YES
proof of Transformed_CSR_04_LengthOfFiniteLists_nokinds_iGM.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 152 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 0 ms]
(4) QDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) AND
    (7) QDP
        (8) UsableRulesProof [EQUIVALENT, 0 ms]
        (9) QDP
        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) QDP
        (13) UsableRulesProof [EQUIVALENT, 0 ms]
        (14) QDP
        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (16) YES
    (17) QDP
        (18) UsableRulesProof [EQUIVALENT, 0 ms]
        (19) QDP
        (20) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (21) YES
    (22) QDP
        (23) UsableRulesProof [EQUIVALENT, 0 ms]
        (24) QDP
        (25) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (26) YES
    (27) QDP
        (28) UsableRulesProof [EQUIVALENT, 0 ms]
        (29) QDP
        (30) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (31) YES
    (32) QDP
        (33) UsableRulesProof [EQUIVALENT, 0 ms]
        (34) QDP
        (35) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (36) YES
    (37) QDP
        (38) UsableRulesProof [EQUIVALENT, 0 ms]
        (39) QDP
        (40) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (41) YES
    (42) QDP
        (43) UsableRulesProof [EQUIVALENT, 0 ms]
        (44) QDP
        (45) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (46) YES
    (47) QDP
        (48) MRRProof [EQUIVALENT, 60 ms]
        (49) QDP
        (50) QDPOrderProof [EQUIVALENT, 122 ms]
        (51) QDP
        (52) QDPOrderProof [EQUIVALENT, 164 ms]
        (53) QDP
        (54) QDPOrderProof [EQUIVALENT, 154 ms]
        (55) QDP
        (56) QDPOrderProof [EQUIVALENT, 395 ms]
        (57) QDP
        (58) QDPOrderProof [EQUIVALENT, 125 ms]
        (59) QDP
        (60) QDPOrderProof [EQUIVALENT, 125 ms]
        (61) QDP
        (62) QDPOrderProof [EQUIVALENT, 117 ms]
        (63) QDP
        (64) QDPOrderProof [EQUIVALENT, 133 ms]
        (65) QDP
        (66) QDPOrderProof [EQUIVALENT, 219 ms]
        (67) QDP
        (68) QDPOrderProof [EQUIVALENT, 97 ms]
        (69) QDP
        (70) QDPOrderProof [EQUIVALENT, 196 ms]
        (71) QDP
        (72) QDPOrderProof [EQUIVALENT, 204 ms]
        (73) QDP
        (74) TransformationProof [EQUIVALENT, 0 ms]
        (75) QDP
        (76) TransformationProof [EQUIVALENT, 0 ms]
        (77) QDP
        (78) TransformationProof [EQUIVALENT, 0 ms]
        (79) QDP
        (80) DependencyGraphProof [EQUIVALENT, 0 ms]
        (81) QDP
        (82) TransformationProof [EQUIVALENT, 0 ms]
        (83) QDP
        (84) DependencyGraphProof [EQUIVALENT, 0 ms]
        (85) QDP
        (86) TransformationProof [EQUIVALENT, 8 ms]
        (87) QDP
        (88) DependencyGraphProof [EQUIVALENT, 0 ms]
        (89) QDP
        (90) QDPOrderProof [EQUIVALENT, 329 ms]
        (91) QDP
        (92) QDPOrderProof [EQUIVALENT, 232 ms]
        (93) QDP
        (94) QDPOrderProof [EQUIVALENT, 179 ms]
        (95) QDP
        (96) QDPOrderProof [EQUIVALENT, 374 ms]
        (97) QDP
        (98) QDPOrderProof [EQUIVALENT, 388 ms]
        (99) QDP
        (100) QDPOrderProof [EQUIVALENT, 246 ms]
        (101) QDP
        (102) QDPOrderProof [EQUIVALENT, 395 ms]
        (103) QDP
        (104) QDPOrderProof [EQUIVALENT, 571 ms]
        (105) QDP
        (106) QDPOrderProof [EQUIVALENT, 193 ms]
        (107) QDP
        (108) QDPOrderProof [EQUIVALENT, 429 ms]
        (109) QDP
        (110) QDPOrderProof [EQUIVALENT, 190 ms]
        (111) QDP
        (112) QDPOrderProof [EQUIVALENT, 152 ms]
        (113) QDP
        (114) NonMonReductionPairProof [EQUIVALENT, 3667 ms]
        (115) QDP
        (116) QDPOrderProof [EQUIVALENT, 48 ms]
        (117) QDP
        (118) QDPOrderProof [EQUIVALENT, 0 ms]
        (119) QDP
        (120) PisEmptyProof [EQUIVALENT, 0 ms]
        (121) YES


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

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

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(isNatList(V1))
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(V)) -> mark(isNatList(V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

Q is empty.

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

(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(U11(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(active(x_1)) = x_1
   POL(and(x_1, x_2)) = 2*x_1 + x_2
   POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(isNat(x_1)) = x_1
   POL(isNatIList(x_1)) = 1 + 2*x_1
   POL(isNatList(x_1)) = x_1
   POL(length(x_1)) = 1 + 2*x_1
   POL(mark(x_1)) = x_1
   POL(nil) = 2
   POL(s(x_1)) = x_1
   POL(tt) = 0
   POL(zeros) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   active(isNat(length(V1))) -> mark(isNatList(V1))
   active(isNatIList(V)) -> mark(isNatList(V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)




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

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

Q is empty.

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

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

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

   ACTIVE(zeros) -> MARK(cons(0, zeros))
   ACTIVE(zeros) -> CONS(0, zeros)
   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   ACTIVE(U11(tt, L)) -> S(length(L))
   ACTIVE(U11(tt, L)) -> LENGTH(L)
   ACTIVE(and(tt, X)) -> MARK(X)
   ACTIVE(isNat(0)) -> MARK(tt)
   ACTIVE(isNat(s(V1))) -> MARK(isNat(V1))
   ACTIVE(isNat(s(V1))) -> ISNAT(V1)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(and(isNat(V1), isNatIList(V2)))
   ACTIVE(isNatIList(cons(V1, V2))) -> AND(isNat(V1), isNatIList(V2))
   ACTIVE(isNatIList(cons(V1, V2))) -> ISNAT(V1)
   ACTIVE(isNatIList(cons(V1, V2))) -> ISNATILIST(V2)
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(and(isNat(V1), isNatList(V2)))
   ACTIVE(isNatList(cons(V1, V2))) -> AND(isNat(V1), isNatList(V2))
   ACTIVE(isNatList(cons(V1, V2))) -> ISNAT(V1)
   ACTIVE(isNatList(cons(V1, V2))) -> ISNATLIST(V2)
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   ACTIVE(length(cons(N, L))) -> U11^1(and(isNatList(L), isNat(N)), L)
   ACTIVE(length(cons(N, L))) -> AND(isNatList(L), isNat(N))
   ACTIVE(length(cons(N, L))) -> ISNATLIST(L)
   ACTIVE(length(cons(N, L))) -> ISNAT(N)
   MARK(zeros) -> ACTIVE(zeros)
   MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2))
   MARK(cons(X1, X2)) -> CONS(mark(X1), X2)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(0) -> ACTIVE(0)
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   MARK(U11(X1, X2)) -> U11^1(mark(X1), X2)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(tt) -> ACTIVE(tt)
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   MARK(s(X)) -> S(mark(X))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(length(X)) -> LENGTH(mark(X))
   MARK(length(X)) -> MARK(X)
   MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), X2))
   MARK(and(X1, X2)) -> AND(mark(X1), X2)
   MARK(and(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   MARK(nil) -> ACTIVE(nil)
   CONS(mark(X1), X2) -> CONS(X1, X2)
   CONS(X1, mark(X2)) -> CONS(X1, X2)
   CONS(active(X1), X2) -> CONS(X1, X2)
   CONS(X1, active(X2)) -> CONS(X1, X2)
   U11^1(mark(X1), X2) -> U11^1(X1, X2)
   U11^1(X1, mark(X2)) -> U11^1(X1, X2)
   U11^1(active(X1), X2) -> U11^1(X1, X2)
   U11^1(X1, active(X2)) -> U11^1(X1, X2)
   S(mark(X)) -> S(X)
   S(active(X)) -> S(X)
   LENGTH(mark(X)) -> LENGTH(X)
   LENGTH(active(X)) -> LENGTH(X)
   AND(mark(X1), X2) -> AND(X1, X2)
   AND(X1, mark(X2)) -> AND(X1, X2)
   AND(active(X1), X2) -> AND(X1, X2)
   AND(X1, active(X2)) -> AND(X1, X2)
   ISNAT(mark(X)) -> ISNAT(X)
   ISNAT(active(X)) -> ISNAT(X)
   ISNATLIST(mark(X)) -> ISNATLIST(X)
   ISNATLIST(active(X)) -> ISNATLIST(X)
   ISNATILIST(mark(X)) -> ISNATILIST(X)
   ISNATILIST(active(X)) -> ISNATILIST(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 9 SCCs with 23 less nodes.
----------------------------------------

(6)
Complex Obligation (AND)

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

(7)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(active(X)) -> ISNATILIST(X)
   ISNATILIST(mark(X)) -> ISNATILIST(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

(8) 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.
----------------------------------------

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

   ISNATILIST(active(X)) -> ISNATILIST(X)
   ISNATILIST(mark(X)) -> ISNATILIST(X)

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

(10) 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:
*ISNATILIST(active(X)) -> ISNATILIST(X)
The graph contains the following edges 1 > 1


*ISNATILIST(mark(X)) -> ISNATILIST(X)
The graph contains the following edges 1 > 1


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

(11)
YES

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

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(active(X)) -> ISNATLIST(X)
   ISNATLIST(mark(X)) -> ISNATLIST(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

(13) 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.
----------------------------------------

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

   ISNATLIST(active(X)) -> ISNATLIST(X)
   ISNATLIST(mark(X)) -> ISNATLIST(X)

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

(15) 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:
*ISNATLIST(active(X)) -> ISNATLIST(X)
The graph contains the following edges 1 > 1


*ISNATLIST(mark(X)) -> ISNATLIST(X)
The graph contains the following edges 1 > 1


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

(16)
YES

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

(17)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNAT(active(X)) -> ISNAT(X)
   ISNAT(mark(X)) -> ISNAT(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

(18) 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.
----------------------------------------

