNATILIST(n__cons(y0, n__nil)) -> ISNATILIST(nil))
   (ISNATILIST(n__cons(y0, n__take(x0, x1))) -> ISNATILIST(take(activate(x0), activate(x1))),ISNATILIST(n__cons(y0, n__take(x0, x1))) -> ISNATILIST(take(activate(x0), activate(x1))))
   (ISNATILIST(n__cons(y0, x0)) -> ISNATILIST(x0),ISNATILIST(n__cons(y0, x0)) -> ISNATILIST(x0))


----------------------------------------

(67)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(n__cons(y0, n__0)) -> ISNATILIST(0)
   ISNATILIST(n__cons(y0, n__s(x0))) -> ISNATILIST(s(activate(x0)))
   ISNATILIST(n__cons(y0, n__length(x0))) -> ISNATILIST(length(activate(x0)))
   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(zeros)
   ISNATILIST(n__cons(y0, n__cons(x0, x1))) -> ISNATILIST(cons(activate(x0), x1))
   ISNATILIST(n__cons(y0, n__nil)) -> ISNATILIST(nil)
   ISNATILIST(n__cons(y0, n__take(x0, x1))) -> ISNATILIST(take(activate(x0), activate(x1)))
   ISNATILIST(n__cons(y0, x0)) -> ISNATILIST(x0)

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(68) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule ISNATILIST(n__cons(y0, n__0)) -> ISNATILIST(0) at position [0] we obtained the following new rules [LPAR04]:

   (ISNATILIST(n__cons(y0, n__0)) -> ISNATILIST(n__0),ISNATILIST(n__cons(y0, n__0)) -> ISNATILIST(n__0))


----------------------------------------

(69)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(n__cons(y0, n__s(x0))) -> ISNATILIST(s(activate(x0)))
   ISNATILIST(n__cons(y0, n__length(x0))) -> ISNATILIST(length(activate(x0)))
   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(zeros)
   ISNATILIST(n__cons(y0, n__cons(x0, x1))) -> ISNATILIST(cons(activate(x0), x1))
   ISNATILIST(n__cons(y0, n__nil)) -> ISNATILIST(nil)
   ISNATILIST(n__cons(y0, n__take(x0, x1))) -> ISNATILIST(take(activate(x0), activate(x1)))
   ISNATILIST(n__cons(y0, x0)) -> ISNATILIST(x0)
   ISNATILIST(n__cons(y0, n__0)) -> ISNATILIST(n__0)

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(70) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(71)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(n__cons(y0, n__s(x0))) -> ISNATILIST(s(activate(x0)))
   ISNATILIST(n__cons(y0, n__length(x0))) -> ISNATILIST(length(activate(x0)))
   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(zeros)
   ISNATILIST(n__cons(y0, n__cons(x0, x1))) -> ISNATILIST(cons(activate(x0), x1))
   ISNATILIST(n__cons(y0, n__nil)) -> ISNATILIST(nil)
   ISNATILIST(n__cons(y0, n__take(x0, x1))) -> ISNATILIST(take(activate(x0), activate(x1)))
   ISNATILIST(n__cons(y0, x0)) -> ISNATILIST(x0)

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(72) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(zeros) at position [0] we obtained the following new rules [LPAR04]:

   (ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(cons(0, n__zeros)),ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(cons(0, n__zeros)))
   (ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__zeros),ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__zeros))


----------------------------------------

(73)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(n__cons(y0, n__s(x0))) -> ISNATILIST(s(activate(x0)))
   ISNATILIST(n__cons(y0, n__length(x0))) -> ISNATILIST(length(activate(x0)))
   ISNATILIST(n__cons(y0, n__cons(x0, x1))) -> ISNATILIST(cons(activate(x0), x1))
   ISNATILIST(n__cons(y0, n__nil)) -> ISNATILIST(nil)
   ISNATILIST(n__cons(y0, n__take(x0, x1))) -> ISNATILIST(take(activate(x0), activate(x1)))
   ISNATILIST(n__cons(y0, x0)) -> ISNATILIST(x0)
   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(cons(0, n__zeros))
   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__zeros)

