/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 140 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) AND
    (5) QDP
        (6) UsableRulesProof [EQUIVALENT, 0 ms]
        (7) QDP
        (8) QReductionProof [EQUIVALENT, 0 ms]
        (9) QDP
        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) QDP
        (13) UsableRulesProof [EQUIVALENT, 0 ms]
        (14) QDP
        (15) QReductionProof [EQUIVALENT, 0 ms]
        (16) QDP
        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (18) YES
    (19) QDP
        (20) UsableRulesProof [EQUIVALENT, 0 ms]
        (21) QDP
        (22) QReductionProof [EQUIVALENT, 0 ms]
        (23) QDP
        (24) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (25) YES
    (26) QDP
        (27) UsableRulesProof [EQUIVALENT, 0 ms]
        (28) QDP
        (29) QReductionProof [EQUIVALENT, 0 ms]
        (30) QDP
        (31) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (32) YES
    (33) QDP
        (34) UsableRulesProof [EQUIVALENT, 0 ms]
        (35) QDP
        (36) QReductionProof [EQUIVALENT, 0 ms]
        (37) QDP
        (38) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (39) YES
    (40) QDP
        (41) UsableRulesProof [EQUIVALENT, 0 ms]
        (42) QDP
        (43) QReductionProof [EQUIVALENT, 0 ms]
        (44) QDP
        (45) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (46) YES
    (47) QDP
        (48) UsableRulesProof [EQUIVALENT, 0 ms]
        (49) QDP
        (50) QReductionProof [EQUIVALENT, 0 ms]
        (51) QDP
        (52) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (53) YES
    (54) QDP
        (55) UsableRulesProof [EQUIVALENT, 0 ms]
        (56) QDP
        (57) QReductionProof [EQUIVALENT, 0 ms]
        (58) QDP
        (59) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (60) YES
    (61) QDP
        (62) UsableRulesProof [EQUIVALENT, 0 ms]
        (63) QDP
        (64) QReductionProof [EQUIVALENT, 0 ms]
        (65) QDP
        (66) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (67) YES
    (68) QDP
        (69) UsableRulesProof [EQUIVALENT, 0 ms]
        (70) QDP
        (71) QReductionProof [EQUIVALENT, 0 ms]
        (72) QDP
        (73) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (74) YES
    (75) QDP
        (76) UsableRulesProof [EQUIVALENT, 0 ms]
        (77) QDP
        (78) QReductionProof [EQUIVALENT, 0 ms]
        (79) QDP
        (80) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (81) YES
    (82) QDP
        (83) UsableRulesProof [EQUIVALENT, 0 ms]
        (84) QDP
        (85) QReductionProof [EQUIVALENT, 0 ms]
        (86) QDP
        (87) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (88) YES
    (89) QDP
        (90) UsableRulesProof [EQUIVALENT, 0 ms]
        (91) QDP
        (92) QReductionProof [EQUIVALENT, 0 ms]
        (93) QDP
        (94) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (95) YES
    (96) QDP
        (97) UsableRulesProof [EQUIVALENT, 0 ms]
        (98) QDP
        (99) QReductionProof [EQUIVALENT, 0 ms]
        (100) QDP
        (101) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (102) YES
    (103) QDP
        (104) UsableRulesProof [EQUIVALENT, 0 ms]
        (105) QDP
        (106) QReductionProof [EQUIVALENT, 0 ms]
        (107) QDP
        (108) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (109) YES
    (110) QDP
        (111) UsableRulesProof [EQUIVALENT, 0 ms]
        (112) QDP
        (113) QDPOrderProof [EQUIVALENT, 457 ms]
        (114) QDP
        (115) QDPOrderProof [EQUIVALENT, 969 ms]
        (116) QDP
        (117) QDPOrderProof [EQUIVALENT, 876 ms]
        (118) QDP
        (119) DependencyGraphProof [EQUIVALENT, 0 ms]
        (120) QDP
        (121) QDPOrderProof [EQUIVALENT, 335 ms]
        (122) QDP
        (123) QDPQMonotonicMRRProof [EQUIVALENT, 140 ms]
        (124) QDP
        (125) QDPQMonotonicMRRProof [EQUIVALENT, 201 ms]
        (126) QDP
        (127) QDPOrderProof [EQUIVALENT, 629 ms]
        (128) QDP
        (129) QDPQMonotonicMRRProof [EQUIVALENT, 123 ms]
        (130) QDP
        (131) QDPOrderProof [EQUIVALENT, 234 ms]
        (132) QDP
        (133) QDPOrderProof [EQUIVALENT, 547 ms]
        (134) QDP
        (135) DependencyGraphProof [EQUIVALENT, 0 ms]
        (136) QDP
        (137) QDPOrderProof [EQUIVALENT, 487 ms]
        (138) QDP
        (139) DependencyGraphProof [EQUIVALENT, 0 ms]
        (140) QDP
        (141) UsableRulesProof [EQUIVALENT, 0 ms]
        (142) QDP
        (143) QReductionProof [EQUIVALENT, 0 ms]
        (144) QDP
        (145) UsableRulesReductionPairsProof [EQUIVALENT, 5 ms]
        (146) QDP
        (147) DependencyGraphProof [EQUIVALENT, 0 ms]
        (148) QDP
        (149) QReductionProof [EQUIVALENT, 0 ms]
        (150) QDP
        (151) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (152) YES


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

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

   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, N)) -> mark(N)
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   active(U61(tt)) -> mark(0)
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X)) -> active(U12(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X)) -> active(U32(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(0) -> active(0)
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(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)
   U12(mark(X)) -> U12(X)
   U12(active(X)) -> U12(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X)) -> U32(X)
   U32(active(X)) -> U32(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))