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

   ISNAT(active(X)) -> ISNAT(X)
   ISNAT(mark(X)) -> ISNAT(X)

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

(20) 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:
*ISNAT(active(X)) -> ISNAT(X)
The graph contains the following edges 1 > 1


*ISNAT(mark(X)) -> ISNAT(X)
The graph contains the following edges 1 > 1


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

(21)
YES

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

(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   AND(X1, mark(X2)) -> AND(X1, X2)
   AND(mark(X1), X2) -> AND(X1, X2)
   AND(active(X1), X2) -> AND(X1, X2)
   AND(X1, active(X2)) -> AND(X1, X2)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

(23) 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.
----------------------------------------

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

   AND(X1, mark(X2)) -> AND(X1, X2)
   AND(mark(X1), X2) -> AND(X1, X2)
   AND(active(X1), X2) -> AND(X1, X2)
   AND(X1, active(X2)) -> AND(X1, X2)

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

(25) 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:
*AND(X1, mark(X2)) -> AND(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*AND(mark(X1), X2) -> AND(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*AND(active(X1), X2) -> AND(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*AND(X1, active(X2)) -> AND(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


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

(26)
YES

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

(27)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LENGTH(active(X)) -> LENGTH(X)
   LENGTH(mark(X)) -> LENGTH(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

(28) 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.
----------------------------------------

(29)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LENGTH(active(X)) -> LENGTH(X)
   LENGTH(mark(X)) -> LENGTH(X)

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

(30) 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:
*LENGTH(active(X)) -> LENGTH(X)
The graph contains the following edges 1 > 1


*LENGTH(mark(X)) -> LENGTH(X)
The graph contains the following edges 1 > 1


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

(31)
YES

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

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

   S(active(X)) -> S(X)
   S(mark(X)) -> S(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

(33) 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.
----------------------------------------

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

   S(active(X)) -> S(X)
   S(mark(X)) -> S(X)

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

(35) 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:
*S(active(X)) -> S(X)
The graph contains the following edges 1 > 1


*S(mark(X)) -> S(X)
The graph contains the following edges 1 > 1


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

(36)
YES

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

(37)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U11^1(X1, mark(X2)) -> U11^1(X1, X2)
   U11^1(mark(X1), X2) -> U11^1(X1, X2)
   U11^1(active(X1), X2) -> U11^1(X1, X2)
   U11^1(X1, active(X2)) -> U11^1(X1, X2)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

(38) 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.
----------------------------------------

(39)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U11^1(X1, mark(X2)) -> U11^1(X1, X2)
   U11^1(mark(X1), X2) -> U11^1(X1, X2)
   U11^1(active(X1), X2) -> U11^1(X1, X2)
   U11^1(X1, active(X2)) -> U11^1(X1, X2)

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

(40) 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:
*U11^1(X1, mark(X2)) -> U11^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*U11^1(mark(X1), X2) -> U11^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U11^1(active(X1), X2) -> U11^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U11^1(X1, active(X2)) -> U11^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


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

(41)
YES

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

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

   CONS(X1, mark(X2)) -> CONS(X1, X2)
   CONS(mark(X1), X2) -> CONS(X1, X2)
   CONS(active(X1), X2) -> CONS(X1, X2)
   CONS(X1, active(X2)) -> CONS(X1, X2)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

(43) 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.
----------------------------------------

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

   CONS(X1, mark(X2)) -> CONS(X1, X2)
   CONS(mark(X1), X2) -> CONS(X1, X2)
   CONS(active(X1), X2) -> CONS(X1, X2)
   CONS(X1, active(X2)) -> CONS(X1, X2)

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

(45) 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:
*CONS(X1, mark(X2)) -> CONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*CONS(mark(X1), X2) -> CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*CONS(active(X1), X2) -> CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*CONS(X1, active(X2)) -> CONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


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

(46)
YES

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

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

   MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2))
   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(and(tt, X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   ACTIVE(isNat(s(V1))) -> MARK(isNat(V1))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(and(isNat(V1), isNatIList(V2)))
   MARK(length(X)) -> MARK(X)
   MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), X2))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(and(isNat(V1), isNatList(V2)))
   MARK(and(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

(48) MRRProof (EQUIVALENT)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented dependency pairs:

   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(length(X)) -> MARK(X)


Used ordering: Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = x_1
   POL(MARK(x_1)) = x_1
   POL(U11(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(active(x_1)) = x_1
   POL(and(x_1, x_2)) = 2*x_1 + x_2
   POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(isNat(x_1)) = x_1
   POL(isNatIList(x_1)) = 1 + x_1
   POL(isNatList(x_1)) = x_1
   POL(length(x_1)) = 1 + 2*x_1
   POL(mark(x_1)) = x_1
   POL(nil) = 2
   POL(s(x_1)) = x_1
   POL(tt) = 0
   POL(zeros) = 0


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

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

   MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2))
   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(and(tt, X)) -> MARK(X)
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   ACTIVE(isNat(s(V1))) -> MARK(isNat(V1))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(and(isNat(V1), isNatIList(V2)))
   MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), X2))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(and(isNat(V1), isNatList(V2)))
   MARK(and(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(cons(X1, X2)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = 2x_1 + 2
POL( U11_2(x_1, x_2) ) = x_1 + 2
POL( and_2(x_1, x_2) ) = 2x_1 + x_2
POL( cons_2(x_1, x_2) ) = 2x_1 + 1
POL( length_1(x_1) ) = 2
POL( s_1(x_1) ) = x_1
POL( mark_1(x_1) ) = x_1
POL( active_1(x_1) ) = x_1
POL( zeros ) = 1
POL( 0 ) = 0
POL( tt ) = 0
POL( isNat_1(x_1) ) = 0
POL( isNatIList_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 0
POL( nil ) = 0
POL( MARK_1(x_1) ) = 2x_1 + 2

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

   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   mark(zeros) -> active(zeros)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(and(tt, X)) -> mark(X)
   mark(s(X)) -> active(s(mark(X)))
   active(isNat(s(V1))) -> mark(isNat(V1))
   mark(length(X)) -> active(length(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   mark(isNat(X)) -> active(isNat(X))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(nil) -> active(nil)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   and(X1, mark(X2)) -> and(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   active(isNat(0)) -> mark(tt)


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

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

   MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2))
   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(and(tt, X)) -> MARK(X)
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   ACTIVE(isNat(s(V1))) -> MARK(isNat(V1))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(and(isNat(V1), isNatIList(V2)))
   MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), X2))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(and(isNat(V1), isNatList(V2)))
   MARK(and(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = x_1
POL( U11_2(x_1, x_2) ) = 0
POL( and_2(x_1, x_2) ) = x_1 + 2x_2
POL( cons_2(x_1, x_2) ) = 0
POL( length_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1
POL( mark_1(x_1) ) = 2x_1
POL( active_1(x_1) ) = x_1
POL( zeros ) = 1
POL( 0 ) = 0
POL( tt ) = 0
POL( isNat_1(x_1) ) = 0
POL( isNatIList_1(x_1) ) = 0
POL( isNatList_1(x_1) ) = 0
POL( nil ) = 0
POL( MARK_1(x_1) ) = 2x_1

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

   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   mark(zeros) -> active(zeros)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(and(tt, X)) -> mark(X)
   mark(s(X)) -> active(s(mark(X)))
   active(isNat(s(V1))) -> mark(isNat(V1))
   mark(length(X)) -> active(length(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   mark(isNat(X)) -> active(isNat(X))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(nil) -> active(nil)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   and(X1, mark(X2)) -> and(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   active(isNat(0)) -> mark(tt)


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

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

   MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2))
   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(and(tt, X)) -> MARK(X)
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   ACTIVE(isNat(s(V1))) -> MARK(isNat(V1))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(and(isNat(V1), isNatIList(V2)))
   MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), X2))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(and(isNat(V1), isNatList(V2)))
   MARK(and(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2))
   MARK(s(X)) -> ACTIVE(s(mark(X)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Combined order from the following AFS and order.
MARK(x1)  =  MARK

cons(x1, x2)  =  cons

ACTIVE(x1)  =  x1

mark(x1)  =  mark(x1)

U11(x1, x2)  =  U11

tt  =  tt

s(x1)  =  s

length(x1)  =  length

and(x1, x2)  =  and

isNat(x1)  =  isNat

isNatIList(x1)  =  isNatIList

isNatList(x1)  =  isNatList

active(x1)  =  active(x1)

zeros  =  zeros

0  =  0

nil  =  nil


Recursive path order with status [RPO].
Quasi-Precedence: active_1 > [MARK, U11, length, and, isNat, isNatIList, isNatList] > mark_1 > [cons, zeros] > 0 > s
active_1 > [MARK, U11, length, and, isNat, isNatIList, isNatList] > mark_1 > tt > s
active_1 > [MARK, U11, length, and, isNat, isNatIList, isNatList] > mark_1 > nil > s

Status: MARK: multiset status
cons: multiset status
mark_1: multiset status
U11: multiset status
tt: multiset status
s: multiset status
length: multiset status
and: multiset status
isNat: multiset status
isNatIList: multiset status
isNatList: multiset status
active_1: [1]
zeros: multiset status
0: multiset status
nil: multiset status


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

   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   and(X1, mark(X2)) -> and(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)


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

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

   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(and(tt, X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(isNat(V1))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(and(isNat(V1), isNatIList(V2)))
   MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), X2))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(and(isNat(V1), isNatList(V2)))
   MARK(and(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(and(tt, X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(ACTIVE(x_1)) =  	[[5A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U11(x_1, x_2)) =  	[[3A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2
>>>

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

   <<<
 POL(MARK(x_1)) =  	[[4A]] 	 +  	[[2A]] 	* 	x_1
>>>

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

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

   <<<
 POL(mark(x_1)) =  	[[5A]] 	 +  	[[0A]] 	* 	x_1
>>>

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

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

   <<<
 POL(isNatIList(x_1)) =  	[[3A]] 	 +  	[[5A]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[3A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[5A]] 	* 	x_2
>>>

   <<<
 POL(isNatList(x_1)) =  	[[3A]] 	 +  	[[5A]] 	* 	x_1
>>>

   <<<
 POL(active(x_1)) =  	[[5A]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(0) =  	[[3A]]
>>>

