26.02/8.13	YES
26.18/8.15	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
26.18/8.15	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
26.18/8.15	
26.18/8.15	
26.18/8.15	Termination w.r.t. Q of the given QTRS could be proven:
26.18/8.15	
26.18/8.15	(0) QTRS
26.18/8.15	(1) DependencyPairsProof [EQUIVALENT, 117 ms]
26.18/8.15	(2) QDP
26.18/8.15	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
26.18/8.15	(4) AND
26.18/8.15	    (5) QDP
26.18/8.15	        (6) UsableRulesProof [EQUIVALENT, 0 ms]
26.18/8.15	        (7) QDP
26.18/8.15	        (8) QReductionProof [EQUIVALENT, 0 ms]
26.18/8.15	        (9) QDP
26.18/8.15	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
26.18/8.15	        (11) YES
26.18/8.15	    (12) QDP
26.18/8.15	        (13) UsableRulesProof [EQUIVALENT, 0 ms]
26.18/8.15	        (14) QDP
26.18/8.15	        (15) QReductionProof [EQUIVALENT, 0 ms]
26.18/8.15	        (16) QDP
26.18/8.15	        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
26.18/8.15	        (18) YES
26.18/8.15	    (19) QDP
26.18/8.15	        (20) UsableRulesProof [EQUIVALENT, 0 ms]
26.18/8.15	        (21) QDP
26.18/8.15	        (22) QReductionProof [EQUIVALENT, 0 ms]
26.18/8.15	        (23) QDP
26.18/8.15	        (24) QDPSizeChangeProof [EQUIVALENT, 0 ms]
26.18/8.15	        (25) YES
26.18/8.15	    (26) QDP
26.18/8.15	        (27) UsableRulesProof [EQUIVALENT, 0 ms]
26.18/8.15	        (28) QDP
26.18/8.15	        (29) QReductionProof [EQUIVALENT, 0 ms]
26.18/8.15	        (30) QDP
26.18/8.15	        (31) QDPSizeChangeProof [EQUIVALENT, 0 ms]
26.18/8.15	        (32) YES
26.18/8.15	    (33) QDP
26.18/8.15	        (34) UsableRulesProof [EQUIVALENT, 0 ms]
26.18/8.15	        (35) QDP
26.18/8.15	        (36) QReductionProof [EQUIVALENT, 0 ms]
26.18/8.15	        (37) QDP
26.18/8.15	        (38) QDPSizeChangeProof [EQUIVALENT, 0 ms]
26.18/8.15	        (39) YES
26.18/8.15	    (40) QDP
26.18/8.15	        (41) UsableRulesProof [EQUIVALENT, 0 ms]
26.18/8.15	        (42) QDP
26.18/8.15	        (43) QReductionProof [EQUIVALENT, 0 ms]
26.18/8.15	        (44) QDP
26.18/8.15	        (45) QDPSizeChangeProof [EQUIVALENT, 0 ms]
26.18/8.15	        (46) YES
26.18/8.15	    (47) QDP
26.18/8.15	        (48) UsableRulesProof [EQUIVALENT, 0 ms]
26.18/8.15	        (49) QDP
26.18/8.15	        (50) QReductionProof [EQUIVALENT, 0 ms]
26.18/8.15	        (51) QDP
26.18/8.15	        (52) QDPSizeChangeProof [EQUIVALENT, 0 ms]
26.18/8.15	        (53) YES
26.18/8.15	    (54) QDP
26.18/8.15	        (55) UsableRulesProof [EQUIVALENT, 0 ms]
26.18/8.15	        (56) QDP
26.18/8.15	        (57) QReductionProof [EQUIVALENT, 0 ms]
26.18/8.15	        (58) QDP
26.18/8.15	        (59) QDPSizeChangeProof [EQUIVALENT, 0 ms]
26.18/8.15	        (60) YES
26.18/8.15	    (61) QDP
26.18/8.15	        (62) UsableRulesProof [EQUIVALENT, 0 ms]
26.18/8.15	        (63) QDP
26.18/8.15	        (64) QReductionProof [EQUIVALENT, 0 ms]
26.18/8.15	        (65) QDP
26.18/8.15	        (66) QDPSizeChangeProof [EQUIVALENT, 0 ms]
26.18/8.15	        (67) YES
26.18/8.15	    (68) QDP
26.18/8.15	        (69) UsableRulesProof [EQUIVALENT, 0 ms]
26.18/8.15	        (70) QDP
26.18/8.15	        (71) QDPOrderProof [EQUIVALENT, 41 ms]
26.18/8.15	        (72) QDP
26.18/8.15	        (73) QDPOrderProof [EQUIVALENT, 36 ms]
26.18/8.15	        (74) QDP
26.18/8.15	        (75) QDPOrderProof [EQUIVALENT, 31 ms]
26.18/8.15	        (76) QDP
26.18/8.15	        (77) QDPOrderProof [EQUIVALENT, 0 ms]
26.18/8.15	        (78) QDP
26.18/8.15	        (79) QDPOrderProof [EQUIVALENT, 21 ms]
26.18/8.15	        (80) QDP