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(74) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(75)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(n__cons(y0, n__s(x0))) -> ISNATILIST(s(activate(x0)))
   ISNATILIST(n__cons(y0, n__length(x0))) -> ISNATILIST(length(activate(x0)))
   ISNATILIST(n__cons(y0, n__cons(x0, x1))) -> ISNATILIST(cons(activate(x0), x1))
   ISNATILIST(n__cons(y0, n__nil)) -> ISNATILIST(nil)
   ISNATILIST(n__cons(y0, n__take(x0, x1))) -> ISNATILIST(take(activate(x0), activate(x1)))
   ISNATILIST(n__cons(y0, x0)) -> ISNATILIST(x0)
   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(cons(0, n__zeros))

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(76) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule ISNATILIST(n__cons(y0, n__nil)) -> ISNATILIST(nil) at position [0] we obtained the following new rules [LPAR04]:

   (ISNATILIST(n__cons(y0, n__nil)) -> ISNATILIST(n__nil),ISNATILIST(n__cons(y0, n__nil)) -> ISNATILIST(n__nil))


----------------------------------------

(77)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(n__cons(y0, n__s(x0))) -> ISNATILIST(s(activate(x0)))
   ISNATILIST(n__cons(y0, n__length(x0))) -> ISNATILIST(length(activate(x0)))
   ISNATILIST(n__cons(y0, n__cons(x0, x1))) -> ISNATILIST(cons(activate(x0), x1))
   ISNATILIST(n__cons(y0, n__take(x0, x1))) -> ISNATILIST(take(activate(x0), activate(x1)))
   ISNATILIST(n__cons(y0, x0)) -> ISNATILIST(x0)
   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(cons(0, n__zeros))
   ISNATILIST(n__cons(y0, n__nil)) -> ISNATILIST(n__nil)

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(78) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(79)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(n__cons(y0, n__s(x0))) -> ISNATILIST(s(activate(x0)))
   ISNATILIST(n__cons(y0, n__length(x0))) -> ISNATILIST(length(activate(x0)))
   ISNATILIST(n__cons(y0, n__cons(x0, x1))) -> ISNATILIST(cons(activate(x0), x1))
   ISNATILIST(n__cons(y0, n__take(x0, x1))) -> ISNATILIST(take(activate(x0), activate(x1)))
   ISNATILIST(n__cons(y0, x0)) -> ISNATILIST(x0)
   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(cons(0, n__zeros))

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(80) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(cons(0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

   (ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__cons(0, n__zeros)),ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__cons(0, n__zeros)))
   (ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(cons(n__0, n__zeros)),ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(cons(n__0, n__zeros)))


----------------------------------------

(81)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(n__cons(y0, n__s(x0))) -> ISNATILIST(s(activate(x0)))
   ISNATILIST(n__cons(y0, n__length(x0))) -> ISNATILIST(length(activate(x0)))
   ISNATILIST(n__cons(y0, n__cons(x0, x1))) -> ISNATILIST(cons(activate(x0), x1))
   ISNATILIST(n__cons(y0, n__take(x0, x1))) -> ISNATILIST(take(activate(x0), activate(x1)))
   ISNATILIST(n__cons(y0, x0)) -> ISNATILIST(x0)
   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__cons(0, n__zeros))
   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(cons(n__0, n__zeros))

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(82) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(cons(n__0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

   (ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__cons(n__0, n__zeros)),ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__cons(n__0, n__zeros)))


----------------------------------------

(83)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(n__cons(y0, n__s(x0))) -> ISNATILIST(s(activate(x0)))
   ISNATILIST(n__cons(y0, n__length(x0))) -> ISNATILIST(length(activate(x0)))
   ISNATILIST(n__cons(y0, n__cons(x0, x1))) -> ISNATILIST(cons(activate(x0), x1))
   ISNATILIST(n__cons(y0, n__take(x0, x1))) -> ISNATILIST(take(activate(x0), activate(x1)))
   ISNATILIST(n__cons(y0, x0)) -> ISNATILIST(x0)
   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__cons(0, n__zeros))
   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(84) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ISNATILIST(n__cons(y0, n__cons(x0, x1))) -> ISNATILIST(cons(activate(x0), x1))
   ISNATILIST(n__cons(y0, x0)) -> ISNATILIST(x0)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ISNATILIST_1(x_1) ) = x_1 + 2
POL( cons_2(x_1, x_2) ) = x_2 + 1
POL( isNat_1(x_1) ) = 1
POL( isNatList_1(x_1) ) = 1
POL( length_1(x_1) ) = 2
POL( s_1(x_1) ) = 1
POL( take_2(x_1, x_2) ) = 1
POL( uLength_2(x_1, x_2) ) = 2
POL( uTake2_4(x_1, ..., x_4) ) = max{0, 2x_1 - 1}
POL( and_2(x_1, x_2) ) = max{0, x_1 + x_2 - 1}
POL( isNatIList_1(x_1) ) = 1
POL( n__take_2(x_1, x_2) ) = 0
POL( activate_1(x_1) ) = 2x_1 + 2
POL( n__0 ) = 0
POL( 0 ) = 0
POL( n__s_1(x_1) ) = 0
POL( n__length_1(x_1) ) = 1
POL( n__zeros ) = 1
POL( zeros ) = 2
POL( n__cons_2(x_1, x_2) ) = x_2 + 1
POL( n__nil ) = 0
POL( nil ) = 0
POL( uTake1_1(x_1) ) = max{0, x_1 - 2}
POL( tt ) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   s(X) -> n__s(X)
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   length(X) -> n__length(X)
   cons(X1, X2) -> n__cons(X1, X2)
   take(0, IL) -> uTake1(isNatIList(IL))
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   take(X1, X2) -> n__take(X1, X2)
   isNatIList(n__zeros) -> tt
   uTake1(tt) -> nil
   isNatIList(IL) -> isNatList(activate(IL))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNat(n__0) -> tt
   and(tt, T) -> T
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   uLength(tt, L) -> s(length(activate(L)))
   nil -> n__nil