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

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

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

   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
   ACTIVE(U11(tt, V2)) -> U12^1(isNat(V2))
   ACTIVE(U11(tt, V2)) -> ISNAT(V2)
   ACTIVE(U12(tt)) -> MARK(tt)
   ACTIVE(U21(tt)) -> MARK(tt)
   ACTIVE(U31(tt, V2)) -> MARK(U32(isNat(V2)))
   ACTIVE(U31(tt, V2)) -> U32^1(isNat(V2))
   ACTIVE(U31(tt, V2)) -> ISNAT(V2)
   ACTIVE(U32(tt)) -> MARK(tt)
   ACTIVE(U41(tt, N)) -> MARK(N)
   ACTIVE(U51(tt, M, N)) -> MARK(U52(isNat(N), M, N))
   ACTIVE(U51(tt, M, N)) -> U52^1(isNat(N), M, N)
   ACTIVE(U51(tt, M, N)) -> ISNAT(N)
   ACTIVE(U52(tt, M, N)) -> MARK(s(plus(N, M)))
   ACTIVE(U52(tt, M, N)) -> S(plus(N, M))
   ACTIVE(U52(tt, M, N)) -> PLUS(N, M)
   ACTIVE(U61(tt)) -> MARK(0)
   ACTIVE(U71(tt, M, N)) -> MARK(U72(isNat(N), M, N))
   ACTIVE(U71(tt, M, N)) -> U72^1(isNat(N), M, N)
   ACTIVE(U71(tt, M, N)) -> ISNAT(N)
   ACTIVE(U72(tt, M, N)) -> MARK(plus(x(N, M), N))
   ACTIVE(U72(tt, M, N)) -> PLUS(x(N, M), N)
   ACTIVE(U72(tt, M, N)) -> X(N, M)
   ACTIVE(isNat(0)) -> MARK(tt)
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
   ACTIVE(isNat(plus(V1, V2))) -> U11^1(isNat(V1), V2)
   ACTIVE(isNat(plus(V1, V2))) -> ISNAT(V1)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   ACTIVE(isNat(s(V1))) -> U21^1(isNat(V1))
   ACTIVE(isNat(s(V1))) -> ISNAT(V1)
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2))
   ACTIVE(isNat(x(V1, V2))) -> U31^1(isNat(V1), V2)
   ACTIVE(isNat(x(V1, V2))) -> ISNAT(V1)
   ACTIVE(plus(N, 0)) -> MARK(U41(isNat(N), N))
   ACTIVE(plus(N, 0)) -> U41^1(isNat(N), N)
   ACTIVE(plus(N, 0)) -> ISNAT(N)
   ACTIVE(plus(N, s(M))) -> MARK(U51(isNat(M), M, N))
   ACTIVE(plus(N, s(M))) -> U51^1(isNat(M), M, N)
   ACTIVE(plus(N, s(M))) -> ISNAT(M)
   ACTIVE(x(N, 0)) -> MARK(U61(isNat(N)))
   ACTIVE(x(N, 0)) -> U61^1(isNat(N))
   ACTIVE(x(N, 0)) -> ISNAT(N)
   ACTIVE(x(N, s(M))) -> MARK(U71(isNat(M), M, N))
   ACTIVE(x(N, s(M))) -> U71^1(isNat(M), M, N)
   ACTIVE(x(N, s(M))) -> ISNAT(M)
   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(U12(X)) -> ACTIVE(U12(mark(X)))
   MARK(U12(X)) -> U12^1(mark(X))
   MARK(U12(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   MARK(U21(X)) -> ACTIVE(U21(mark(X)))
   MARK(U21(X)) -> U21^1(mark(X))
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U31(X1, X2)) -> U31^1(mark(X1), X2)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X)) -> ACTIVE(U32(mark(X)))
   MARK(U32(X)) -> U32^1(mark(X))
   MARK(U32(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   MARK(U41(X1, X2)) -> U41^1(mark(X1), X2)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2, X3)) -> ACTIVE(U51(mark(X1), X2, X3))
   MARK(U51(X1, X2, X3)) -> U51^1(mark(X1), X2, X3)
   MARK(U51(X1, X2, X3)) -> MARK(X1)
   MARK(U52(X1, X2, X3)) -> ACTIVE(U52(mark(X1), X2, X3))
   MARK(U52(X1, X2, X3)) -> U52^1(mark(X1), X2, X3)
   MARK(U52(X1, X2, X3)) -> MARK(X1)
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   MARK(s(X)) -> S(mark(X))
   MARK(s(X)) -> MARK(X)
   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
   MARK(plus(X1, X2)) -> PLUS(mark(X1), mark(X2))
   MARK(plus(X1, X2)) -> MARK(X1)
   MARK(plus(X1, X2)) -> MARK(X2)
   MARK(U61(X)) -> ACTIVE(U61(mark(X)))
   MARK(U61(X)) -> U61^1(mark(X))
   MARK(U61(X)) -> MARK(X)
   MARK(0) -> ACTIVE(0)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   MARK(U71(X1, X2, X3)) -> U71^1(mark(X1), X2, X3)
   MARK(U71(X1, X2, X3)) -> MARK(X1)
   MARK(U72(X1, X2, X3)) -> ACTIVE(U72(mark(X1), X2, X3))
   MARK(U72(X1, X2, X3)) -> U72^1(mark(X1), X2, X3)
   MARK(U72(X1, X2, X3)) -> MARK(X1)
   MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2)))
   MARK(x(X1, X2)) -> X(mark(X1), mark(X2))
   MARK(x(X1, X2)) -> MARK(X1)
   MARK(x(X1, X2)) -> MARK(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)
   U12^1(mark(X)) -> U12^1(X)
   U12^1(active(X)) -> U12^1(X)
   ISNAT(mark(X)) -> ISNAT(X)
   ISNAT(active(X)) -> ISNAT(X)
   U21^1(mark(X)) -> U21^1(X)
   U21^1(active(X)) -> U21^1(X)
   U31^1(mark(X1), X2) -> U31^1(X1, X2)
   U31^1(X1, mark(X2)) -> U31^1(X1, X2)
   U31^1(active(X1), X2) -> U31^1(X1, X2)
   U31^1(X1, active(X2)) -> U31^1(X1, X2)
   U32^1(mark(X)) -> U32^1(X)
   U32^1(active(X)) -> U32^1(X)
   U41^1(mark(X1), X2) -> U41^1(X1, X2)
   U41^1(X1, mark(X2)) -> U41^1(X1, X2)
   U41^1(active(X1), X2) -> U41^1(X1, X2)
   U41^1(X1, active(X2)) -> U41^1(X1, X2)
   U51^1(mark(X1), X2, X3) -> U51^1(X1, X2, X3)
   U51^1(X1, mark(X2), X3) -> U51^1(X1, X2, X3)
   U51^1(X1, X2, mark(X3)) -> U51^1(X1, X2, X3)
   U51^1(active(X1), X2, X3) -> U51^1(X1, X2, X3)
   U51^1(X1, active(X2), X3) -> U51^1(X1, X2, X3)
   U51^1(X1, X2, active(X3)) -> U51^1(X1, X2, X3)
   U52^1(mark(X1), X2, X3) -> U52^1(X1, X2, X3)
   U52^1(X1, mark(X2), X3) -> U52^1(X1, X2, X3)
   U52^1(X1, X2, mark(X3)) -> U52^1(X1, X2, X3)
   U52^1(active(X1), X2, X3) -> U52^1(X1, X2, X3)
   U52^1(X1, active(X2), X3) -> U52^1(X1, X2, X3)
   U52^1(X1, X2, active(X3)) -> U52^1(X1, X2, X3)
   S(mark(X)) -> S(X)
   S(active(X)) -> S(X)
   PLUS(mark(X1), X2) -> PLUS(X1, X2)
   PLUS(X1, mark(X2)) -> PLUS(X1, X2)
   PLUS(active(X1), X2) -> PLUS(X1, X2)
   PLUS(X1, active(X2)) -> PLUS(X1, X2)
   U61^1(mark(X)) -> U61^1(X)
   U61^1(active(X)) -> U61^1(X)
   U71^1(mark(X1), X2, X3) -> U71^1(X1, X2, X3)
   U71^1(X1, mark(X2), X3) -> U71^1(X1, X2, X3)
   U71^1(X1, X2, mark(X3)) -> U71^1(X1, X2, X3)
   U71^1(active(X1), X2, X3) -> U71^1(X1, X2, X3)
   U71^1(X1, active(X2), X3) -> U71^1(X1, X2, X3)
   U71^1(X1, X2, active(X3)) -> U71^1(X1, X2, X3)
   U72^1(mark(X1), X2, X3) -> U72^1(X1, X2, X3)
   U72^1(X1, mark(X2), X3) -> U72^1(X1, X2, X3)
   U72^1(X1, X2, mark(X3)) -> U72^1(X1, X2, X3)
   U72^1(active(X1), X2, X3) -> U72^1(X1, X2, X3)
   U72^1(X1, active(X2), X3) -> U72^1(X1, X2, X3)
   U72^1(X1, X2, active(X3)) -> U72^1(X1, X2, X3)
   X(mark(X1), X2) -> X(X1, X2)
   X(X1, mark(X2)) -> X(X1, X2)
   X(active(X1), X2) -> X(X1, X2)
   X(X1, active(X2)) -> X(X1, X2)

The TRS R consists of the following rules:

   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, N)) -> mark(N)
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   active(U61(tt)) -> mark(0)
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X)) -> active(U12(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X)) -> active(U32(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(0) -> active(0)
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(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)
   U12(mark(X)) -> U12(X)
   U12(active(X)) -> U12(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X)) -> U32(X)
   U32(active(X)) -> U32(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 16 SCCs with 47 less nodes.
----------------------------------------

(4)
Complex Obligation (AND)

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

(5)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, N)) -> mark(N)
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   active(U61(tt)) -> mark(0)
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X)) -> active(U12(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X)) -> active(U32(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(0) -> active(0)
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(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)
   U12(mark(X)) -> U12(X)
   U12(active(X)) -> U12(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X)) -> U32(X)
   U32(active(X)) -> U32(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

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

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

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))


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

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

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

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))

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


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


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


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


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

(11)
YES

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

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

   U72^1(X1, mark(X2), X3) -> U72^1(X1, X2, X3)
   U72^1(mark(X1), X2, X3) -> U72^1(X1, X2, X3)
   U72^1(X1, X2, mark(X3)) -> U72^1(X1, X2, X3)
   U72^1(active(X1), X2, X3) -> U72^1(X1, X2, X3)
   U72^1(X1, active(X2), X3) -> U72^1(X1, X2, X3)
   U72^1(X1, X2, active(X3)) -> U72^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, N)) -> mark(N)
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   active(U61(tt)) -> mark(0)
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X)) -> active(U12(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X)) -> active(U32(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(0) -> active(0)
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(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)
   U12(mark(X)) -> U12(X)
   U12(active(X)) -> U12(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X)) -> U32(X)
   U32(active(X)) -> U32(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

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

   U72^1(X1, mark(X2), X3) -> U72^1(X1, X2, X3)
   U72^1(mark(X1), X2, X3) -> U72^1(X1, X2, X3)
   U72^1(X1, X2, mark(X3)) -> U72^1(X1, X2, X3)
   U72^1(active(X1), X2, X3) -> U72^1(X1, X2, X3)
   U72^1(X1, active(X2), X3) -> U72^1(X1, X2, X3)
   U72^1(X1, X2, active(X3)) -> U72^1(X1, X2, X3)

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))


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

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

   U72^1(X1, mark(X2), X3) -> U72^1(X1, X2, X3)
   U72^1(mark(X1), X2, X3) -> U72^1(X1, X2, X3)
   U72^1(X1, X2, mark(X3)) -> U72^1(X1, X2, X3)
   U72^1(active(X1), X2, X3) -> U72^1(X1, X2, X3)
   U72^1(X1, active(X2), X3) -> U72^1(X1, X2, X3)
   U72^1(X1, X2, active(X3)) -> U72^1(X1, X2, X3)

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))

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

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

From the DPs we obtained the following set of size-change graphs:
*U72^1(X1, mark(X2), X3) -> U72^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U72^1(mark(X1), X2, X3) -> U72^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U72^1(X1, X2, mark(X3)) -> U72^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U72^1(active(X1), X2, X3) -> U72^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U72^1(X1, active(X2), X3) -> U72^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U72^1(X1, X2, active(X3)) -> U72^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