26.18/8.15	        (81) DependencyGraphProof [EQUIVALENT, 0 ms]
26.18/8.15	        (82) QDP
26.18/8.15	        (83) QDPQMonotonicMRRProof [EQUIVALENT, 109 ms]
26.18/8.15	        (84) QDP
26.18/8.15	        (85) DependencyGraphProof [EQUIVALENT, 0 ms]
26.18/8.15	        (86) QDP
26.18/8.15	        (87) QDPQMonotonicMRRProof [EQUIVALENT, 74 ms]
26.18/8.15	        (88) QDP
26.18/8.15	        (89) DependencyGraphProof [EQUIVALENT, 0 ms]
26.18/8.15	        (90) QDP
26.18/8.15	        (91) QDPQMonotonicMRRProof [EQUIVALENT, 78 ms]
26.18/8.15	        (92) QDP
26.18/8.15	        (93) QDPQMonotonicMRRProof [EQUIVALENT, 92 ms]
26.18/8.15	        (94) QDP
26.18/8.15	        (95) UsableRulesProof [EQUIVALENT, 0 ms]
26.18/8.15	        (96) QDP
26.18/8.15	        (97) QReductionProof [EQUIVALENT, 0 ms]
26.18/8.15	        (98) QDP
26.18/8.15	        (99) QDPSizeChangeProof [EQUIVALENT, 0 ms]
26.18/8.15	        (100) YES
26.18/8.15	
26.18/8.15	
26.18/8.15	----------------------------------------
26.18/8.15	
26.18/8.15	(0)
26.18/8.15	Obligation:
26.18/8.15	Q restricted rewrite system:
26.18/8.15	The TRS R consists of the following rules:
26.18/8.15	
26.18/8.15	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.15	   active(U12(tt)) -> mark(tt)
26.18/8.15	   active(U21(tt)) -> mark(tt)
26.18/8.15	   active(U31(tt, N)) -> mark(N)
26.18/8.15	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.15	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.15	   active(isNat(0)) -> mark(tt)
26.18/8.15	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.15	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.15	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.15	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.15	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.15	   mark(tt) -> active(tt)
26.18/8.15	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.15	   mark(isNat(X)) -> active(isNat(X))
26.18/8.15	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.15	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.15	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.15	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.15	   mark(s(X)) -> active(s(mark(X)))
26.18/8.15	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.15	   mark(0) -> active(0)
26.18/8.15	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.15	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.15	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.15	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.15	   U12(mark(X)) -> U12(X)
26.18/8.15	   U12(active(X)) -> U12(X)
26.18/8.15	   isNat(mark(X)) -> isNat(X)
26.18/8.15	   isNat(active(X)) -> isNat(X)
26.18/8.15	   U21(mark(X)) -> U21(X)
26.18/8.15	   U21(active(X)) -> U21(X)
26.18/8.15	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.15	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.15	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.15	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.15	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.15	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.15	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.15	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.15	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.15	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.15	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.15	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.15	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.15	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.15	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.15	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.15	   s(mark(X)) -> s(X)