----------------------------------------

(85)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(n__cons(y0, n__s(x0))) -> ISNATILIST(s(activate(x0)))
   ISNATILIST(n__cons(y0, n__length(x0))) -> ISNATILIST(length(activate(x0)))
   ISNATILIST(n__cons(y0, n__take(x0, x1))) -> ISNATILIST(take(activate(x0), activate(x1)))
   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__cons(0, n__zeros))
   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(86) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
----------------------------------------

(87)
Complex Obligation (AND)

----------------------------------------

(88)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__cons(0, n__zeros))
   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(89) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(90)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__cons(0, n__zeros))
   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

   0 -> n__0

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(91) MNOCProof (EQUIVALENT)
We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
----------------------------------------

(92)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__cons(0, n__zeros))
   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

   0 -> n__0

The set Q consists of the following terms:

   0

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(93) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__cons(0, n__zeros)) at position [0,0] we obtained the following new rules [LPAR04]:

   (ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__cons(n__0, n__zeros)),ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__cons(n__0, n__zeros)))


----------------------------------------

(94)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

   0 -> n__0

The set Q consists of the following terms:

   0

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(95) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(96)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__cons(n__0, n__zeros))

R is empty.
The set Q consists of the following terms:

   0

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(97) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   0


----------------------------------------

(98)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__cons(n__0, n__zeros))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(99) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule ISNATILIST(n__cons(y0, n__zeros)) -> ISNATILIST(n__cons(n__0, n__zeros)) we obtained the following new rules [LPAR04]:

   (ISNATILIST(n__cons(n__0, n__zeros)) -> ISNATILIST(n__cons(n__0, n__zeros)),ISNATILIST(n__cons(n__0, n__zeros)) -> ISNATILIST(n__cons(n__0, n__zeros)))


----------------------------------------

(100)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(n__cons(n__0, n__zeros)) -> ISNATILIST(n__cons(n__0, n__zeros))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(101)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(n__cons(y0, n__s(x0))) -> ISNATILIST(s(activate(x0)))
   ISNATILIST(n__cons(y0, n__length(x0))) -> ISNATILIST(length(activate(x0)))
   ISNATILIST(n__cons(y0, n__take(x0, x1))) -> ISNATILIST(take(activate(x0), activate(x1)))