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

(18)
YES

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

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

   U71^1(X1, mark(X2), X3) -> U71^1(X1, X2, X3)
   U71^1(mark(X1), X2, X3) -> U71^1(X1, X2, X3)
   U71^1(X1, X2, mark(X3)) -> U71^1(X1, X2, X3)
   U71^1(active(X1), X2, X3) -> U71^1(X1, X2, X3)
   U71^1(X1, active(X2), X3) -> U71^1(X1, X2, X3)
   U71^1(X1, X2, active(X3)) -> U71^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, N)) -> mark(N)
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   active(U61(tt)) -> mark(0)
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X)) -> active(U12(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X)) -> active(U32(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(0) -> active(0)
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(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)
   U12(mark(X)) -> U12(X)
   U12(active(X)) -> U12(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X)) -> U32(X)
   U32(active(X)) -> U32(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

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

   U71^1(X1, mark(X2), X3) -> U71^1(X1, X2, X3)
   U71^1(mark(X1), X2, X3) -> U71^1(X1, X2, X3)
   U71^1(X1, X2, mark(X3)) -> U71^1(X1, X2, X3)
   U71^1(active(X1), X2, X3) -> U71^1(X1, X2, X3)
   U71^1(X1, active(X2), X3) -> U71^1(X1, X2, X3)
   U71^1(X1, X2, active(X3)) -> U71^1(X1, X2, X3)

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))


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

(23)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U71^1(X1, mark(X2), X3) -> U71^1(X1, X2, X3)
   U71^1(mark(X1), X2, X3) -> U71^1(X1, X2, X3)
   U71^1(X1, X2, mark(X3)) -> U71^1(X1, X2, X3)
   U71^1(active(X1), X2, X3) -> U71^1(X1, X2, X3)
   U71^1(X1, active(X2), X3) -> U71^1(X1, X2, X3)
   U71^1(X1, X2, active(X3)) -> U71^1(X1, X2, X3)

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))

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

(24) 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:
*U71^1(X1, mark(X2), X3) -> U71^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U71^1(mark(X1), X2, X3) -> U71^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U71^1(X1, X2, mark(X3)) -> U71^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U71^1(active(X1), X2, X3) -> U71^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U71^1(X1, active(X2), X3) -> U71^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U71^1(X1, X2, active(X3)) -> U71^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


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

(25)
YES

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

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

   U61^1(active(X)) -> U61^1(X)
   U61^1(mark(X)) -> U61^1(X)

The TRS R consists of the following rules:

   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, N)) -> mark(N)
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   active(U61(tt)) -> mark(0)
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X)) -> active(U12(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X)) -> active(U32(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(0) -> active(0)
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(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)
   U12(mark(X)) -> U12(X)
   U12(active(X)) -> U12(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X)) -> U32(X)
   U32(active(X)) -> U32(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

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

   U61^1(active(X)) -> U61^1(X)
   U61^1(mark(X)) -> U61^1(X)

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))


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

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

   U61^1(active(X)) -> U61^1(X)
   U61^1(mark(X)) -> U61^1(X)

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))

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

(31) 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:
*U61^1(active(X)) -> U61^1(X)
The graph contains the following edges 1 > 1


*U61^1(mark(X)) -> U61^1(X)
The graph contains the following edges 1 > 1


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

(32)
YES

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

(33)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, N)) -> mark(N)
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   active(U61(tt)) -> mark(0)
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X)) -> active(U12(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X)) -> active(U32(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(0) -> active(0)
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(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)
   U12(mark(X)) -> U12(X)
   U12(active(X)) -> U12(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X)) -> U32(X)
   U32(active(X)) -> U32(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

(35)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))


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

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

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

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))

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

(38) 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:
*PLUS(X1, mark(X2)) -> PLUS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


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


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


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


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

(39)
YES

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

(40)
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(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, N)) -> mark(N)
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   active(U61(tt)) -> mark(0)
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X)) -> active(U12(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X)) -> active(U32(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(0) -> active(0)
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(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)
   U12(mark(X)) -> U12(X)
   U12(active(X)) -> U12(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X)) -> U32(X)
   U32(active(X)) -> U32(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

(42)
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.
The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))


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

(44)
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.
The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))

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:
*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


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

(46)
YES

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

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

   U52^1(X1, mark(X2), X3) -> U52^1(X1, X2, X3)
   U52^1(mark(X1), X2, X3) -> U52^1(X1, X2, X3)
   U52^1(X1, X2, mark(X3)) -> U52^1(X1, X2, X3)
   U52^1(active(X1), X2, X3) -> U52^1(X1, X2, X3)
   U52^1(X1, active(X2), X3) -> U52^1(X1, X2, X3)
   U52^1(X1, X2, active(X3)) -> U52^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, N)) -> mark(N)
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   active(U61(tt)) -> mark(0)
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X)) -> active(U12(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X)) -> active(U32(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(0) -> active(0)
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(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)
   U12(mark(X)) -> U12(X)
   U12(active(X)) -> U12(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X)) -> U32(X)
   U32(active(X)) -> U32(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

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

   U52^1(X1, mark(X2), X3) -> U52^1(X1, X2, X3)
   U52^1(mark(X1), X2, X3) -> U52^1(X1, X2, X3)
   U52^1(X1, X2, mark(X3)) -> U52^1(X1, X2, X3)
   U52^1(active(X1), X2, X3) -> U52^1(X1, X2, X3)
   U52^1(X1, active(X2), X3) -> U52^1(X1, X2, X3)
   U52^1(X1, X2, active(X3)) -> U52^1(X1, X2, X3)

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))


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

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

   U52^1(X1, mark(X2), X3) -> U52^1(X1, X2, X3)
   U52^1(mark(X1), X2, X3) -> U52^1(X1, X2, X3)
   U52^1(X1, X2, mark(X3)) -> U52^1(X1, X2, X3)
   U52^1(active(X1), X2, X3) -> U52^1(X1, X2, X3)
   U52^1(X1, active(X2), X3) -> U52^1(X1, X2, X3)
   U52^1(X1, X2, active(X3)) -> U52^1(X1, X2, X3)

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))

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

(52) 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:
*U52^1(X1, mark(X2), X3) -> U52^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U52^1(mark(X1), X2, X3) -> U52^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U52^1(X1, X2, mark(X3)) -> U52^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U52^1(active(X1), X2, X3) -> U52^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U52^1(X1, active(X2), X3) -> U52^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U52^1(X1, X2, active(X3)) -> U52^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


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

(53)
YES

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

(54)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U51^1(X1, mark(X2), X3) -> U51^1(X1, X2, X3)
   U51^1(mark(X1), X2, X3) -> U51^1(X1, X2, X3)
   U51^1(X1, X2, mark(X3)) -> U51^1(X1, X2, X3)
   U51^1(active(X1), X2, X3) -> U51^1(X1, X2, X3)
   U51^1(X1, active(X2), X3) -> U51^1(X1, X2, X3)
   U51^1(X1, X2, active(X3)) -> U51^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, N)) -> mark(N)
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   active(U61(tt)) -> mark(0)
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X)) -> active(U12(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X)) -> active(U32(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(0) -> active(0)
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(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)
   U12(mark(X)) -> U12(X)
   U12(active(X)) -> U12(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X)) -> U32(X)
   U32(active(X)) -> U32(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

(56)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U51^1(X1, mark(X2), X3) -> U51^1(X1, X2, X3)
   U51^1(mark(X1), X2, X3) -> U51^1(X1, X2, X3)
   U51^1(X1, X2, mark(X3)) -> U51^1(X1, X2, X3)
   U51^1(active(X1), X2, X3) -> U51^1(X1, X2, X3)
   U51^1(X1, active(X2), X3) -> U51^1(X1, X2, X3)
   U51^1(X1, X2, active(X3)) -> U51^1(X1, X2, X3)

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))


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

(58)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U51^1(X1, mark(X2), X3) -> U51^1(X1, X2, X3)
   U51^1(mark(X1), X2, X3) -> U51^1(X1, X2, X3)
   U51^1(X1, X2, mark(X3)) -> U51^1(X1, X2, X3)
   U51^1(active(X1), X2, X3) -> U51^1(X1, X2, X3)
   U51^1(X1, active(X2), X3) -> U51^1(X1, X2, X3)
   U51^1(X1, X2, active(X3)) -> U51^1(X1, X2, X3)

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))

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

(59) 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:
*U51^1(X1, mark(X2), X3) -> U51^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U51^1(mark(X1), X2, X3) -> U51^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U51^1(X1, X2, mark(X3)) -> U51^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U51^1(active(X1), X2, X3) -> U51^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U51^1(X1, active(X2), X3) -> U51^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U51^1(X1, X2, active(X3)) -> U51^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


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

(60)
YES

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

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

   U41^1(X1, mark(X2)) -> U41^1(X1, X2)
   U41^1(mark(X1), X2) -> U41^1(X1, X2)
   U41^1(active(X1), X2) -> U41^1(X1, X2)
   U41^1(X1, active(X2)) -> U41^1(X1, X2)

The TRS R consists of the following rules:

   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, N)) -> mark(N)
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   active(U61(tt)) -> mark(0)
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X)) -> active(U12(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X)) -> active(U32(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(0) -> active(0)
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(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)
   U12(mark(X)) -> U12(X)
   U12(active(X)) -> U12(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X)) -> U32(X)
   U32(active(X)) -> U32(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

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

   U41^1(X1, mark(X2)) -> U41^1(X1, X2)
   U41^1(mark(X1), X2) -> U41^1(X1, X2)
   U41^1(active(X1), X2) -> U41^1(X1, X2)
   U41^1(X1, active(X2)) -> U41^1(X1, X2)

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))


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

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

   U41^1(X1, mark(X2)) -> U41^1(X1, X2)
   U41^1(mark(X1), X2) -> U41^1(X1, X2)
   U41^1(active(X1), X2) -> U41^1(X1, X2)
   U41^1(X1, active(X2)) -> U41^1(X1, X2)

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))

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

(66) 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:
*U41^1(X1, mark(X2)) -> U41^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*U41^1(mark(X1), X2) -> U41^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U41^1(active(X1), X2) -> U41^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U41^1(X1, active(X2)) -> U41^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


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

(67)
YES

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

(68)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U32^1(active(X)) -> U32^1(X)
   U32^1(mark(X)) -> U32^1(X)

The TRS R consists of the following rules:

   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, N)) -> mark(N)
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   active(U61(tt)) -> mark(0)
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X)) -> active(U12(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X)) -> active(U32(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(0) -> active(0)
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(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)
   U12(mark(X)) -> U12(X)
   U12(active(X)) -> U12(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X)) -> U32(X)
   U32(active(X)) -> U32(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

(70)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U32^1(active(X)) -> U32^1(X)
   U32^1(mark(X)) -> U32^1(X)

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))


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

(72)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U32^1(active(X)) -> U32^1(X)
   U32^1(mark(X)) -> U32^1(X)

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))

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

(73) 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:
*U32^1(active(X)) -> U32^1(X)
The graph contains the following edges 1 > 1


*U32^1(mark(X)) -> U32^1(X)
The graph contains the following edges 1 > 1


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

(74)
YES

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

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

   U31^1(X1, mark(X2)) -> U31^1(X1, X2)
   U31^1(mark(X1), X2) -> U31^1(X1, X2)
   U31^1(active(X1), X2) -> U31^1(X1, X2)
   U31^1(X1, active(X2)) -> U31^1(X1, X2)

The TRS R consists of the following rules:

   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, N)) -> mark(N)
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   active(U61(tt)) -> mark(0)
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X)) -> active(U12(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X)) -> active(U32(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(0) -> active(0)
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(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)
   U12(mark(X)) -> U12(X)
   U12(active(X)) -> U12(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X)) -> U32(X)
   U32(active(X)) -> U32(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

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

   U31^1(X1, mark(X2)) -> U31^1(X1, X2)
   U31^1(mark(X1), X2) -> U31^1(X1, X2)
   U31^1(active(X1), X2) -> U31^1(X1, X2)
   U31^1(X1, active(X2)) -> U31^1(X1, X2)

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))


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

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

   U31^1(X1, mark(X2)) -> U31^1(X1, X2)
   U31^1(mark(X1), X2) -> U31^1(X1, X2)
   U31^1(active(X1), X2) -> U31^1(X1, X2)
   U31^1(X1, active(X2)) -> U31^1(X1, X2)

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))

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

(80) 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:
*U31^1(X1, mark(X2)) -> U31^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*U31^1(mark(X1), X2) -> U31^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U31^1(active(X1), X2) -> U31^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U31^1(X1, active(X2)) -> U31^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


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

(81)
YES

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

(82)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U21^1(active(X)) -> U21^1(X)
   U21^1(mark(X)) -> U21^1(X)

The TRS R consists of the following rules:

   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, N)) -> mark(N)
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   active(U61(tt)) -> mark(0)
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X)) -> active(U12(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X)) -> active(U32(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(0) -> active(0)
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(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)
   U12(mark(X)) -> U12(X)
   U12(active(X)) -> U12(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X)) -> U32(X)
   U32(active(X)) -> U32(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

(84)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U21^1(active(X)) -> U21^1(X)
   U21^1(mark(X)) -> U21^1(X)

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))


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

(86)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U21^1(active(X)) -> U21^1(X)
   U21^1(mark(X)) -> U21^1(X)

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))

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

(87) 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:
*U21^1(active(X)) -> U21^1(X)
The graph contains the following edges 1 > 1


*U21^1(mark(X)) -> U21^1(X)
The graph contains the following edges 1 > 1


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

(88)
YES

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

(89)
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(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, N)) -> mark(N)
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   active(U61(tt)) -> mark(0)
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X)) -> active(U12(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X)) -> active(U32(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(0) -> active(0)
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(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)
   U12(mark(X)) -> U12(X)
   U12(active(X)) -> U12(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X)) -> U32(X)
   U32(active(X)) -> U32(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

(91)
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.
The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))


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

(93)
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.
The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))

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

(94) 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


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

(95)
YES

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

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

   U12^1(active(X)) -> U12^1(X)
   U12^1(mark(X)) -> U12^1(X)

The TRS R consists of the following rules:

   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, N)) -> mark(N)
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   active(U61(tt)) -> mark(0)
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X)) -> active(U12(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X)) -> active(U32(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(0) -> active(0)
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(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)
   U12(mark(X)) -> U12(X)
   U12(active(X)) -> U12(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X)) -> U32(X)
   U32(active(X)) -> U32(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

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

   U12^1(active(X)) -> U12^1(X)
   U12^1(mark(X)) -> U12^1(X)

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))


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

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

   U12^1(active(X)) -> U12^1(X)
   U12^1(mark(X)) -> U12^1(X)

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))

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

(101) 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:
*U12^1(active(X)) -> U12^1(X)
The graph contains the following edges 1 > 1


*U12^1(mark(X)) -> U12^1(X)
The graph contains the following edges 1 > 1


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

(102)
YES

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

(103)
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(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, N)) -> mark(N)
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   active(U61(tt)) -> mark(0)
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X)) -> active(U12(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X)) -> active(U32(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(0) -> active(0)
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(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)
   U12(mark(X)) -> U12(X)
   U12(active(X)) -> U12(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X)) -> U32(X)
   U32(active(X)) -> U32(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

(105)
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.
The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))


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

(107)
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.
The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))

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

(108) 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


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

(109)
YES

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

(110)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X)) -> ACTIVE(U12(mark(X)))
   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
   MARK(U12(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U31(tt, V2)) -> MARK(U32(isNat(V2)))
   MARK(U32(X)) -> ACTIVE(U32(mark(X)))
   ACTIVE(U41(tt, N)) -> MARK(N)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(U21(X)) -> ACTIVE(U21(mark(X)))
   ACTIVE(U51(tt, M, N)) -> MARK(U52(isNat(N), M, N))
   MARK(U52(X1, X2, X3)) -> ACTIVE(U52(mark(X1), X2, X3))
   ACTIVE(U52(tt, M, N)) -> MARK(s(plus(N, M)))
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   ACTIVE(U71(tt, M, N)) -> MARK(U72(isNat(N), M, N))
   MARK(U72(X1, X2, X3)) -> ACTIVE(U72(mark(X1), X2, X3))
   ACTIVE(U72(tt, M, N)) -> MARK(plus(x(N, M), N))
   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
   ACTIVE(plus(N, 0)) -> MARK(U41(isNat(N), N))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(plus(N, s(M))) -> MARK(U51(isNat(M), M, N))
   MARK(U51(X1, X2, X3)) -> ACTIVE(U51(mark(X1), X2, X3))
   ACTIVE(x(N, 0)) -> MARK(U61(isNat(N)))
   MARK(U61(X)) -> ACTIVE(U61(mark(X)))
   ACTIVE(x(N, s(M))) -> MARK(U71(isNat(M), M, N))
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   MARK(U71(X1, X2, X3)) -> MARK(X1)
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2, X3)) -> MARK(X1)
   MARK(U52(X1, X2, X3)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(plus(X1, X2)) -> MARK(X1)
   MARK(plus(X1, X2)) -> MARK(X2)
   MARK(U61(X)) -> MARK(X)
   MARK(U72(X1, X2, X3)) -> MARK(X1)
   MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2)))
   MARK(x(X1, X2)) -> MARK(X1)
   MARK(x(X1, X2)) -> MARK(X2)
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2))

The TRS R consists of the following rules:

   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, N)) -> mark(N)
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   active(U61(tt)) -> mark(0)
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U12(X)) -> active(U12(mark(X)))
   mark(isNat(X)) -> active(isNat(X))
   mark(U21(X)) -> active(U21(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X)) -> active(U32(mark(X)))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(U61(X)) -> active(U61(mark(X)))
   mark(0) -> active(0)
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(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)
   U12(mark(X)) -> U12(X)
   U12(active(X)) -> U12(X)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U21(mark(X)) -> U21(X)
   U21(active(X)) -> U21(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X)) -> U32(X)
   U32(active(X)) -> U32(X)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U61(mark(X)) -> U61(X)
   U61(active(X)) -> U61(X)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   x(mark(X1), X2) -> x(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