26.18/8.15	   s(active(X)) -> s(X)
26.18/8.15	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.15	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.15	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.15	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.15	
26.18/8.15	The set Q consists of the following terms:
26.18/8.15	
26.18/8.15	   active(U11(tt, x0))
26.18/8.15	   active(U12(tt))
26.18/8.15	   active(U21(tt))
26.18/8.15	   active(U31(tt, x0))
26.18/8.15	   active(U41(tt, x0, x1))
26.18/8.15	   active(U42(tt, x0, x1))
26.18/8.15	   active(isNat(0))
26.18/8.15	   active(isNat(plus(x0, x1)))
26.18/8.15	   active(isNat(s(x0)))
26.18/8.15	   active(plus(x0, 0))
26.18/8.15	   active(plus(x0, s(x1)))
26.18/8.15	   mark(U11(x0, x1))
26.18/8.15	   mark(tt)
26.18/8.15	   mark(U12(x0))
26.18/8.15	   mark(isNat(x0))
26.18/8.15	   mark(U21(x0))
26.18/8.15	   mark(U31(x0, x1))
26.18/8.15	   mark(U41(x0, x1, x2))
26.18/8.15	   mark(U42(x0, x1, x2))
26.18/8.15	   mark(s(x0))
26.18/8.15	   mark(plus(x0, x1))
26.18/8.15	   mark(0)
26.18/8.15	   U11(mark(x0), x1)
26.18/8.15	   U11(x0, mark(x1))
26.18/8.15	   U11(active(x0), x1)
26.18/8.15	   U11(x0, active(x1))
26.18/8.15	   U12(mark(x0))
26.18/8.15	   U12(active(x0))
26.18/8.15	   isNat(mark(x0))
26.18/8.15	   isNat(active(x0))
26.18/8.15	   U21(mark(x0))
26.18/8.15	   U21(active(x0))
26.18/8.15	   U31(mark(x0), x1)
26.18/8.15	   U31(x0, mark(x1))
26.18/8.15	   U31(active(x0), x1)
26.18/8.15	   U31(x0, active(x1))
26.18/8.15	   U41(mark(x0), x1, x2)
26.18/8.15	   U41(x0, mark(x1), x2)
26.18/8.15	   U41(x0, x1, mark(x2))
26.18/8.15	   U41(active(x0), x1, x2)
26.18/8.15	   U41(x0, active(x1), x2)
26.18/8.15	   U41(x0, x1, active(x2))
26.18/8.15	   U42(mark(x0), x1, x2)
26.18/8.15	   U42(x0, mark(x1), x2)
26.18/8.15	   U42(x0, x1, mark(x2))
26.18/8.15	   U42(active(x0), x1, x2)
26.18/8.15	   U42(x0, active(x1), x2)
26.18/8.15	   U42(x0, x1, active(x2))
26.18/8.15	   s(mark(x0))
26.18/8.15	   s(active(x0))
26.18/8.15	   plus(mark(x0), x1)
26.18/8.15	   plus(x0, mark(x1))
26.18/8.15	   plus(active(x0), x1)
26.18/8.15	   plus(x0, active(x1))
26.18/8.15	
26.18/8.15	
26.18/8.15	----------------------------------------
26.18/8.15	
26.18/8.15	(1) DependencyPairsProof (EQUIVALENT)
26.18/8.15	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
26.18/8.15	----------------------------------------
26.18/8.15	
26.18/8.15	(2)
26.18/8.15	Obligation:
26.18/8.15	Q DP problem:
26.18/8.15	The TRS P consists of the following rules:
26.18/8.15	
26.18/8.15	   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
26.18/8.15	   ACTIVE(U11(tt, V2)) -> U12^1(isNat(V2))
26.18/8.15	   ACTIVE(U11(tt, V2)) -> ISNAT(V2)
26.18/8.15	   ACTIVE(U12(tt)) -> MARK(tt)
26.18/8.15	   ACTIVE(U21(tt)) -> MARK(tt)
26.18/8.15	   ACTIVE(U31(tt, N)) -> MARK(N)
26.18/8.15	   ACTIVE(U41(tt, M, N)) -> MARK(U42(isNat(N), M, N))
26.18/8.15	   ACTIVE(U41(tt, M, N)) -> U42^1(isNat(N), M, N)
26.18/8.15	   ACTIVE(U41(tt, M, N)) -> ISNAT(N)
26.18/8.15	   ACTIVE(U42(tt, M, N)) -> MARK(s(plus(N, M)))
26.18/8.15	   ACTIVE(U42(tt, M, N)) -> S(plus(N, M))
26.18/8.15	   ACTIVE(U42(tt, M, N)) -> PLUS(N, M)
26.18/8.15	   ACTIVE(isNat(0)) -> MARK(tt)
26.18/8.15	   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
26.18/8.15	   ACTIVE(isNat(plus(V1, V2))) -> U11^1(isNat(V1), V2)
26.18/8.15	   ACTIVE(isNat(plus(V1, V2))) -> ISNAT(V1)
26.18/8.15	   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