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


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

   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   mark(zeros) -> active(zeros)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(and(tt, X)) -> mark(X)
   mark(s(X)) -> active(s(mark(X)))
   active(isNat(s(V1))) -> mark(isNat(V1))
   mark(length(X)) -> active(length(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   mark(isNat(X)) -> active(isNat(X))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(nil) -> active(nil)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   and(X1, mark(X2)) -> and(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   active(isNat(0)) -> mark(tt)


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

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

   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(isNat(V1))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(and(isNat(V1), isNatIList(V2)))
   MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), X2))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(and(isNat(V1), isNatList(V2)))
   MARK(and(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = 0
POL( MARK_1(x_1) ) = 2x_1
POL( s_1(x_1) ) = 2x_1
POL( length_1(x_1) ) = max{0, -2}
POL( active_1(x_1) ) = x_1 + 1
POL( mark_1(x_1) ) = 2x_1 + 2
POL( U11_2(x_1, x_2) ) = max{0, -1}
POL( and_2(x_1, x_2) ) = 2x_1
POL( cons_2(x_1, x_2) ) = max{0, x_1 - 1}
POL( zeros ) = 0
POL( 0 ) = 0
POL( tt ) = 0
POL( isNat_1(x_1) ) = 0
POL( isNatIList_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 0
POL( nil ) = 1

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

   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   and(X1, mark(X2)) -> and(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)


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

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

   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(isNat(V1))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(and(isNat(V1), isNatIList(V2)))
   MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), X2))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(and(isNat(V1), isNatList(V2)))
   MARK(and(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(and(isNat(V1), isNatIList(V2)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2}
POL( MARK_1(x_1) ) = max{0, -1}
POL( s_1(x_1) ) = x_1
POL( length_1(x_1) ) = 0
POL( active_1(x_1) ) = 2
POL( mark_1(x_1) ) = x_1
POL( U11_2(x_1, x_2) ) = 0
POL( and_2(x_1, x_2) ) = 1
POL( cons_2(x_1, x_2) ) = x_1 + 2
POL( zeros ) = 2
POL( 0 ) = 2
POL( tt ) = 0
POL( isNat_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = x_1 + 1
POL( isNatList_1(x_1) ) = 2
POL( nil ) = 2

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

   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   and(X1, mark(X2)) -> and(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)


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

(61)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(isNat(V1))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), X2))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(and(isNat(V1), isNatList(V2)))
   MARK(and(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), X2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Combined order from the following AFS and order.
ACTIVE(x1)  =  x1

U11(x1, x2)  =  U11

tt  =  tt

MARK(x1)  =  MARK

s(x1)  =  s

length(x1)  =  length

mark(x1)  =  mark

isNat(x1)  =  isNat

and(x1, x2)  =  and

isNatList(x1)  =  isNatList

cons(x1, x2)  =  cons(x1, x2)

active(x1)  =  active

zeros  =  zeros

0  =  0

isNatIList(x1)  =  x1

nil  =  nil


Recursive path order with status [RPO].
Quasi-Precedence: mark > tt > [U11, MARK, length, isNat, isNatList] > [s, and, active, zeros, 0, nil]
mark > cons_2 > [U11, MARK, length, isNat, isNatList] > [s, and, active, zeros, 0, nil]

Status: U11: multiset status
tt: multiset status
MARK: multiset status
s: multiset status
length: multiset status
mark: multiset status
isNat: multiset status
and: []
isNatList: multiset status
cons_2: multiset status
active: multiset status
zeros: multiset status
0: multiset status
nil: multiset status


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

   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   and(X1, mark(X2)) -> and(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)


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

(63)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(isNat(V1))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(and(isNat(V1), isNatList(V2)))
   MARK(and(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(isNatList(cons(V1, V2))) -> MARK(and(isNat(V1), isNatList(V2)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = 2x_1 + 1
POL( MARK_1(x_1) ) = 2x_1 + 1
POL( s_1(x_1) ) = x_1
POL( length_1(x_1) ) = 2
POL( active_1(x_1) ) = x_1 + 1
POL( mark_1(x_1) ) = 2x_1
POL( U11_2(x_1, x_2) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 1}
POL( zeros ) = 2
POL( 0 ) = 2
POL( tt ) = 0
POL( and_2(x_1, x_2) ) = 2x_1
POL( isNat_1(x_1) ) = 0
POL( isNatIList_1(x_1) ) = 2x_1 + 2
POL( isNatList_1(x_1) ) = 2
POL( nil ) = 1

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

   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   and(X1, mark(X2)) -> and(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)


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

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

   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(isNat(V1))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(and(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(and(X1, X2)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = x_1 + 1
POL( MARK_1(x_1) ) = x_1 + 1
POL( s_1(x_1) ) = x_1
POL( length_1(x_1) ) = 2
POL( active_1(x_1) ) = x_1 + 2
POL( mark_1(x_1) ) = x_1 + 2
POL( U11_2(x_1, x_2) ) = 2
POL( cons_2(x_1, x_2) ) = 2x_1 + x_2
POL( zeros ) = 2
POL( 0 ) = 0
POL( tt ) = 0
POL( and_2(x_1, x_2) ) = x_1 + 2
POL( isNat_1(x_1) ) = 1
POL( isNatIList_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( nil ) = 2

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

   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)


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

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

   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(isNat(V1))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = 2x_1
POL( MARK_1(x_1) ) = 2
POL( s_1(x_1) ) = 2x_1
POL( length_1(x_1) ) = 1
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( mark_1(x_1) ) = x_1
POL( U11_2(x_1, x_2) ) = 1
POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 + 1
POL( zeros ) = 2
POL( 0 ) = 2
POL( tt ) = 0
POL( and_2(x_1, x_2) ) = x_2
POL( isNat_1(x_1) ) = 1
POL( isNatIList_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 0
POL( nil ) = 0

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

   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)


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

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

   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(isNat(V1))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(isNat(s(V1))) -> MARK(isNat(V1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2}
POL( MARK_1(x_1) ) = max{0, 2x_1 - 2}
POL( s_1(x_1) ) = x_1 + 1
POL( length_1(x_1) ) = 0
POL( active_1(x_1) ) = x_1
POL( mark_1(x_1) ) = x_1
POL( U11_2(x_1, x_2) ) = 1
POL( cons_2(x_1, x_2) ) = 1
POL( zeros ) = 1
POL( 0 ) = 1
POL( tt ) = 1
POL( and_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( isNat_1(x_1) ) = x_1 + 2
POL( isNatIList_1(x_1) ) = 2x_1 + 2
POL( isNatList_1(x_1) ) = 1
POL( nil ) = 0

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

   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)


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

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

   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(isNat(X)) -> ACTIVE(isNat(X))
The remaining pairs can at least be oriented weakly.
Used ordering:  Combined order from the following AFS and order.
ACTIVE(x1)  =  x1

U11(x1, x2)  =  U11

tt  =  tt

MARK(x1)  =  MARK

s(x1)  =  x1

length(x1)  =  length

mark(x1)  =  mark

isNat(x1)  =  isNat

cons(x1, x2)  =  cons(x2)

and(x1, x2)  =  and(x2)

isNatList(x1)  =  isNatList(x1)

active(x1)  =  active

zeros  =  zeros

0  =  0

isNatIList(x1)  =  x1

nil  =  nil


Recursive path order with status [RPO].
Quasi-Precedence: [U11, MARK, length] > [mark, isNat, cons_1, isNatList_1, active, zeros, 0, nil]
tt > [mark, isNat, cons_1, isNatList_1, active, zeros, 0, nil]
and_1 > [mark, isNat, cons_1, isNatList_1, active, zeros, 0, nil]

Status: U11: multiset status
tt: multiset status
MARK: multiset status
length: multiset status
mark: multiset status
isNat: multiset status
cons_1: multiset status
and_1: multiset status
isNatList_1: [1]
active: []
zeros: multiset status
0: multiset status
nil: multiset status


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

   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)


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

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

   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

(74) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2)) at position [0] we obtained the following new rules [LPAR04]:

   (MARK(U11(x0, x1)) -> ACTIVE(U11(x0, x1)),MARK(U11(x0, x1)) -> ACTIVE(U11(x0, x1)))
   (MARK(U11(y0, mark(x1))) -> ACTIVE(U11(mark(y0), x1)),MARK(U11(y0, mark(x1))) -> ACTIVE(U11(mark(y0), x1)))
   (MARK(U11(y0, active(x1))) -> ACTIVE(U11(mark(y0), x1)),MARK(U11(y0, active(x1))) -> ACTIVE(U11(mark(y0), x1)))
   (MARK(U11(zeros, y1)) -> ACTIVE(U11(active(zeros), y1)),MARK(U11(zeros, y1)) -> ACTIVE(U11(active(zeros), y1)))
   (MARK(U11(cons(x0, x1), y1)) -> ACTIVE(U11(active(cons(mark(x0), x1)), y1)),MARK(U11(cons(x0, x1), y1)) -> ACTIVE(U11(active(cons(mark(x0), x1)), y1)))
   (MARK(U11(0, y1)) -> ACTIVE(U11(active(0), y1)),MARK(U11(0, y1)) -> ACTIVE(U11(active(0), y1)))
   (MARK(U11(U11(x0, x1), y1)) -> ACTIVE(U11(active(U11(mark(x0), x1)), y1)),MARK(U11(U11(x0, x1), y1)) -> ACTIVE(U11(active(U11(mark(x0), x1)), y1)))
   (MARK(U11(tt, y1)) -> ACTIVE(U11(active(tt), y1)),MARK(U11(tt, y1)) -> ACTIVE(U11(active(tt), y1)))
   (MARK(U11(s(x0), y1)) -> ACTIVE(U11(active(s(mark(x0))), y1)),MARK(U11(s(x0), y1)) -> ACTIVE(U11(active(s(mark(x0))), y1)))
   (MARK(U11(length(x0), y1)) -> ACTIVE(U11(active(length(mark(x0))), y1)),MARK(U11(length(x0), y1)) -> ACTIVE(U11(active(length(mark(x0))), y1)))
   (MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1)),MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1)))
   (MARK(U11(isNat(x0), y1)) -> ACTIVE(U11(active(isNat(x0)), y1)),MARK(U11(isNat(x0), y1)) -> ACTIVE(U11(active(isNat(x0)), y1)))
   (MARK(U11(isNatList(x0), y1)) -> ACTIVE(U11(active(isNatList(x0)), y1)),MARK(U11(isNatList(x0), y1)) -> ACTIVE(U11(active(isNatList(x0)), y1)))
   (MARK(U11(isNatIList(x0), y1)) -> ACTIVE(U11(active(isNatIList(x0)), y1)),MARK(U11(isNatIList(x0), y1)) -> ACTIVE(U11(active(isNatIList(x0)), y1)))
   (MARK(U11(nil, y1)) -> ACTIVE(U11(active(nil), y1)),MARK(U11(nil, y1)) -> ACTIVE(U11(active(nil), y1)))