(112)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X)) -> ACTIVE(U12(mark(X)))
   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
   MARK(U12(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U31(tt, V2)) -> MARK(U32(isNat(V2)))
   MARK(U32(X)) -> ACTIVE(U32(mark(X)))
   ACTIVE(U41(tt, N)) -> MARK(N)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(U21(X)) -> ACTIVE(U21(mark(X)))
   ACTIVE(U51(tt, M, N)) -> MARK(U52(isNat(N), M, N))
   MARK(U52(X1, X2, X3)) -> ACTIVE(U52(mark(X1), X2, X3))
   ACTIVE(U52(tt, M, N)) -> MARK(s(plus(N, M)))
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   ACTIVE(U71(tt, M, N)) -> MARK(U72(isNat(N), M, N))
   MARK(U72(X1, X2, X3)) -> ACTIVE(U72(mark(X1), X2, X3))
   ACTIVE(U72(tt, M, N)) -> MARK(plus(x(N, M), N))
   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
   ACTIVE(plus(N, 0)) -> MARK(U41(isNat(N), N))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(plus(N, s(M))) -> MARK(U51(isNat(M), M, N))
   MARK(U51(X1, X2, X3)) -> ACTIVE(U51(mark(X1), X2, X3))
   ACTIVE(x(N, 0)) -> MARK(U61(isNat(N)))
   MARK(U61(X)) -> ACTIVE(U61(mark(X)))
   ACTIVE(x(N, s(M))) -> MARK(U71(isNat(M), M, N))
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   MARK(U71(X1, X2, X3)) -> MARK(X1)
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2, X3)) -> MARK(X1)
   MARK(U52(X1, X2, X3)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(plus(X1, X2)) -> MARK(X1)
   MARK(plus(X1, X2)) -> MARK(X2)
   MARK(U61(X)) -> MARK(X)
   MARK(U72(X1, X2, X3)) -> MARK(X1)
   MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2)))
   MARK(x(X1, X2)) -> MARK(X1)
   MARK(x(X1, X2)) -> MARK(X2)
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2))

The TRS R consists of the following rules:

   mark(U12(X)) -> active(U12(mark(X)))
   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   mark(U32(X)) -> active(U32(mark(X)))
   active(U41(tt, N)) -> mark(N)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   mark(U61(X)) -> active(U61(mark(X)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U12(active(X)) -> U12(X)
   U12(mark(X)) -> U12(X)
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U32(tt)) -> mark(tt)
   active(U61(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   U32(active(X)) -> U32(X)
   U32(mark(X)) -> U32(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)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U12(X)) -> ACTIVE(U12(mark(X)))
   MARK(U32(X)) -> ACTIVE(U32(mark(X)))
   MARK(U21(X)) -> ACTIVE(U21(mark(X)))
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   MARK(U61(X)) -> ACTIVE(U61(mark(X)))
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( U11_2(x_1, x_2) ) = 1
POL( U12_1(x_1) ) = max{0, -2}
POL( U21_1(x_1) ) = max{0, -2}
POL( U31_2(x_1, x_2) ) = 1
POL( U32_1(x_1) ) = max{0, -2}
POL( U41_2(x_1, x_2) ) = 1
POL( U51_3(x_1, ..., x_3) ) = 1
POL( U52_3(x_1, ..., x_3) ) = 1
POL( U61_1(x_1) ) = max{0, -2}
POL( U71_3(x_1, ..., x_3) ) = 1
POL( U72_3(x_1, ..., x_3) ) = 1
POL( plus_2(x_1, x_2) ) = 1
POL( s_1(x_1) ) = 0
POL( x_2(x_1, x_2) ) = 1
POL( mark_1(x_1) ) = 2
POL( active_1(x_1) ) = max{0, -2}
POL( tt ) = 0
POL( isNat_1(x_1) ) = 1
POL( 0 ) = 0
POL( MARK_1(x_1) ) = 2

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

   U12(active(X)) -> U12(X)
   U12(mark(X)) -> U12(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)
   U32(active(X)) -> U32(X)
   U32(mark(X)) -> U32(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)


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

(114)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
   MARK(U12(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U31(tt, V2)) -> MARK(U32(isNat(V2)))
   ACTIVE(U41(tt, N)) -> MARK(N)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   ACTIVE(U51(tt, M, N)) -> MARK(U52(isNat(N), M, N))
   MARK(U52(X1, X2, X3)) -> ACTIVE(U52(mark(X1), X2, X3))
   ACTIVE(U52(tt, M, N)) -> MARK(s(plus(N, M)))
   ACTIVE(U71(tt, M, N)) -> MARK(U72(isNat(N), M, N))
   MARK(U72(X1, X2, X3)) -> ACTIVE(U72(mark(X1), X2, X3))
   ACTIVE(U72(tt, M, N)) -> MARK(plus(x(N, M), N))
   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
   ACTIVE(plus(N, 0)) -> MARK(U41(isNat(N), N))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   ACTIVE(plus(N, s(M))) -> MARK(U51(isNat(M), M, N))
   MARK(U51(X1, X2, X3)) -> ACTIVE(U51(mark(X1), X2, X3))
   ACTIVE(x(N, 0)) -> MARK(U61(isNat(N)))
   ACTIVE(x(N, s(M))) -> MARK(U71(isNat(M), M, N))
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   MARK(U71(X1, X2, X3)) -> MARK(X1)
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X)) -> MARK(X)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2, X3)) -> MARK(X1)
   MARK(U52(X1, X2, X3)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(plus(X1, X2)) -> MARK(X1)
   MARK(plus(X1, X2)) -> MARK(X2)
   MARK(U61(X)) -> MARK(X)
   MARK(U72(X1, X2, X3)) -> MARK(X1)
   MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2)))
   MARK(x(X1, X2)) -> MARK(X1)
   MARK(x(X1, X2)) -> MARK(X2)
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2))

The TRS R consists of the following rules:

   mark(U12(X)) -> active(U12(mark(X)))
   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   mark(U32(X)) -> active(U32(mark(X)))
   active(U41(tt, N)) -> mark(N)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   mark(U61(X)) -> active(U61(mark(X)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U12(active(X)) -> U12(X)
   U12(mark(X)) -> U12(X)
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U32(tt)) -> mark(tt)
   active(U61(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   U32(active(X)) -> U32(X)
   U32(mark(X)) -> U32(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)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U41(tt, N)) -> MARK(N)
   ACTIVE(U52(tt, M, N)) -> MARK(s(plus(N, M)))
   ACTIVE(U72(tt, M, N)) -> MARK(plus(x(N, M), N))
   ACTIVE(plus(N, s(M))) -> MARK(U51(isNat(M), M, N))
   ACTIVE(x(N, 0)) -> MARK(U61(isNat(N)))
   ACTIVE(x(N, s(M))) -> MARK(U71(isNat(M), M, N))
   MARK(U71(X1, X2, X3)) -> MARK(X1)
   MARK(U41(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2, X3)) -> MARK(X1)
   MARK(U52(X1, X2, X3)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(plus(X1, X2)) -> MARK(X1)
   MARK(plus(X1, X2)) -> MARK(X2)
   MARK(U72(X1, X2, X3)) -> MARK(X1)
   MARK(x(X1, X2)) -> MARK(X1)
   MARK(x(X1, X2)) -> MARK(X2)
The remaining pairs can at least be oriented weakly.
Used ordering:  Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)

U11(x1, x2)  =  x1

tt  =  tt

MARK(x1)  =  MARK(x1)

U12(x1)  =  x1

isNat(x1)  =  isNat

mark(x1)  =  x1

U31(x1, x2)  =  x1

U32(x1)  =  x1

U41(x1, x2)  =  U41(x1, x2)

plus(x1, x2)  =  plus(x1, x2)

s(x1)  =  s(x1)

U21(x1)  =  x1

U51(x1, x2, x3)  =  U51(x1, x2, x3)

U52(x1, x2, x3)  =  U52(x1, x2, x3)

U71(x1, x2, x3)  =  U71(x1, x2, x3)

U72(x1, x2, x3)  =  U72(x1, x2, x3)

x(x1, x2)  =  x(x1, x2)

0  =  0

U61(x1)  =  x1

active(x1)  =  x1


Recursive path order with status [RPO].
Quasi-Precedence: [U71_3, U72_3, x_2] > [U41_2, plus_2, U51_3, U52_3] > [ACTIVE_1, MARK_1] > [tt, isNat, 0] > s_1

Status: ACTIVE_1: multiset status
tt: multiset status
MARK_1: multiset status
isNat: multiset status
U41_2: [1,2]
plus_2: [2,1]
s_1: [1]
U51_3: [2,3,1]
U52_3: [2,3,1]
U71_3: [2,3,1]
U72_3: [2,3,1]
x_2: [2,1]
0: multiset status