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(102) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ISNATILIST(n__cons(y0, n__s(x0))) -> ISNATILIST(s(activate(x0)))
   ISNATILIST(n__cons(y0, n__length(x0))) -> ISNATILIST(length(activate(x0)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ISNATILIST_1(x_1) ) = x_1
POL( cons_2(x_1, x_2) ) = 2x_1 + 2x_2
POL( isNat_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( length_1(x_1) ) = 2
POL( s_1(x_1) ) = 1
POL( take_2(x_1, x_2) ) = x_2
POL( uLength_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( uTake2_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_3 + 2x_4 - 2}
POL( and_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( isNatIList_1(x_1) ) = 2
POL( n__take_2(x_1, x_2) ) = x_2
POL( activate_1(x_1) ) = x_1
POL( n__0 ) = 0
POL( 0 ) = 0
POL( n__s_1(x_1) ) = 1
POL( n__length_1(x_1) ) = 2
POL( n__zeros ) = 0
POL( zeros ) = 0
POL( n__cons_2(x_1, x_2) ) = 2x_1 + 2x_2
POL( n__nil ) = 0
POL( nil ) = 0
POL( uTake1_1(x_1) ) = max{0, -2}
POL( tt ) = 2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X
   s(X) -> n__s(X)
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   length(X) -> n__length(X)
   take(0, IL) -> uTake1(isNatIList(IL))
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   take(X1, X2) -> n__take(X1, X2)
   cons(X1, X2) -> n__cons(X1, X2)
   isNatIList(n__zeros) -> tt
   uTake1(tt) -> nil
   isNatIList(IL) -> isNatList(activate(IL))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNat(n__0) -> tt
   and(tt, T) -> T
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   uLength(tt, L) -> s(length(activate(L)))
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   0 -> n__0
   nil -> n__nil


----------------------------------------

(103)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(n__cons(y0, n__take(x0, x1))) -> ISNATILIST(take(activate(x0), activate(x1)))

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(104)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LENGTH(cons(N, L)) -> ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
   ULENGTH(tt, L) -> LENGTH(activate(L))

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(105) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ULENGTH(tt, L) -> LENGTH(activate(L))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(LENGTH(x_1)) =  	[[4A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(ULENGTH(x_1, x_2)) =  	[[4A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2
>>>

   <<<
 POL(and(x_1, x_2)) =  	[[3A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(isNat(x_1)) =  	[[5A]] 	 +  	[[2A]] 	* 	x_1
>>>

   <<<
 POL(isNatList(x_1)) =  	[[3A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(activate(x_1)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(tt) =  	[[5A]]
>>>

   <<<
 POL(n__0) =  	[[2A]]
>>>

   <<<
 POL(n__s(x_1)) =  	[[5A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(n__length(x_1)) =  	[[3A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[2A]]
>>>

   <<<
 POL(s(x_1)) =  	[[5A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(length(x_1)) =  	[[3A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(n__zeros) =  	[[1A]]
>>>

   <<<
 POL(zeros) =  	[[2A]]
>>>

   <<<
 POL(n__cons(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(n__nil) =  	[[4A]]
>>>

   <<<
 POL(nil) =  	[[4A]]
>>>

   <<<
 POL(n__take(x_1, x_2)) =  	[[4A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(take(x_1, x_2)) =  	[[4A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(isNatIList(x_1)) =  	[[5A]] 	 +  	[[2A]] 	* 	x_1
>>>

   <<<
 POL(uTake1(x_1)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(uTake2(x_1, x_2, x_3, x_4)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2 	 +  	[[-I]] 	* 	x_3 	 +  	[[2A]] 	* 	x_4
>>>

   <<<
 POL(uLength(x_1, x_2)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   and(tt, T) -> T
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   cons(X1, X2) -> n__cons(X1, X2)
   take(X1, X2) -> n__take(X1, X2)
   take(0, IL) -> uTake1(isNatIList(IL))
   isNatIList(n__zeros) -> tt
   uTake1(tt) -> nil
   isNatIList(IL) -> isNatList(activate(IL))
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   0 -> n__0
   nil -> n__nil


----------------------------------------

(106)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LENGTH(cons(N, L)) -> ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))

The TRS R consists of the following rules:

   and(tt, T) -> T
   isNatIList(IL) -> isNatList(activate(IL))
   isNat(n__0) -> tt
   isNat(n__s(N)) -> isNat(activate(N))
   isNat(n__length(L)) -> isNatList(activate(L))
   isNatIList(n__zeros) -> tt
   isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   isNatList(n__nil) -> tt
   isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L)))
   isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL)))
   zeros -> cons(0, n__zeros)
   take(0, IL) -> uTake1(isNatIList(IL))
   uTake1(tt) -> nil
   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
   uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL)))
   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L))
   uLength(tt, L) -> s(length(activate(L)))
   0 -> n__0
   s(X) -> n__s(X)
   length(X) -> n__length(X)
   zeros -> n__zeros
   cons(X1, X2) -> n__cons(X1, X2)
   nil -> n__nil
   take(X1, X2) -> n__take(X1, X2)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))
   activate(n__length(X)) -> length(activate(X))
   activate(n__zeros) -> zeros
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__nil) -> nil
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(107) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
----------------------------------------

(108)
TRUE