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

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

   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(U11(x0, x1)) -> ACTIVE(U11(x0, x1))
   MARK(U11(y0, mark(x1))) -> ACTIVE(U11(mark(y0), x1))
   MARK(U11(y0, active(x1))) -> ACTIVE(U11(mark(y0), x1))
   MARK(U11(zeros, y1)) -> ACTIVE(U11(active(zeros), y1))
   MARK(U11(cons(x0, x1), y1)) -> ACTIVE(U11(active(cons(mark(x0), x1)), y1))
   MARK(U11(0, y1)) -> ACTIVE(U11(active(0), y1))
   MARK(U11(U11(x0, x1), y1)) -> ACTIVE(U11(active(U11(mark(x0), x1)), y1))
   MARK(U11(tt, y1)) -> ACTIVE(U11(active(tt), y1))
   MARK(U11(s(x0), y1)) -> ACTIVE(U11(active(s(mark(x0))), y1))
   MARK(U11(length(x0), y1)) -> ACTIVE(U11(active(length(mark(x0))), y1))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
   MARK(U11(isNat(x0), y1)) -> ACTIVE(U11(active(isNat(x0)), y1))
   MARK(U11(isNatList(x0), y1)) -> ACTIVE(U11(active(isNatList(x0)), y1))
   MARK(U11(isNatIList(x0), y1)) -> ACTIVE(U11(active(isNatIList(x0)), y1))
   MARK(U11(nil, y1)) -> ACTIVE(U11(active(nil), y1))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

(76) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule MARK(length(X)) -> ACTIVE(length(mark(X))) at position [0] we obtained the following new rules [LPAR04]:

   (MARK(length(x0)) -> ACTIVE(length(x0)),MARK(length(x0)) -> ACTIVE(length(x0)))
   (MARK(length(zeros)) -> ACTIVE(length(active(zeros))),MARK(length(zeros)) -> ACTIVE(length(active(zeros))))
   (MARK(length(cons(x0, x1))) -> ACTIVE(length(active(cons(mark(x0), x1)))),MARK(length(cons(x0, x1))) -> ACTIVE(length(active(cons(mark(x0), x1)))))
   (MARK(length(0)) -> ACTIVE(length(active(0))),MARK(length(0)) -> ACTIVE(length(active(0))))
   (MARK(length(U11(x0, x1))) -> ACTIVE(length(active(U11(mark(x0), x1)))),MARK(length(U11(x0, x1))) -> ACTIVE(length(active(U11(mark(x0), x1)))))
   (MARK(length(tt)) -> ACTIVE(length(active(tt))),MARK(length(tt)) -> ACTIVE(length(active(tt))))
   (MARK(length(s(x0))) -> ACTIVE(length(active(s(mark(x0))))),MARK(length(s(x0))) -> ACTIVE(length(active(s(mark(x0))))))
   (MARK(length(length(x0))) -> ACTIVE(length(active(length(mark(x0))))),MARK(length(length(x0))) -> ACTIVE(length(active(length(mark(x0))))))
   (MARK(length(and(x0, x1))) -> ACTIVE(length(active(and(mark(x0), x1)))),MARK(length(and(x0, x1))) -> ACTIVE(length(active(and(mark(x0), x1)))))
   (MARK(length(isNat(x0))) -> ACTIVE(length(active(isNat(x0)))),MARK(length(isNat(x0))) -> ACTIVE(length(active(isNat(x0)))))
   (MARK(length(isNatList(x0))) -> ACTIVE(length(active(isNatList(x0)))),MARK(length(isNatList(x0))) -> ACTIVE(length(active(isNatList(x0)))))
   (MARK(length(isNatIList(x0))) -> ACTIVE(length(active(isNatIList(x0)))),MARK(length(isNatIList(x0))) -> ACTIVE(length(active(isNatIList(x0)))))
   (MARK(length(nil)) -> ACTIVE(length(active(nil))),MARK(length(nil)) -> ACTIVE(length(active(nil))))


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

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

   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(U11(x0, x1)) -> ACTIVE(U11(x0, x1))
   MARK(U11(y0, mark(x1))) -> ACTIVE(U11(mark(y0), x1))
   MARK(U11(y0, active(x1))) -> ACTIVE(U11(mark(y0), x1))
   MARK(U11(zeros, y1)) -> ACTIVE(U11(active(zeros), y1))
   MARK(U11(cons(x0, x1), y1)) -> ACTIVE(U11(active(cons(mark(x0), x1)), y1))
   MARK(U11(0, y1)) -> ACTIVE(U11(active(0), y1))
   MARK(U11(U11(x0, x1), y1)) -> ACTIVE(U11(active(U11(mark(x0), x1)), y1))
   MARK(U11(tt, y1)) -> ACTIVE(U11(active(tt), y1))
   MARK(U11(s(x0), y1)) -> ACTIVE(U11(active(s(mark(x0))), y1))
   MARK(U11(length(x0), y1)) -> ACTIVE(U11(active(length(mark(x0))), y1))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
   MARK(U11(isNat(x0), y1)) -> ACTIVE(U11(active(isNat(x0)), y1))
   MARK(U11(isNatList(x0), y1)) -> ACTIVE(U11(active(isNatList(x0)), y1))
   MARK(U11(isNatIList(x0), y1)) -> ACTIVE(U11(active(isNatIList(x0)), y1))
   MARK(U11(nil, y1)) -> ACTIVE(U11(active(nil), y1))
   MARK(length(x0)) -> ACTIVE(length(x0))
   MARK(length(zeros)) -> ACTIVE(length(active(zeros)))
   MARK(length(cons(x0, x1))) -> ACTIVE(length(active(cons(mark(x0), x1))))
   MARK(length(0)) -> ACTIVE(length(active(0)))
   MARK(length(U11(x0, x1))) -> ACTIVE(length(active(U11(mark(x0), x1))))
   MARK(length(tt)) -> ACTIVE(length(active(tt)))
   MARK(length(s(x0))) -> ACTIVE(length(active(s(mark(x0)))))
   MARK(length(length(x0))) -> ACTIVE(length(active(length(mark(x0)))))
   MARK(length(and(x0, x1))) -> ACTIVE(length(active(and(mark(x0), x1))))
   MARK(length(isNat(x0))) -> ACTIVE(length(active(isNat(x0))))
   MARK(length(isNatList(x0))) -> ACTIVE(length(active(isNatList(x0))))
   MARK(length(isNatIList(x0))) -> ACTIVE(length(active(isNatIList(x0))))
   MARK(length(nil)) -> ACTIVE(length(active(nil)))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

(78) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule MARK(length(0)) -> ACTIVE(length(active(0))) at position [0] we obtained the following new rules [LPAR04]:

   (MARK(length(0)) -> ACTIVE(length(0)),MARK(length(0)) -> ACTIVE(length(0)))


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

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

   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(U11(x0, x1)) -> ACTIVE(U11(x0, x1))
   MARK(U11(y0, mark(x1))) -> ACTIVE(U11(mark(y0), x1))
   MARK(U11(y0, active(x1))) -> ACTIVE(U11(mark(y0), x1))
   MARK(U11(zeros, y1)) -> ACTIVE(U11(active(zeros), y1))
   MARK(U11(cons(x0, x1), y1)) -> ACTIVE(U11(active(cons(mark(x0), x1)), y1))
   MARK(U11(0, y1)) -> ACTIVE(U11(active(0), y1))
   MARK(U11(U11(x0, x1), y1)) -> ACTIVE(U11(active(U11(mark(x0), x1)), y1))
   MARK(U11(tt, y1)) -> ACTIVE(U11(active(tt), y1))
   MARK(U11(s(x0), y1)) -> ACTIVE(U11(active(s(mark(x0))), y1))
   MARK(U11(length(x0), y1)) -> ACTIVE(U11(active(length(mark(x0))), y1))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
   MARK(U11(isNat(x0), y1)) -> ACTIVE(U11(active(isNat(x0)), y1))
   MARK(U11(isNatList(x0), y1)) -> ACTIVE(U11(active(isNatList(x0)), y1))
   MARK(U11(isNatIList(x0), y1)) -> ACTIVE(U11(active(isNatIList(x0)), y1))
   MARK(U11(nil, y1)) -> ACTIVE(U11(active(nil), y1))
   MARK(length(x0)) -> ACTIVE(length(x0))
   MARK(length(zeros)) -> ACTIVE(length(active(zeros)))
   MARK(length(cons(x0, x1))) -> ACTIVE(length(active(cons(mark(x0), x1))))
   MARK(length(U11(x0, x1))) -> ACTIVE(length(active(U11(mark(x0), x1))))
   MARK(length(tt)) -> ACTIVE(length(active(tt)))
   MARK(length(s(x0))) -> ACTIVE(length(active(s(mark(x0)))))
   MARK(length(length(x0))) -> ACTIVE(length(active(length(mark(x0)))))
   MARK(length(and(x0, x1))) -> ACTIVE(length(active(and(mark(x0), x1))))
   MARK(length(isNat(x0))) -> ACTIVE(length(active(isNat(x0))))
   MARK(length(isNatList(x0))) -> ACTIVE(length(active(isNatList(x0))))
   MARK(length(isNatIList(x0))) -> ACTIVE(length(active(isNatIList(x0))))
   MARK(length(nil)) -> ACTIVE(length(active(nil)))
   MARK(length(0)) -> ACTIVE(length(0))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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