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

   mark(U12(X)) -> active(U12(mark(X)))
   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   mark(U32(X)) -> active(U32(mark(X)))
   active(U41(tt, N)) -> mark(N)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   mark(U61(X)) -> active(U61(mark(X)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   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)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U12(active(X)) -> U12(X)
   U12(mark(X)) -> U12(X)
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U32(tt)) -> mark(tt)
   active(U61(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   U32(active(X)) -> U32(X)
   U32(mark(X)) -> U32(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)


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

(116)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
   MARK(U12(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U31(tt, V2)) -> MARK(U32(isNat(V2)))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   ACTIVE(U51(tt, M, N)) -> MARK(U52(isNat(N), M, N))
   MARK(U52(X1, X2, X3)) -> ACTIVE(U52(mark(X1), X2, X3))
   ACTIVE(U71(tt, M, N)) -> MARK(U72(isNat(N), M, N))
   MARK(U72(X1, X2, X3)) -> ACTIVE(U72(mark(X1), X2, X3))
   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
   ACTIVE(plus(N, 0)) -> MARK(U41(isNat(N), N))
   MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2))
   MARK(U51(X1, X2, X3)) -> ACTIVE(U51(mark(X1), X2, X3))
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X)) -> MARK(X)
   MARK(U61(X)) -> MARK(X)
   MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2)))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2))

The TRS R consists of the following rules:

   mark(U12(X)) -> active(U12(mark(X)))
   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   mark(U32(X)) -> active(U32(mark(X)))
   active(U41(tt, N)) -> mark(N)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   mark(U61(X)) -> active(U61(mark(X)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U12(active(X)) -> U12(X)
   U12(mark(X)) -> U12(X)
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U32(tt)) -> mark(tt)
   active(U61(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   U32(active(X)) -> U32(X)
   U32(mark(X)) -> U32(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)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U41(X1, X2)) -> ACTIVE(U41(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

U12(x1)  =  U12

isNat(x1)  =  isNat

mark(x1)  =  mark

U31(x1, x2)  =  U31

U32(x1)  =  U32

plus(x1, x2)  =  plus

s(x1)  =  s

U21(x1)  =  U21

U51(x1, x2, x3)  =  U51

U52(x1, x2, x3)  =  U52

U71(x1, x2, x3)  =  U71

U72(x1, x2, x3)  =  U72

0  =  0

U41(x1, x2)  =  U41

U61(x1)  =  U61(x1)

x(x1, x2)  =  x

active(x1)  =  x1


Recursive path order with status [RPO].
Quasi-Precedence: [mark, U32, 0] > U12 > tt > [U11, MARK, isNat, U31, plus, U21, U51, U52, U71, U72, x] > U41
[mark, U32, 0] > s > [U11, MARK, isNat, U31, plus, U21, U51, U52, U71, U72, x] > U41
[mark, U32, 0] > U61_1 > [U11, MARK, isNat, U31, plus, U21, U51, U52, U71, U72, x] > U41

Status: U11: []
tt: multiset status
MARK: []
U12: multiset status
isNat: []
mark: []
U31: []
U32: multiset status
plus: []
s: multiset status
U21: []
U51: []
U52: []
U71: []
U72: []
0: multiset status
U41: multiset status
U61_1: multiset status
x: []


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)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)


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

(118)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
   MARK(U12(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U31(tt, V2)) -> MARK(U32(isNat(V2)))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   ACTIVE(U51(tt, M, N)) -> MARK(U52(isNat(N), M, N))
   MARK(U52(X1, X2, X3)) -> ACTIVE(U52(mark(X1), X2, X3))
   ACTIVE(U71(tt, M, N)) -> MARK(U72(isNat(N), M, N))
   MARK(U72(X1, X2, X3)) -> ACTIVE(U72(mark(X1), X2, X3))
   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
   ACTIVE(plus(N, 0)) -> MARK(U41(isNat(N), N))
   MARK(U51(X1, X2, X3)) -> ACTIVE(U51(mark(X1), X2, X3))
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X)) -> MARK(X)
   MARK(U61(X)) -> MARK(X)
   MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2)))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2))

The TRS R consists of the following rules:

   mark(U12(X)) -> active(U12(mark(X)))
   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   mark(U32(X)) -> active(U32(mark(X)))
   active(U41(tt, N)) -> mark(N)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   mark(U61(X)) -> active(U61(mark(X)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U12(active(X)) -> U12(X)
   U12(mark(X)) -> U12(X)
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U32(tt)) -> mark(tt)
   active(U61(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   U32(active(X)) -> U32(X)
   U32(mark(X)) -> U32(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)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

(120)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
   ACTIVE(U31(tt, V2)) -> MARK(U32(isNat(V2)))
   MARK(U32(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U51(tt, M, N)) -> MARK(U52(isNat(N), M, N))
   MARK(U52(X1, X2, X3)) -> ACTIVE(U52(mark(X1), X2, X3))
   ACTIVE(U71(tt, M, N)) -> MARK(U72(isNat(N), M, N))
   MARK(U72(X1, X2, X3)) -> ACTIVE(U72(mark(X1), X2, X3))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2, X3)) -> ACTIVE(U51(mark(X1), X2, X3))
   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
   MARK(U61(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
   MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2)))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2))

The TRS R consists of the following rules:

   mark(U12(X)) -> active(U12(mark(X)))
   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   mark(U32(X)) -> active(U32(mark(X)))
   active(U41(tt, N)) -> mark(N)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   mark(U61(X)) -> active(U61(mark(X)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U12(active(X)) -> U12(X)
   U12(mark(X)) -> U12(X)
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U32(tt)) -> mark(tt)
   active(U61(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   U32(active(X)) -> U32(X)
   U32(mark(X)) -> U32(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)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U51(X1, X2, X3)) -> ACTIVE(U51(mark(X1), X2, X3))
   MARK(U61(X)) -> MARK(X)
   MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U11_2(x_1, x_2) ) = 2x_1 + 1
POL( U31_2(x_1, x_2) ) = 2x_1 + 1
POL( U51_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3 + 2
POL( U52_3(x_1, ..., x_3) ) = 1
POL( U71_3(x_1, ..., x_3) ) = x_1 + x_2 + x_3 + 2
POL( U72_3(x_1, ..., x_3) ) = x_1 + 1
POL( plus_2(x_1, x_2) ) = 1
POL( x_2(x_1, x_2) ) = 2x_1 + x_2 + 1
POL( mark_1(x_1) ) = 0
POL( U12_1(x_1) ) = x_1 + 1
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( tt ) = 1
POL( isNat_1(x_1) ) = 0
POL( U32_1(x_1) ) = x_1 + 1
POL( U41_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( s_1(x_1) ) = 0
POL( U21_1(x_1) ) = 2x_1 + 1
POL( 0 ) = 0
POL( U61_1(x_1) ) = x_1 + 2
POL( MARK_1(x_1) ) = max{0, 2x_1 - 2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


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

(122)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
   ACTIVE(U31(tt, V2)) -> MARK(U32(isNat(V2)))
   MARK(U32(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U51(tt, M, N)) -> MARK(U52(isNat(N), M, N))
   MARK(U52(X1, X2, X3)) -> ACTIVE(U52(mark(X1), X2, X3))
   ACTIVE(U71(tt, M, N)) -> MARK(U72(isNat(N), M, N))
   MARK(U72(X1, X2, X3)) -> ACTIVE(U72(mark(X1), X2, X3))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
   MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2)))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2))

The TRS R consists of the following rules:

   mark(U12(X)) -> active(U12(mark(X)))
   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   mark(U32(X)) -> active(U32(mark(X)))
   active(U41(tt, N)) -> mark(N)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   mark(U61(X)) -> active(U61(mark(X)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U12(active(X)) -> U12(X)
   U12(mark(X)) -> U12(X)
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U32(tt)) -> mark(tt)
   active(U61(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   U32(active(X)) -> U32(X)
   U32(mark(X)) -> U32(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)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

(123) QDPQMonotonicMRRProof (EQUIVALENT)
By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.

Strictly oriented dependency pairs:

   ACTIVE(U51(tt, M, N)) -> MARK(U52(isNat(N), M, N))
   ACTIVE(U71(tt, M, N)) -> MARK(U72(isNat(N), M, N))


Used ordering: Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = 2*x_1
   POL(MARK(x_1)) = 0
   POL(U11(x_1, x_2)) = 0
   POL(U12(x_1)) = 0
   POL(U21(x_1)) = 0
   POL(U31(x_1, x_2)) = 0
   POL(U32(x_1)) = 0
   POL(U41(x_1, x_2)) = 0
   POL(U51(x_1, x_2, x_3)) = 2
   POL(U52(x_1, x_2, x_3)) = 0
   POL(U61(x_1)) = 0
   POL(U71(x_1, x_2, x_3)) = 2
   POL(U72(x_1, x_2, x_3)) = 0
   POL(active(x_1)) = 0
   POL(isNat(x_1)) = 0
   POL(mark(x_1)) = 0
   POL(plus(x_1, x_2)) = 0
   POL(s(x_1)) = 0
   POL(tt) = 0
   POL(x(x_1, x_2)) = 0


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

(124)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
   ACTIVE(U31(tt, V2)) -> MARK(U32(isNat(V2)))
   MARK(U32(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U52(X1, X2, X3)) -> ACTIVE(U52(mark(X1), X2, X3))
   MARK(U72(X1, X2, X3)) -> ACTIVE(U72(mark(X1), X2, X3))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
   MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2)))
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2))

The TRS R consists of the following rules:

   mark(U12(X)) -> active(U12(mark(X)))
   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   mark(U32(X)) -> active(U32(mark(X)))
   active(U41(tt, N)) -> mark(N)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   mark(U61(X)) -> active(U61(mark(X)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U12(active(X)) -> U12(X)
   U12(mark(X)) -> U12(X)
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U32(tt)) -> mark(tt)
   active(U61(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   U32(active(X)) -> U32(X)
   U32(mark(X)) -> U32(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)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

(125) QDPQMonotonicMRRProof (EQUIVALENT)
By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.