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

   MARK(s(X)) -> MARK(X)
   MARK(U11(x0, x1)) -> ACTIVE(U11(x0, x1))
   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(y0, mark(x1))) -> ACTIVE(U11(mark(y0), x1))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(U11(y0, active(x1))) -> ACTIVE(U11(mark(y0), x1))
   MARK(U11(zeros, y1)) -> ACTIVE(U11(active(zeros), y1))
   MARK(U11(cons(x0, x1), y1)) -> ACTIVE(U11(active(cons(mark(x0), x1)), y1))
   MARK(U11(0, y1)) -> ACTIVE(U11(active(0), y1))
   MARK(U11(U11(x0, x1), y1)) -> ACTIVE(U11(active(U11(mark(x0), x1)), y1))
   MARK(U11(tt, y1)) -> ACTIVE(U11(active(tt), y1))
   MARK(U11(s(x0), y1)) -> ACTIVE(U11(active(s(mark(x0))), y1))
   MARK(U11(length(x0), y1)) -> ACTIVE(U11(active(length(mark(x0))), y1))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
   MARK(U11(isNat(x0), y1)) -> ACTIVE(U11(active(isNat(x0)), y1))
   MARK(U11(isNatList(x0), y1)) -> ACTIVE(U11(active(isNatList(x0)), y1))
   MARK(U11(isNatIList(x0), y1)) -> ACTIVE(U11(active(isNatIList(x0)), y1))
   MARK(U11(nil, y1)) -> ACTIVE(U11(active(nil), y1))
   MARK(length(x0)) -> ACTIVE(length(x0))
   MARK(length(zeros)) -> ACTIVE(length(active(zeros)))
   MARK(length(cons(x0, x1))) -> ACTIVE(length(active(cons(mark(x0), x1))))
   MARK(length(U11(x0, x1))) -> ACTIVE(length(active(U11(mark(x0), x1))))
   MARK(length(tt)) -> ACTIVE(length(active(tt)))
   MARK(length(s(x0))) -> ACTIVE(length(active(s(mark(x0)))))
   MARK(length(length(x0))) -> ACTIVE(length(active(length(mark(x0)))))
   MARK(length(and(x0, x1))) -> ACTIVE(length(active(and(mark(x0), x1))))
   MARK(length(isNat(x0))) -> ACTIVE(length(active(isNat(x0))))
   MARK(length(isNatList(x0))) -> ACTIVE(length(active(isNatList(x0))))
   MARK(length(isNatIList(x0))) -> ACTIVE(length(active(isNatIList(x0))))
   MARK(length(nil)) -> ACTIVE(length(active(nil)))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

(82) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule MARK(length(tt)) -> ACTIVE(length(active(tt))) at position [0] we obtained the following new rules [LPAR04]:

   (MARK(length(tt)) -> ACTIVE(length(tt)),MARK(length(tt)) -> ACTIVE(length(tt)))


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

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

   MARK(s(X)) -> MARK(X)
   MARK(U11(x0, x1)) -> ACTIVE(U11(x0, x1))
   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(y0, mark(x1))) -> ACTIVE(U11(mark(y0), x1))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(U11(y0, active(x1))) -> ACTIVE(U11(mark(y0), x1))
   MARK(U11(zeros, y1)) -> ACTIVE(U11(active(zeros), y1))
   MARK(U11(cons(x0, x1), y1)) -> ACTIVE(U11(active(cons(mark(x0), x1)), y1))
   MARK(U11(0, y1)) -> ACTIVE(U11(active(0), y1))
   MARK(U11(U11(x0, x1), y1)) -> ACTIVE(U11(active(U11(mark(x0), x1)), y1))
   MARK(U11(tt, y1)) -> ACTIVE(U11(active(tt), y1))
   MARK(U11(s(x0), y1)) -> ACTIVE(U11(active(s(mark(x0))), y1))
   MARK(U11(length(x0), y1)) -> ACTIVE(U11(active(length(mark(x0))), y1))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
   MARK(U11(isNat(x0), y1)) -> ACTIVE(U11(active(isNat(x0)), y1))
   MARK(U11(isNatList(x0), y1)) -> ACTIVE(U11(active(isNatList(x0)), y1))
   MARK(U11(isNatIList(x0), y1)) -> ACTIVE(U11(active(isNatIList(x0)), y1))
   MARK(U11(nil, y1)) -> ACTIVE(U11(active(nil), y1))
   MARK(length(x0)) -> ACTIVE(length(x0))
   MARK(length(zeros)) -> ACTIVE(length(active(zeros)))
   MARK(length(cons(x0, x1))) -> ACTIVE(length(active(cons(mark(x0), x1))))
   MARK(length(U11(x0, x1))) -> ACTIVE(length(active(U11(mark(x0), x1))))
   MARK(length(s(x0))) -> ACTIVE(length(active(s(mark(x0)))))
   MARK(length(length(x0))) -> ACTIVE(length(active(length(mark(x0)))))
   MARK(length(and(x0, x1))) -> ACTIVE(length(active(and(mark(x0), x1))))
   MARK(length(isNat(x0))) -> ACTIVE(length(active(isNat(x0))))
   MARK(length(isNatList(x0))) -> ACTIVE(length(active(isNatList(x0))))
   MARK(length(isNatIList(x0))) -> ACTIVE(length(active(isNatIList(x0))))
   MARK(length(nil)) -> ACTIVE(length(active(nil)))
   MARK(length(tt)) -> ACTIVE(length(tt))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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

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

   MARK(s(X)) -> MARK(X)
   MARK(U11(x0, x1)) -> ACTIVE(U11(x0, x1))
   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(y0, mark(x1))) -> ACTIVE(U11(mark(y0), x1))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(U11(y0, active(x1))) -> ACTIVE(U11(mark(y0), x1))
   MARK(U11(zeros, y1)) -> ACTIVE(U11(active(zeros), y1))
   MARK(U11(cons(x0, x1), y1)) -> ACTIVE(U11(active(cons(mark(x0), x1)), y1))
   MARK(U11(0, y1)) -> ACTIVE(U11(active(0), y1))
   MARK(U11(U11(x0, x1), y1)) -> ACTIVE(U11(active(U11(mark(x0), x1)), y1))
   MARK(U11(tt, y1)) -> ACTIVE(U11(active(tt), y1))
   MARK(U11(s(x0), y1)) -> ACTIVE(U11(active(s(mark(x0))), y1))
   MARK(U11(length(x0), y1)) -> ACTIVE(U11(active(length(mark(x0))), y1))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
   MARK(U11(isNat(x0), y1)) -> ACTIVE(U11(active(isNat(x0)), y1))
   MARK(U11(isNatList(x0), y1)) -> ACTIVE(U11(active(isNatList(x0)), y1))
   MARK(U11(isNatIList(x0), y1)) -> ACTIVE(U11(active(isNatIList(x0)), y1))
   MARK(U11(nil, y1)) -> ACTIVE(U11(active(nil), y1))
   MARK(length(x0)) -> ACTIVE(length(x0))
   MARK(length(zeros)) -> ACTIVE(length(active(zeros)))
   MARK(length(cons(x0, x1))) -> ACTIVE(length(active(cons(mark(x0), x1))))
   MARK(length(U11(x0, x1))) -> ACTIVE(length(active(U11(mark(x0), x1))))
   MARK(length(s(x0))) -> ACTIVE(length(active(s(mark(x0)))))
   MARK(length(length(x0))) -> ACTIVE(length(active(length(mark(x0)))))
   MARK(length(and(x0, x1))) -> ACTIVE(length(active(and(mark(x0), x1))))
   MARK(length(isNat(x0))) -> ACTIVE(length(active(isNat(x0))))
   MARK(length(isNatList(x0))) -> ACTIVE(length(active(isNatList(x0))))
   MARK(length(isNatIList(x0))) -> ACTIVE(length(active(isNatIList(x0))))
   MARK(length(nil)) -> ACTIVE(length(active(nil)))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

(86) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule MARK(length(nil)) -> ACTIVE(length(active(nil))) at position [0] we obtained the following new rules [LPAR04]:

   (MARK(length(nil)) -> ACTIVE(length(nil)),MARK(length(nil)) -> ACTIVE(length(nil)))


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

(87)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(s(X)) -> MARK(X)
   MARK(U11(x0, x1)) -> ACTIVE(U11(x0, x1))
   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(y0, mark(x1))) -> ACTIVE(U11(mark(y0), x1))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(U11(y0, active(x1))) -> ACTIVE(U11(mark(y0), x1))
   MARK(U11(zeros, y1)) -> ACTIVE(U11(active(zeros), y1))
   MARK(U11(cons(x0, x1), y1)) -> ACTIVE(U11(active(cons(mark(x0), x1)), y1))
   MARK(U11(0, y1)) -> ACTIVE(U11(active(0), y1))
   MARK(U11(U11(x0, x1), y1)) -> ACTIVE(U11(active(U11(mark(x0), x1)), y1))
   MARK(U11(tt, y1)) -> ACTIVE(U11(active(tt), y1))
   MARK(U11(s(x0), y1)) -> ACTIVE(U11(active(s(mark(x0))), y1))
   MARK(U11(length(x0), y1)) -> ACTIVE(U11(active(length(mark(x0))), y1))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
   MARK(U11(isNat(x0), y1)) -> ACTIVE(U11(active(isNat(x0)), y1))
   MARK(U11(isNatList(x0), y1)) -> ACTIVE(U11(active(isNatList(x0)), y1))
   MARK(U11(isNatIList(x0), y1)) -> ACTIVE(U11(active(isNatIList(x0)), y1))
   MARK(U11(nil, y1)) -> ACTIVE(U11(active(nil), y1))
   MARK(length(x0)) -> ACTIVE(length(x0))
   MARK(length(zeros)) -> ACTIVE(length(active(zeros)))
   MARK(length(cons(x0, x1))) -> ACTIVE(length(active(cons(mark(x0), x1))))
   MARK(length(U11(x0, x1))) -> ACTIVE(length(active(U11(mark(x0), x1))))
   MARK(length(s(x0))) -> ACTIVE(length(active(s(mark(x0)))))
   MARK(length(length(x0))) -> ACTIVE(length(active(length(mark(x0)))))
   MARK(length(and(x0, x1))) -> ACTIVE(length(active(and(mark(x0), x1))))
   MARK(length(isNat(x0))) -> ACTIVE(length(active(isNat(x0))))
   MARK(length(isNatList(x0))) -> ACTIVE(length(active(isNatList(x0))))
   MARK(length(isNatIList(x0))) -> ACTIVE(length(active(isNatIList(x0))))
   MARK(length(nil)) -> ACTIVE(length(nil))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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