Strictly oriented dependency pairs:

   MARK(U52(X1, X2, X3)) -> ACTIVE(U52(mark(X1), X2, X3))
   MARK(U72(X1, X2, X3)) -> ACTIVE(U72(mark(X1), X2, X3))
   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
   MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2)))


Used ordering: Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = 2*x_1
   POL(MARK(x_1)) = 2
   POL(U11(x_1, x_2)) = 1
   POL(U12(x_1)) = 0
   POL(U21(x_1)) = 0
   POL(U31(x_1, x_2)) = 1
   POL(U32(x_1)) = 0
   POL(U41(x_1, x_2)) = 0
   POL(U51(x_1, x_2, x_3)) = 0
   POL(U52(x_1, x_2, x_3)) = 0
   POL(U61(x_1)) = 0
   POL(U71(x_1, x_2, x_3)) = 0
   POL(U72(x_1, x_2, x_3)) = 0
   POL(active(x_1)) = 0
   POL(isNat(x_1)) = 1
   POL(mark(x_1)) = 0
   POL(plus(x_1, x_2)) = 0
   POL(s(x_1)) = 0
   POL(tt) = 0
   POL(x(x_1, x_2)) = 0


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

(126)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
   ACTIVE(U31(tt, V2)) -> MARK(U32(isNat(V2)))
   MARK(U32(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U31(X1, X2)) -> MARK(X1)
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2))

The TRS R consists of the following rules:

   mark(U12(X)) -> active(U12(mark(X)))
   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   mark(U32(X)) -> active(U32(mark(X)))
   active(U41(tt, N)) -> mark(N)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   mark(U61(X)) -> active(U61(mark(X)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U12(active(X)) -> U12(X)
   U12(mark(X)) -> U12(X)
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U32(tt)) -> mark(tt)
   active(U61(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   U32(active(X)) -> U32(X)
   U32(mark(X)) -> U32(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)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U31(tt, V2)) -> MARK(U32(isNat(V2)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

   <<<
 POL(U12(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

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

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

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

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

   <<<
 POL(U31(x_1, x_2)) =  	[[-I]] 	 +  	[[5A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(U32(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

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

   <<<
 POL(U21(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(x(x_1, x_2)) =  	[[0A]] 	 +  	[[5A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

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

   <<<
 POL(U41(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U51(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U52(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U71(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[5A]] 	* 	x_3
>>>

   <<<
 POL(U72(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[5A]] 	* 	x_3
>>>

   <<<
 POL(0) =  	[[4A]]
>>>

   <<<
 POL(U61(x_1)) =  	[[4A]] 	 +  	[[-I]] 	* 	x_1
>>>


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

   mark(U12(X)) -> active(U12(mark(X)))
   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   mark(U32(X)) -> active(U32(mark(X)))
   active(U41(tt, N)) -> mark(N)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   mark(U61(X)) -> active(U61(mark(X)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   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)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U12(active(X)) -> U12(X)
   U12(mark(X)) -> U12(X)
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U32(tt)) -> mark(tt)
   active(U61(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   U32(active(X)) -> U32(X)
   U32(mark(X)) -> U32(X)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)


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

(128)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
   MARK(U32(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U31(X1, X2)) -> MARK(X1)
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2))

The TRS R consists of the following rules:

   mark(U12(X)) -> active(U12(mark(X)))
   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   mark(U32(X)) -> active(U32(mark(X)))
   active(U41(tt, N)) -> mark(N)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   mark(U61(X)) -> active(U61(mark(X)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U12(active(X)) -> U12(X)
   U12(mark(X)) -> U12(X)
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U32(tt)) -> mark(tt)
   active(U61(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   U32(active(X)) -> U32(X)
   U32(mark(X)) -> U32(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)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

(129) QDPQMonotonicMRRProof (EQUIVALENT)
By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.

Strictly oriented dependency pairs:

   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))


Used ordering: Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = 2*x_1
   POL(MARK(x_1)) = 2
   POL(U11(x_1, x_2)) = 1
   POL(U12(x_1)) = 0
   POL(U21(x_1)) = 0
   POL(U31(x_1, x_2)) = 0
   POL(U32(x_1)) = 0
   POL(U41(x_1, x_2)) = 0
   POL(U51(x_1, x_2, x_3)) = 0
   POL(U52(x_1, x_2, x_3)) = 0
   POL(U61(x_1)) = 0
   POL(U71(x_1, x_2, x_3)) = 0
   POL(U72(x_1, x_2, x_3)) = 0
   POL(active(x_1)) = 0
   POL(isNat(x_1)) = 1
   POL(mark(x_1)) = 0
   POL(plus(x_1, x_2)) = 0
   POL(s(x_1)) = 0
   POL(tt) = 0
   POL(x(x_1, x_2)) = 0


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

(130)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
   MARK(U32(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> MARK(X1)
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2))

The TRS R consists of the following rules:

   mark(U12(X)) -> active(U12(mark(X)))
   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   mark(U32(X)) -> active(U32(mark(X)))
   active(U41(tt, N)) -> mark(N)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   mark(U61(X)) -> active(U61(mark(X)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U12(active(X)) -> U12(X)
   U12(mark(X)) -> U12(X)
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U32(tt)) -> mark(tt)
   active(U61(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   U32(active(X)) -> U32(X)
   U32(mark(X)) -> U32(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)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U32(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U11_2(x_1, x_2) ) = 2x_1 + 1
POL( mark_1(x_1) ) = 0
POL( U12_1(x_1) ) = x_1 + 1
POL( active_1(x_1) ) = 2
POL( tt ) = 0
POL( isNat_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = x_1 + 1
POL( U32_1(x_1) ) = x_1 + 2
POL( U41_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U51_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 1}
POL( U52_3(x_1, ..., x_3) ) = x_1 + x_3
POL( s_1(x_1) ) = max{0, x_1 - 1}
POL( plus_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U71_3(x_1, ..., x_3) ) = x_2 + 2
POL( U72_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2}
POL( x_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U21_1(x_1) ) = x_1 + 1
POL( 0 ) = 0
POL( U61_1(x_1) ) = 2
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


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

(132)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(U21(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> MARK(X1)
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2))

The TRS R consists of the following rules:

   mark(U12(X)) -> active(U12(mark(X)))
   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   mark(U32(X)) -> active(U32(mark(X)))
   active(U41(tt, N)) -> mark(N)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   mark(U61(X)) -> active(U61(mark(X)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U12(active(X)) -> U12(X)
   U12(mark(X)) -> U12(X)
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U32(tt)) -> mark(tt)
   active(U61(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   U32(active(X)) -> U32(X)
   U32(mark(X)) -> U32(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)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U31(X1, X2)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

   <<<
 POL(U12(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

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

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

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

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

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

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

   <<<
 POL(U21(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U31(x_1, x_2)) =  	[[0A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

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

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

   <<<
 POL(U32(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(U41(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U51(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U52(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U71(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U72(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

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

   <<<
 POL(U61(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>


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

   mark(U12(X)) -> active(U12(mark(X)))
   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   mark(U32(X)) -> active(U32(mark(X)))
   active(U41(tt, N)) -> mark(N)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   mark(U61(X)) -> active(U61(mark(X)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   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)
   U12(active(X)) -> U12(X)
   U12(mark(X)) -> U12(X)
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U32(tt)) -> mark(tt)
   active(U61(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   U32(active(X)) -> U32(X)
   U32(mark(X)) -> U32(X)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)


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

(134)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U12(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(U21(X)) -> MARK(X)
   ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2))

The TRS R consists of the following rules:

   mark(U12(X)) -> active(U12(mark(X)))
   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   mark(U32(X)) -> active(U32(mark(X)))
   active(U41(tt, N)) -> mark(N)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   mark(U61(X)) -> active(U61(mark(X)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U12(active(X)) -> U12(X)
   U12(mark(X)) -> U12(X)
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U32(tt)) -> mark(tt)
   active(U61(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   U32(active(X)) -> U32(X)
   U32(mark(X)) -> U32(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)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

(136)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
   MARK(U12(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(U21(X)) -> MARK(X)