(89)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(s(X)) -> MARK(X)
   MARK(U11(x0, x1)) -> ACTIVE(U11(x0, x1))
   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(y0, mark(x1))) -> ACTIVE(U11(mark(y0), x1))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(U11(y0, active(x1))) -> ACTIVE(U11(mark(y0), x1))
   MARK(U11(zeros, y1)) -> ACTIVE(U11(active(zeros), y1))
   MARK(U11(cons(x0, x1), y1)) -> ACTIVE(U11(active(cons(mark(x0), x1)), y1))
   MARK(U11(0, y1)) -> ACTIVE(U11(active(0), y1))
   MARK(U11(U11(x0, x1), y1)) -> ACTIVE(U11(active(U11(mark(x0), x1)), y1))
   MARK(U11(tt, y1)) -> ACTIVE(U11(active(tt), y1))
   MARK(U11(s(x0), y1)) -> ACTIVE(U11(active(s(mark(x0))), y1))
   MARK(U11(length(x0), y1)) -> ACTIVE(U11(active(length(mark(x0))), y1))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
   MARK(U11(isNat(x0), y1)) -> ACTIVE(U11(active(isNat(x0)), y1))
   MARK(U11(isNatList(x0), y1)) -> ACTIVE(U11(active(isNatList(x0)), y1))
   MARK(U11(isNatIList(x0), y1)) -> ACTIVE(U11(active(isNatIList(x0)), y1))
   MARK(U11(nil, y1)) -> ACTIVE(U11(active(nil), y1))
   MARK(length(x0)) -> ACTIVE(length(x0))
   MARK(length(zeros)) -> ACTIVE(length(active(zeros)))
   MARK(length(cons(x0, x1))) -> ACTIVE(length(active(cons(mark(x0), x1))))
   MARK(length(U11(x0, x1))) -> ACTIVE(length(active(U11(mark(x0), x1))))
   MARK(length(s(x0))) -> ACTIVE(length(active(s(mark(x0)))))
   MARK(length(length(x0))) -> ACTIVE(length(active(length(mark(x0)))))
   MARK(length(and(x0, x1))) -> ACTIVE(length(active(and(mark(x0), x1))))
   MARK(length(isNat(x0))) -> ACTIVE(length(active(isNat(x0))))
   MARK(length(isNatList(x0))) -> ACTIVE(length(active(isNatList(x0))))
   MARK(length(isNatIList(x0))) -> ACTIVE(length(active(isNatIList(x0))))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U11(U11(x0, x1), y1)) -> ACTIVE(U11(active(U11(mark(x0), x1)), y1))
   MARK(U11(length(x0), y1)) -> ACTIVE(U11(active(length(mark(x0))), y1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = x_1 + 1
POL( MARK_1(x_1) ) = 2x_1
POL( U11_2(x_1, x_2) ) = x_1 + 1
POL( mark_1(x_1) ) = x_1
POL( active_1(x_1) ) = x_1
POL( s_1(x_1) ) = x_1
POL( length_1(x_1) ) = 1
POL( cons_2(x_1, x_2) ) = 0
POL( zeros ) = 0
POL( 0 ) = 0
POL( tt ) = 0
POL( and_2(x_1, x_2) ) = x_2
POL( isNat_1(x_1) ) = 0
POL( isNatIList_1(x_1) ) = 0
POL( isNatList_1(x_1) ) = 0
POL( nil ) = 0

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

   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   mark(zeros) -> active(zeros)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(and(tt, X)) -> mark(X)
   mark(s(X)) -> active(s(mark(X)))
   active(isNat(s(V1))) -> mark(isNat(V1))
   mark(length(X)) -> active(length(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   mark(isNat(X)) -> active(isNat(X))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(nil) -> active(nil)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   and(X1, mark(X2)) -> and(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   active(isNat(0)) -> mark(tt)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)


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

(91)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(s(X)) -> MARK(X)
   MARK(U11(x0, x1)) -> ACTIVE(U11(x0, x1))
   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(y0, mark(x1))) -> ACTIVE(U11(mark(y0), x1))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(U11(y0, active(x1))) -> ACTIVE(U11(mark(y0), x1))
   MARK(U11(zeros, y1)) -> ACTIVE(U11(active(zeros), y1))
   MARK(U11(cons(x0, x1), y1)) -> ACTIVE(U11(active(cons(mark(x0), x1)), y1))
   MARK(U11(0, y1)) -> ACTIVE(U11(active(0), y1))
   MARK(U11(tt, y1)) -> ACTIVE(U11(active(tt), y1))
   MARK(U11(s(x0), y1)) -> ACTIVE(U11(active(s(mark(x0))), y1))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
   MARK(U11(isNat(x0), y1)) -> ACTIVE(U11(active(isNat(x0)), y1))
   MARK(U11(isNatList(x0), y1)) -> ACTIVE(U11(active(isNatList(x0)), y1))
   MARK(U11(isNatIList(x0), y1)) -> ACTIVE(U11(active(isNatIList(x0)), y1))
   MARK(U11(nil, y1)) -> ACTIVE(U11(active(nil), y1))
   MARK(length(x0)) -> ACTIVE(length(x0))
   MARK(length(zeros)) -> ACTIVE(length(active(zeros)))
   MARK(length(cons(x0, x1))) -> ACTIVE(length(active(cons(mark(x0), x1))))
   MARK(length(U11(x0, x1))) -> ACTIVE(length(active(U11(mark(x0), x1))))
   MARK(length(s(x0))) -> ACTIVE(length(active(s(mark(x0)))))
   MARK(length(length(x0))) -> ACTIVE(length(active(length(mark(x0)))))
   MARK(length(and(x0, x1))) -> ACTIVE(length(active(and(mark(x0), x1))))
   MARK(length(isNat(x0))) -> ACTIVE(length(active(isNat(x0))))
   MARK(length(isNatList(x0))) -> ACTIVE(length(active(isNatList(x0))))
   MARK(length(isNatIList(x0))) -> ACTIVE(length(active(isNatIList(x0))))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U11(s(x0), y1)) -> ACTIVE(U11(active(s(mark(x0))), y1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = 0
POL( MARK_1(x_1) ) = max{0, 2x_1 - 2}
POL( U11_2(x_1, x_2) ) = 2x_1 + 1
POL( mark_1(x_1) ) = x_1
POL( active_1(x_1) ) = 2x_1 + 1
POL( s_1(x_1) ) = 2x_1 + 1
POL( length_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = 0
POL( zeros ) = 0
POL( 0 ) = 0
POL( tt ) = 0
POL( and_2(x_1, x_2) ) = 0
POL( isNat_1(x_1) ) = 0
POL( isNatIList_1(x_1) ) = 0
POL( isNatList_1(x_1) ) = 0
POL( nil ) = 0

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

   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   and(X1, mark(X2)) -> and(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)


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

(93)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(s(X)) -> MARK(X)
   MARK(U11(x0, x1)) -> ACTIVE(U11(x0, x1))
   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(y0, mark(x1))) -> ACTIVE(U11(mark(y0), x1))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(U11(y0, active(x1))) -> ACTIVE(U11(mark(y0), x1))
   MARK(U11(zeros, y1)) -> ACTIVE(U11(active(zeros), y1))
   MARK(U11(cons(x0, x1), y1)) -> ACTIVE(U11(active(cons(mark(x0), x1)), y1))
   MARK(U11(0, y1)) -> ACTIVE(U11(active(0), y1))
   MARK(U11(tt, y1)) -> ACTIVE(U11(active(tt), y1))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
   MARK(U11(isNat(x0), y1)) -> ACTIVE(U11(active(isNat(x0)), y1))
   MARK(U11(isNatList(x0), y1)) -> ACTIVE(U11(active(isNatList(x0)), y1))
   MARK(U11(isNatIList(x0), y1)) -> ACTIVE(U11(active(isNatIList(x0)), y1))
   MARK(U11(nil, y1)) -> ACTIVE(U11(active(nil), y1))
   MARK(length(x0)) -> ACTIVE(length(x0))
   MARK(length(zeros)) -> ACTIVE(length(active(zeros)))
   MARK(length(cons(x0, x1))) -> ACTIVE(length(active(cons(mark(x0), x1))))
   MARK(length(U11(x0, x1))) -> ACTIVE(length(active(U11(mark(x0), x1))))
   MARK(length(s(x0))) -> ACTIVE(length(active(s(mark(x0)))))
   MARK(length(length(x0))) -> ACTIVE(length(active(length(mark(x0)))))
   MARK(length(and(x0, x1))) -> ACTIVE(length(active(and(mark(x0), x1))))
   MARK(length(isNat(x0))) -> ACTIVE(length(active(isNat(x0))))
   MARK(length(isNatList(x0))) -> ACTIVE(length(active(isNatList(x0))))
   MARK(length(isNatIList(x0))) -> ACTIVE(length(active(isNatIList(x0))))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U11(zeros, y1)) -> ACTIVE(U11(active(zeros), y1))
   MARK(U11(cons(x0, x1), y1)) -> ACTIVE(U11(active(cons(mark(x0), x1)), y1))
   MARK(U11(0, y1)) -> ACTIVE(U11(active(0), y1))
   MARK(U11(tt, y1)) -> ACTIVE(U11(active(tt), y1))
   MARK(U11(isNatList(x0), y1)) -> ACTIVE(U11(active(isNatList(x0)), y1))
   MARK(U11(isNatIList(x0), y1)) -> ACTIVE(U11(active(isNatIList(x0)), y1))
   MARK(U11(nil, y1)) -> ACTIVE(U11(active(nil), y1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = 2
POL( MARK_1(x_1) ) = x_1 + 2
POL( U11_2(x_1, x_2) ) = x_1
POL( mark_1(x_1) ) = 2x_1 + 2
POL( active_1(x_1) ) = x_1
POL( s_1(x_1) ) = x_1
POL( length_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = x_2 + 1
POL( zeros ) = 1
POL( 0 ) = 1
POL( tt ) = 2
POL( and_2(x_1, x_2) ) = x_2
POL( isNat_1(x_1) ) = 0
POL( isNatIList_1(x_1) ) = 1
POL( isNatList_1(x_1) ) = x_1 + 2
POL( nil ) = 1

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

   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   and(X1, mark(X2)) -> and(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)