The TRS R consists of the following rules:

   mark(U12(X)) -> active(U12(mark(X)))
   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   mark(U32(X)) -> active(U32(mark(X)))
   active(U41(tt, N)) -> mark(N)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   mark(U61(X)) -> active(U61(mark(X)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U12(active(X)) -> U12(X)
   U12(mark(X)) -> U12(X)
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U32(tt)) -> mark(tt)
   active(U61(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   U32(active(X)) -> U32(X)
   U32(mark(X)) -> U32(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)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(U12(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

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

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

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

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

   <<<
 POL(U12(x_1)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1
>>>

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

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

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

   <<<
 POL(U21(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

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

   <<<
 POL(U31(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2
>>>

   <<<
 POL(U32(x_1)) =  	[[2A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(U41(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(U51(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U52(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2 	 +  	[[0A]] 	* 	x_3
>>>

   <<<
 POL(U71(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(U72(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

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

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

   <<<
 POL(U61(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>


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

   mark(U12(X)) -> active(U12(mark(X)))
   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   mark(U32(X)) -> active(U32(mark(X)))
   active(U41(tt, N)) -> mark(N)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   mark(U61(X)) -> active(U61(mark(X)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   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)
   U12(active(X)) -> U12(X)
   U12(mark(X)) -> U12(X)
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U32(tt)) -> mark(tt)
   active(U61(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   U32(active(X)) -> U32(X)
   U32(mark(X)) -> U32(X)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)


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

(138)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
   MARK(U21(X)) -> MARK(X)

The TRS R consists of the following rules:

   mark(U12(X)) -> active(U12(mark(X)))
   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   mark(U32(X)) -> active(U32(mark(X)))
   active(U41(tt, N)) -> mark(N)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   mark(U61(X)) -> active(U61(mark(X)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U12(active(X)) -> U12(X)
   U12(mark(X)) -> U12(X)
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U32(tt)) -> mark(tt)
   active(U61(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   U32(active(X)) -> U32(X)
   U32(mark(X)) -> U32(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)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

(139) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

(140)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U21(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))

The TRS R consists of the following rules:

   mark(U12(X)) -> active(U12(mark(X)))
   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
   mark(U32(X)) -> active(U32(mark(X)))
   active(U41(tt, N)) -> mark(N)
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
   mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3))
   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
   mark(s(X)) -> active(s(mark(X)))
   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
   mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3))
   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   mark(U21(X)) -> active(U21(mark(X)))
   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(plus(N, 0)) -> mark(U41(isNat(N), N))
   mark(U41(X1, X2)) -> active(U41(mark(X1), X2))
   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
   mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3))
   active(x(N, 0)) -> mark(U61(isNat(N)))
   mark(U61(X)) -> active(U61(mark(X)))
   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
   mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3))
   mark(isNat(X)) -> active(isNat(X))
   mark(x(X1, X2)) -> active(x(mark(X1), mark(X2)))
   mark(tt) -> active(tt)
   mark(0) -> active(0)
   x(X1, mark(X2)) -> x(X1, X2)
   x(mark(X1), X2) -> x(X1, X2)
   x(active(X1), X2) -> x(X1, X2)
   x(X1, active(X2)) -> x(X1, X2)
   U12(active(X)) -> U12(X)
   U12(mark(X)) -> U12(X)
   active(U12(tt)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U32(tt)) -> mark(tt)
   active(U61(tt)) -> mark(0)
   active(isNat(0)) -> mark(tt)
   U32(active(X)) -> U32(X)
   U32(mark(X)) -> U32(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)
   U52(X1, mark(X2), X3) -> U52(X1, X2, X3)
   U52(mark(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, X2, mark(X3)) -> U52(X1, X2, X3)
   U52(active(X1), X2, X3) -> U52(X1, X2, X3)
   U52(X1, active(X2), X3) -> U52(X1, X2, X3)
   U52(X1, X2, active(X3)) -> U52(X1, X2, X3)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U72(X1, mark(X2), X3) -> U72(X1, X2, X3)
   U72(mark(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, X2, mark(X3)) -> U72(X1, X2, X3)
   U72(active(X1), X2, X3) -> U72(X1, X2, X3)
   U72(X1, active(X2), X3) -> U72(X1, X2, X3)
   U72(X1, X2, active(X3)) -> U72(X1, X2, X3)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   U21(active(X)) -> U21(X)
   U21(mark(X)) -> U21(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2)) -> U41(X1, X2)
   U41(mark(X1), X2) -> U41(X1, X2)
   U41(active(X1), X2) -> U41(X1, X2)
   U41(X1, active(X2)) -> U41(X1, X2)
   U51(X1, mark(X2), X3) -> U51(X1, X2, X3)
   U51(mark(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, X2, mark(X3)) -> U51(X1, X2, X3)
   U51(active(X1), X2, X3) -> U51(X1, X2, X3)
   U51(X1, active(X2), X3) -> U51(X1, X2, X3)
   U51(X1, X2, active(X3)) -> U51(X1, X2, X3)
   U61(active(X)) -> U61(X)
   U61(mark(X)) -> U61(X)
   U71(X1, mark(X2), X3) -> U71(X1, X2, X3)
   U71(mark(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, X2, mark(X3)) -> U71(X1, X2, X3)
   U71(active(X1), X2, X3) -> U71(X1, X2, X3)
   U71(X1, active(X2), X3) -> U71(X1, X2, X3)
   U71(X1, X2, active(X3)) -> U71(X1, X2, X3)

The set Q consists of the following terms:

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

(142)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U21(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0))
   U12(active(x0))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))

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

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

   active(U11(tt, x0))
   active(U12(tt))
   active(U21(tt))
   active(U31(tt, x0))
   active(U32(tt))
   active(U41(tt, x0))
   active(U51(tt, x0, x1))
   active(U52(tt, x0, x1))
   active(U61(tt))
   active(U71(tt, x0, x1))
   active(U72(tt, x0, x1))
   active(isNat(0))
   active(isNat(plus(x0, x1)))
   active(isNat(s(x0)))
   active(isNat(x(x0, x1)))
   active(plus(x0, 0))
   active(plus(x0, s(x1)))
   active(x(x0, 0))
   active(x(x0, s(x1)))
   mark(U11(x0, x1))
   mark(tt)
   mark(U12(x0))
   mark(isNat(x0))
   mark(U21(x0))
   mark(U31(x0, x1))
   mark(U32(x0))
   mark(U41(x0, x1))
   mark(U51(x0, x1, x2))
   mark(U52(x0, x1, x2))
   mark(s(x0))
   mark(plus(x0, x1))
   mark(U61(x0))
   mark(0)
   mark(U71(x0, x1, x2))
   mark(U72(x0, x1, x2))
   mark(x(x0, x1))
   U12(mark(x0))
   U12(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0))
   U32(active(x0))
   U41(mark(x0), x1)
   U41(x0, mark(x1))
   U41(active(x0), x1)
   U41(x0, active(x1))
   U51(mark(x0), x1, x2)
   U51(x0, mark(x1), x2)
   U51(x0, x1, mark(x2))
   U51(active(x0), x1, x2)
   U51(x0, active(x1), x2)
   U51(x0, x1, active(x2))
   U52(mark(x0), x1, x2)
   U52(x0, mark(x1), x2)
   U52(x0, x1, mark(x2))
   U52(active(x0), x1, x2)
   U52(x0, active(x1), x2)
   U52(x0, x1, active(x2))
   U61(mark(x0))
   U61(active(x0))
   U71(mark(x0), x1, x2)
   U71(x0, mark(x1), x2)
   U71(x0, x1, mark(x2))
   U71(active(x0), x1, x2)
   U71(x0, active(x1), x2)
   U71(x0, x1, active(x2))
   U72(mark(x0), x1, x2)
   U72(x0, mark(x1), x2)
   U72(x0, x1, mark(x2))
   U72(active(x0), x1, x2)
   U72(x0, active(x1), x2)
   U72(x0, x1, active(x2))
   x(mark(x0), x1)
   x(x0, mark(x1))
   x(active(x0), x1)
   x(x0, active(x1))


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

(144)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U21(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))

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

   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))

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

(145) UsableRulesReductionPairsProof (EQUIVALENT)
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
   MARK(U21(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

   POL(ACTIVE(x_1)) = 2 + 2*x_1
   POL(MARK(x_1)) = 2 + 2*x_1
   POL(U11(x_1, x_2)) = x_1 + x_2
   POL(U21(x_1)) = 1 + x_1
   POL(isNat(x_1)) = 2 + 2*x_1
   POL(plus(x_1, x_2)) = 1 + 2*x_1 + x_2
   POL(s(x_1)) = 2 + 2*x_1


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

(146)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNat(X)) -> ACTIVE(isNat(X))
   MARK(U11(X1, X2)) -> MARK(X1)

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

   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))

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

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

(148)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))

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

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

   isNat(mark(x0))
   isNat(active(x0))
   U21(mark(x0))
   U21(active(x0))
   s(mark(x0))
   s(active(x0))
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   plus(active(x0), x1)
   plus(x0, active(x1))


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

(150)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))

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

(151) 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:
*MARK(U11(X1, X2)) -> MARK(X1)
The graph contains the following edges 1 > 1


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

(152)
YES