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

(95)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(s(X)) -> MARK(X)
   MARK(U11(x0, x1)) -> ACTIVE(U11(x0, x1))
   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(y0, mark(x1))) -> ACTIVE(U11(mark(y0), x1))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(U11(y0, active(x1))) -> ACTIVE(U11(mark(y0), x1))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
   MARK(U11(isNat(x0), y1)) -> ACTIVE(U11(active(isNat(x0)), y1))
   MARK(length(x0)) -> ACTIVE(length(x0))
   MARK(length(zeros)) -> ACTIVE(length(active(zeros)))
   MARK(length(cons(x0, x1))) -> ACTIVE(length(active(cons(mark(x0), x1))))
   MARK(length(U11(x0, x1))) -> ACTIVE(length(active(U11(mark(x0), x1))))
   MARK(length(s(x0))) -> ACTIVE(length(active(s(mark(x0)))))
   MARK(length(length(x0))) -> ACTIVE(length(active(length(mark(x0)))))
   MARK(length(and(x0, x1))) -> ACTIVE(length(active(and(mark(x0), x1))))
   MARK(length(isNat(x0))) -> ACTIVE(length(active(isNat(x0))))
   MARK(length(isNatList(x0))) -> ACTIVE(length(active(isNatList(x0))))
   MARK(length(isNatIList(x0))) -> ACTIVE(length(active(isNatIList(x0))))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U11(isNat(x0), y1)) -> ACTIVE(U11(active(isNat(x0)), y1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = 1
POL( MARK_1(x_1) ) = x_1 + 1
POL( U11_2(x_1, x_2) ) = 2x_1
POL( mark_1(x_1) ) = 2x_1 + 1
POL( active_1(x_1) ) = 2x_1
POL( s_1(x_1) ) = 2x_1
POL( length_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = x_2
POL( zeros ) = 1
POL( 0 ) = 1
POL( tt ) = 1
POL( and_2(x_1, x_2) ) = 0
POL( isNat_1(x_1) ) = 1
POL( isNatIList_1(x_1) ) = 0
POL( isNatList_1(x_1) ) = 2
POL( nil ) = 0

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

   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   and(X1, mark(X2)) -> and(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)


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

(97)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(s(X)) -> MARK(X)
   MARK(U11(x0, x1)) -> ACTIVE(U11(x0, x1))
   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(y0, mark(x1))) -> ACTIVE(U11(mark(y0), x1))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(U11(y0, active(x1))) -> ACTIVE(U11(mark(y0), x1))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
   MARK(length(x0)) -> ACTIVE(length(x0))
   MARK(length(zeros)) -> ACTIVE(length(active(zeros)))
   MARK(length(cons(x0, x1))) -> ACTIVE(length(active(cons(mark(x0), x1))))
   MARK(length(U11(x0, x1))) -> ACTIVE(length(active(U11(mark(x0), x1))))
   MARK(length(s(x0))) -> ACTIVE(length(active(s(mark(x0)))))
   MARK(length(length(x0))) -> ACTIVE(length(active(length(mark(x0)))))
   MARK(length(and(x0, x1))) -> ACTIVE(length(active(and(mark(x0), x1))))
   MARK(length(isNat(x0))) -> ACTIVE(length(active(isNat(x0))))
   MARK(length(isNatList(x0))) -> ACTIVE(length(active(isNatList(x0))))
   MARK(length(isNatIList(x0))) -> ACTIVE(length(active(isNatIList(x0))))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(length(isNat(x0))) -> ACTIVE(length(active(isNat(x0))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

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

   <<<
 POL(U11(x_1, x_2)) =  	[[5A]] 	 +  	[[3A]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2
>>>

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

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

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

   <<<
 POL(mark(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[5A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(and(x_1, x_2)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

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

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

   <<<
 POL(active(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

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

   <<<
 POL(0) =  	[[3A]]
>>>

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


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

   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   mark(zeros) -> active(zeros)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(and(tt, X)) -> mark(X)
   mark(s(X)) -> active(s(mark(X)))
   active(isNat(s(V1))) -> mark(isNat(V1))
   mark(length(X)) -> active(length(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   mark(isNat(X)) -> active(isNat(X))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(nil) -> active(nil)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   and(X1, mark(X2)) -> and(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   active(isNat(0)) -> mark(tt)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)


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

(99)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(s(X)) -> MARK(X)
   MARK(U11(x0, x1)) -> ACTIVE(U11(x0, x1))
   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(y0, mark(x1))) -> ACTIVE(U11(mark(y0), x1))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(U11(y0, active(x1))) -> ACTIVE(U11(mark(y0), x1))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
   MARK(length(x0)) -> ACTIVE(length(x0))
   MARK(length(zeros)) -> ACTIVE(length(active(zeros)))
   MARK(length(cons(x0, x1))) -> ACTIVE(length(active(cons(mark(x0), x1))))
   MARK(length(U11(x0, x1))) -> ACTIVE(length(active(U11(mark(x0), x1))))
   MARK(length(s(x0))) -> ACTIVE(length(active(s(mark(x0)))))
   MARK(length(length(x0))) -> ACTIVE(length(active(length(mark(x0)))))
   MARK(length(and(x0, x1))) -> ACTIVE(length(active(and(mark(x0), x1))))
   MARK(length(isNatList(x0))) -> ACTIVE(length(active(isNatList(x0))))
   MARK(length(isNatIList(x0))) -> ACTIVE(length(active(isNatIList(x0))))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(length(isNatIList(x0))) -> ACTIVE(length(active(isNatIList(x0))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(MARK(x_1)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U11(x_1, x_2)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

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

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

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

   <<<
 POL(mark(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

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

   <<<
 POL(isNatList(x_1)) =  	[[3A]] 	 +  	[[4A]] 	* 	x_1
>>>

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

   <<<
 POL(active(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

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

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

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


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

   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   mark(zeros) -> active(zeros)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(and(tt, X)) -> mark(X)
   mark(s(X)) -> active(s(mark(X)))
   active(isNat(s(V1))) -> mark(isNat(V1))
   mark(length(X)) -> active(length(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   mark(isNat(X)) -> active(isNat(X))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(nil) -> active(nil)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   and(X1, mark(X2)) -> and(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   active(isNat(0)) -> mark(tt)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)


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

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

   MARK(s(X)) -> MARK(X)
   MARK(U11(x0, x1)) -> ACTIVE(U11(x0, x1))
   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(y0, mark(x1))) -> ACTIVE(U11(mark(y0), x1))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(U11(y0, active(x1))) -> ACTIVE(U11(mark(y0), x1))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
   MARK(length(x0)) -> ACTIVE(length(x0))
   MARK(length(zeros)) -> ACTIVE(length(active(zeros)))
   MARK(length(cons(x0, x1))) -> ACTIVE(length(active(cons(mark(x0), x1))))
   MARK(length(U11(x0, x1))) -> ACTIVE(length(active(U11(mark(x0), x1))))
   MARK(length(s(x0))) -> ACTIVE(length(active(s(mark(x0)))))
   MARK(length(length(x0))) -> ACTIVE(length(active(length(mark(x0)))))
   MARK(length(and(x0, x1))) -> ACTIVE(length(active(and(mark(x0), x1))))
   MARK(length(isNatList(x0))) -> ACTIVE(length(active(isNatList(x0))))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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.

   MARK(length(isNatList(x0))) -> ACTIVE(length(active(isNatList(x0))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

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

   <<<
 POL(U11(x_1, x_2)) =  	[[3A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

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

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

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

   <<<
 POL(mark(x_1)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[3A]] 	 +  	[[2A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

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

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

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

   <<<
 POL(active(x_1)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1
>>>

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

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

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

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


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

   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   mark(zeros) -> active(zeros)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(and(tt, X)) -> mark(X)
   mark(s(X)) -> active(s(mark(X)))
   active(isNat(s(V1))) -> mark(isNat(V1))
   mark(length(X)) -> active(length(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   mark(isNat(X)) -> active(isNat(X))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(nil) -> active(nil)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   and(X1, mark(X2)) -> and(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   active(isNat(0)) -> mark(tt)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)


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

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

   MARK(s(X)) -> MARK(X)
   MARK(U11(x0, x1)) -> ACTIVE(U11(x0, x1))
   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(y0, mark(x1))) -> ACTIVE(U11(mark(y0), x1))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(U11(y0, active(x1))) -> ACTIVE(U11(mark(y0), x1))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
   MARK(length(x0)) -> ACTIVE(length(x0))
   MARK(length(zeros)) -> ACTIVE(length(active(zeros)))
   MARK(length(cons(x0, x1))) -> ACTIVE(length(active(cons(mark(x0), x1))))
   MARK(length(U11(x0, x1))) -> ACTIVE(length(active(U11(mark(x0), x1))))
   MARK(length(s(x0))) -> ACTIVE(length(active(s(mark(x0)))))
   MARK(length(length(x0))) -> ACTIVE(length(active(length(mark(x0)))))
   MARK(length(and(x0, x1))) -> ACTIVE(length(active(and(mark(x0), x1))))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U11(y0, mark(x1))) -> ACTIVE(U11(mark(y0), x1))
   MARK(U11(y0, active(x1))) -> ACTIVE(U11(mark(y0), x1))
   MARK(length(U11(x0, x1))) -> ACTIVE(length(active(U11(mark(x0), x1))))
   MARK(length(length(x0))) -> ACTIVE(length(active(length(mark(x0)))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U11(x_1, x_2)) =  	[[3A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[3A]] 	* 	x_2
>>>

   <<<
 POL(ACTIVE(x_1)) =  	[[5A]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(length(x_1)) =  	[[4A]] 	 +  	[[2A]] 	* 	x_1
>>>

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

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

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

   <<<
 POL(isNatList(x_1)) =  	[[3A]] 	 +  	[[2A]] 	* 	x_1
>>>

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

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

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

   <<<
 POL(0) =  	[[1A]]
>>>

   <<<
 POL(isNatIList(x_1)) =  	[[5A]] 	 +  	[[3A]] 	* 	x_1
>>>

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


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

   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   mark(zeros) -> active(zeros)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(and(tt, X)) -> mark(X)
   mark(s(X)) -> active(s(mark(X)))
   active(isNat(s(V1))) -> mark(isNat(V1))
   mark(length(X)) -> active(length(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   mark(isNat(X)) -> active(isNat(X))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(nil) -> active(nil)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   and(X1, mark(X2)) -> and(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   active(isNat(0)) -> mark(tt)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)


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

(105)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(s(X)) -> MARK(X)
   MARK(U11(x0, x1)) -> ACTIVE(U11(x0, x1))
   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
   MARK(length(x0)) -> ACTIVE(length(x0))
   MARK(length(zeros)) -> ACTIVE(length(active(zeros)))
   MARK(length(cons(x0, x1))) -> ACTIVE(length(active(cons(mark(x0), x1))))
   MARK(length(s(x0))) -> ACTIVE(length(active(s(mark(x0)))))
   MARK(length(and(x0, x1))) -> ACTIVE(length(active(and(mark(x0), x1))))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(length(cons(x0, x1))) -> ACTIVE(length(active(cons(mark(x0), x1))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

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

   <<<
 POL(U11(x_1, x_2)) =  	[[4A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(ACTIVE(x_1)) =  	[[5A]] 	 +  	[[0A]] 	* 	x_1
>>>

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

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

   <<<
 POL(cons(x_1, x_2)) =  	[[5A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[5A]] 	* 	x_2
>>>

   <<<
 POL(and(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[4A]] 	* 	x_2
>>>

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

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

   <<<
 POL(active(x_1)) =  	[[5A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(mark(x_1)) =  	[[5A]] 	 +  	[[0A]] 	* 	x_1
>>>

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

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

   <<<
 POL(isNatIList(x_1)) =  	[[2A]] 	 +  	[[1A]] 	* 	x_1
>>>

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


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

   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   and(X1, mark(X2)) -> and(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   mark(zeros) -> active(zeros)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(and(tt, X)) -> mark(X)
   mark(s(X)) -> active(s(mark(X)))
   active(isNat(s(V1))) -> mark(isNat(V1))
   mark(length(X)) -> active(length(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   mark(isNat(X)) -> active(isNat(X))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(nil) -> active(nil)
   active(isNat(0)) -> mark(tt)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)


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

(107)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(s(X)) -> MARK(X)
   MARK(U11(x0, x1)) -> ACTIVE(U11(x0, x1))
   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
   MARK(length(x0)) -> ACTIVE(length(x0))
   MARK(length(zeros)) -> ACTIVE(length(active(zeros)))
   MARK(length(s(x0))) -> ACTIVE(length(active(s(mark(x0)))))
   MARK(length(and(x0, x1))) -> ACTIVE(length(active(and(mark(x0), x1))))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(length(and(x0, x1))) -> ACTIVE(length(active(and(mark(x0), x1))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

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

   <<<
 POL(U11(x_1, x_2)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

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

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

   <<<
 POL(length(x_1)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(and(x_1, x_2)) =  	[[3A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

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

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

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

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

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

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

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

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


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

   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   and(X1, mark(X2)) -> and(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   mark(zeros) -> active(zeros)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(and(tt, X)) -> mark(X)
   mark(s(X)) -> active(s(mark(X)))
   active(isNat(s(V1))) -> mark(isNat(V1))
   mark(length(X)) -> active(length(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   mark(isNat(X)) -> active(isNat(X))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(nil) -> active(nil)
   active(isNat(0)) -> mark(tt)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)


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

(109)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(s(X)) -> MARK(X)
   MARK(U11(x0, x1)) -> ACTIVE(U11(x0, x1))
   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
   MARK(length(x0)) -> ACTIVE(length(x0))
   MARK(length(zeros)) -> ACTIVE(length(active(zeros)))
   MARK(length(s(x0))) -> ACTIVE(length(active(s(mark(x0)))))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U11(x0, x1)) -> ACTIVE(U11(x0, x1))
   MARK(length(x0)) -> ACTIVE(length(x0))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

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

   <<<
 POL(U11(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2
>>>

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

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

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

   <<<
 POL(cons(x_1, x_2)) =  	[[5A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[4A]] 	* 	x_2
>>>

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

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

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

   <<<
 POL(active(x_1)) =  	[[5A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(mark(x_1)) =  	[[5A]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(0) =  	[[3A]]
>>>

   <<<
 POL(isNatIList(x_1)) =  	[[0A]] 	 +  	[[5A]] 	* 	x_1
>>>

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


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

   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   and(X1, mark(X2)) -> and(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   mark(zeros) -> active(zeros)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(and(tt, X)) -> mark(X)
   mark(s(X)) -> active(s(mark(X)))
   active(isNat(s(V1))) -> mark(isNat(V1))
   mark(length(X)) -> active(length(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   mark(isNat(X)) -> active(isNat(X))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(nil) -> active(nil)
   active(isNat(0)) -> mark(tt)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)


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

(111)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(s(X)) -> MARK(X)
   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
   MARK(length(zeros)) -> ACTIVE(length(active(zeros)))
   MARK(length(s(x0))) -> ACTIVE(length(active(s(mark(x0)))))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(length(s(x0))) -> ACTIVE(length(active(s(mark(x0)))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( MARK_1(x_1) ) = 1
POL( s_1(x_1) ) = 0
POL( length_1(x_1) ) = x_1 + 1
POL( active_1(x_1) ) = x_1
POL( mark_1(x_1) ) = x_1
POL( U11_2(x_1, x_2) ) = 2
POL( and_2(x_1, x_2) ) = 2x_1 + x_2
POL( isNatList_1(x_1) ) = 0
POL( isNat_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = 1
POL( zeros ) = 1
POL( 0 ) = 1
POL( tt ) = 0
POL( isNatIList_1(x_1) ) = 1
POL( nil ) = 0

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

   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   and(X1, mark(X2)) -> and(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   mark(zeros) -> active(zeros)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(and(tt, X)) -> mark(X)
   mark(s(X)) -> active(s(mark(X)))
   active(isNat(s(V1))) -> mark(isNat(V1))
   mark(length(X)) -> active(length(mark(X)))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   mark(isNat(X)) -> active(isNat(X))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(nil) -> active(nil)
   active(isNat(0)) -> mark(tt)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)


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

(113)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(s(X)) -> MARK(X)
   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
   MARK(length(zeros)) -> ACTIVE(length(active(zeros)))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

(114) NonMonReductionPairProof (EQUIVALENT)
Using the following max-polynomial ordering, we can orient the general usable rules and all rules from P weakly and some rules from P strictly:

Polynomial interpretation with max [POLO,NEGPOLO,MAXPOLO]:

   POL(0) = 1
   POL(ACTIVE(x_1)) = 1 + x_1
   POL(MARK(x_1)) = 1 + x_1
   POL(U11(x_1, x_2)) = max(0, 1 - x_1)
   POL(active(x_1)) = x_1
   POL(and(x_1, x_2)) = max(0, x_1 + x_2)
   POL(cons(x_1, x_2)) = max(0, 0)
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatList(x_1)) = 1
   POL(length(x_1)) = 0
   POL(mark(x_1)) = x_1
   POL(nil) = 1
   POL(s(x_1)) = 1 + x_1
   POL(tt) = 0
   POL(zeros) = 0


The following pairs can be oriented strictly and are deleted.

   MARK(s(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.

   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
   MARK(length(zeros)) -> ACTIVE(length(active(zeros)))
The following rules are usable:
   active(zeros) <-> mark(cons(0, zeros))
   active(U11(tt, L)) <-> mark(s(length(L)))
   active(and(tt, X)) <-> mark(X)
   active(isNat(0)) <-> mark(tt)
   active(isNat(s(V1))) <-> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) <-> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) <-> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) <-> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) <-> active(zeros)
   mark(cons(X1, X2)) <-> active(cons(mark(X1), X2))
   mark(0) <-> active(0)
   mark(U11(X1, X2)) <-> active(U11(mark(X1), X2))
   mark(tt) <-> active(tt)
   mark(s(X)) <-> active(s(mark(X)))
   mark(length(X)) <-> active(length(mark(X)))
   mark(and(X1, X2)) <-> active(and(mark(X1), X2))
   mark(isNat(X)) <-> active(isNat(X))
   mark(isNatList(X)) <-> active(isNatList(X))
   mark(isNatIList(X)) <-> active(isNatIList(X))
   mark(nil) <-> active(nil)
   cons(mark(X1), X2) <-> cons(X1, X2)
   cons(X1, mark(X2)) <-> cons(X1, X2)
   cons(active(X1), X2) <-> cons(X1, X2)
   cons(X1, active(X2)) <-> cons(X1, X2)
   U11(mark(X1), X2) <-> U11(X1, X2)
   U11(X1, mark(X2)) <-> U11(X1, X2)
   U11(active(X1), X2) <-> U11(X1, X2)
   U11(X1, active(X2)) <-> U11(X1, X2)
   s(mark(X)) <-> s(X)
   s(active(X)) <-> s(X)
   length(mark(X)) <-> length(X)
   length(active(X)) <-> length(X)
   and(mark(X1), X2) <-> and(X1, X2)
   and(X1, mark(X2)) <-> and(X1, X2)
   and(active(X1), X2) <-> and(X1, X2)
   and(X1, active(X2)) <-> and(X1, X2)
   isNat(mark(X)) <-> isNat(X)
   isNat(active(X)) <-> isNat(X)
   isNatList(mark(X)) <-> isNatList(X)
   isNatList(active(X)) <-> isNatList(X)
   isNatIList(mark(X)) <-> isNatIList(X)
   isNatIList(active(X)) <-> isNatIList(X)

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

(115)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
   MARK(length(zeros)) -> ACTIVE(length(active(zeros)))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(length(cons(N, L))) -> MARK(U11(and(isNatList(L), isNat(N)), L))
   MARK(length(zeros)) -> ACTIVE(length(active(zeros)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = x_1 + 2
POL( MARK_1(x_1) ) = 2x_1 + 2
POL( s_1(x_1) ) = max{0, -2}
POL( length_1(x_1) ) = 1
POL( active_1(x_1) ) = 2x_1 + 1
POL( mark_1(x_1) ) = 2x_1 + 1
POL( U11_2(x_1, x_2) ) = 0
POL( and_2(x_1, x_2) ) = max{0, x_1 + x_2 - 1}
POL( isNatList_1(x_1) ) = 0
POL( isNat_1(x_1) ) = 1
POL( cons_2(x_1, x_2) ) = 2x_2 + 1
POL( zeros ) = 2
POL( 0 ) = 0
POL( tt ) = 1
POL( isNatIList_1(x_1) ) = x_1 + 2
POL( nil ) = 0

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

   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)


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

(117)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U11(tt, L)) -> MARK(s(length(L)))
   MARK(U11(and(x0, x1), y1)) -> ACTIVE(U11(active(and(mark(x0), x1)), y1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE

MARK(x1)  =  x1

s(x1)  =  s

U11(x1, x2)  =  U11


Knuth-Bendix order [KBO] with precedence:trivial

and weight map:

   s=1
   ACTIVE=2
   U11=3

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

   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)


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

(119)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(nil) -> active(nil)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)

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

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

(121)
YES