26.18/8.15	   ACTIVE(isNat(s(V1))) -> U21^1(isNat(V1))
26.18/8.15	   ACTIVE(isNat(s(V1))) -> ISNAT(V1)
26.18/8.15	   ACTIVE(plus(N, 0)) -> MARK(U31(isNat(N), N))
26.18/8.15	   ACTIVE(plus(N, 0)) -> U31^1(isNat(N), N)
26.18/8.15	   ACTIVE(plus(N, 0)) -> ISNAT(N)
26.18/8.15	   ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N))
26.18/8.15	   ACTIVE(plus(N, s(M))) -> U41^1(isNat(M), M, N)
26.18/8.15	   ACTIVE(plus(N, s(M))) -> ISNAT(M)
26.18/8.15	   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
26.18/8.15	   MARK(U11(X1, X2)) -> U11^1(mark(X1), X2)
26.18/8.15	   MARK(U11(X1, X2)) -> MARK(X1)
26.18/8.15	   MARK(tt) -> ACTIVE(tt)
26.18/8.15	   MARK(U12(X)) -> ACTIVE(U12(mark(X)))
26.18/8.15	   MARK(U12(X)) -> U12^1(mark(X))
26.18/8.15	   MARK(U12(X)) -> MARK(X)
26.18/8.15	   MARK(isNat(X)) -> ACTIVE(isNat(X))
26.18/8.15	   MARK(U21(X)) -> ACTIVE(U21(mark(X)))
26.18/8.15	   MARK(U21(X)) -> U21^1(mark(X))
26.18/8.15	   MARK(U21(X)) -> MARK(X)
26.18/8.15	   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
26.18/8.15	   MARK(U31(X1, X2)) -> U31^1(mark(X1), X2)
26.18/8.15	   MARK(U31(X1, X2)) -> MARK(X1)
26.18/8.15	   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
26.18/8.15	   MARK(U41(X1, X2, X3)) -> U41^1(mark(X1), X2, X3)
26.18/8.15	   MARK(U41(X1, X2, X3)) -> MARK(X1)
26.18/8.15	   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
26.18/8.15	   MARK(U42(X1, X2, X3)) -> U42^1(mark(X1), X2, X3)
26.18/8.15	   MARK(U42(X1, X2, X3)) -> MARK(X1)
26.18/8.15	   MARK(s(X)) -> ACTIVE(s(mark(X)))
26.18/8.15	   MARK(s(X)) -> S(mark(X))
26.18/8.15	   MARK(s(X)) -> MARK(X)
26.18/8.15	   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
26.18/8.15	   MARK(plus(X1, X2)) -> PLUS(mark(X1), mark(X2))
26.18/8.15	   MARK(plus(X1, X2)) -> MARK(X1)
26.18/8.15	   MARK(plus(X1, X2)) -> MARK(X2)
26.18/8.15	   MARK(0) -> ACTIVE(0)
26.18/8.15	   U11^1(mark(X1), X2) -> U11^1(X1, X2)
26.18/8.15	   U11^1(X1, mark(X2)) -> U11^1(X1, X2)
26.18/8.15	   U11^1(active(X1), X2) -> U11^1(X1, X2)
26.18/8.15	   U11^1(X1, active(X2)) -> U11^1(X1, X2)
26.18/8.15	   U12^1(mark(X)) -> U12^1(X)
26.18/8.15	   U12^1(active(X)) -> U12^1(X)
26.18/8.15	   ISNAT(mark(X)) -> ISNAT(X)
26.18/8.15	   ISNAT(active(X)) -> ISNAT(X)
26.18/8.15	   U21^1(mark(X)) -> U21^1(X)
26.18/8.15	   U21^1(active(X)) -> U21^1(X)
26.18/8.15	   U31^1(mark(X1), X2) -> U31^1(X1, X2)
26.18/8.15	   U31^1(X1, mark(X2)) -> U31^1(X1, X2)
26.18/8.15	   U31^1(active(X1), X2) -> U31^1(X1, X2)
26.18/8.15	   U31^1(X1, active(X2)) -> U31^1(X1, X2)
26.18/8.15	   U41^1(mark(X1), X2, X3) -> U41^1(X1, X2, X3)
26.18/8.15	   U41^1(X1, mark(X2), X3) -> U41^1(X1, X2, X3)
26.18/8.15	   U41^1(X1, X2, mark(X3)) -> U41^1(X1, X2, X3)
26.18/8.15	   U41^1(active(X1), X2, X3) -> U41^1(X1, X2, X3)
26.18/8.15	   U41^1(X1, active(X2), X3) -> U41^1(X1, X2, X3)
26.18/8.15	   U41^1(X1, X2, active(X3)) -> U41^1(X1, X2, X3)
26.18/8.15	   U42^1(mark(X1), X2, X3) -> U42^1(X1, X2, X3)
26.18/8.15	   U42^1(X1, mark(X2), X3) -> U42^1(X1, X2, X3)
26.18/8.15	   U42^1(X1, X2, mark(X3)) -> U42^1(X1, X2, X3)
26.18/8.15	   U42^1(active(X1), X2, X3) -> U42^1(X1, X2, X3)
26.18/8.15	   U42^1(X1, active(X2), X3) -> U42^1(X1, X2, X3)
26.18/8.15	   U42^1(X1, X2, active(X3)) -> U42^1(X1, X2, X3)
26.18/8.15	   S(mark(X)) -> S(X)
26.18/8.15	   S(active(X)) -> S(X)
26.18/8.15	   PLUS(mark(X1), X2) -> PLUS(X1, X2)
26.18/8.15	   PLUS(X1, mark(X2)) -> PLUS(X1, X2)
26.18/8.15	   PLUS(active(X1), X2) -> PLUS(X1, X2)
26.18/8.15	   PLUS(X1, active(X2)) -> PLUS(X1, X2)
26.18/8.15	
26.18/8.15	The TRS R consists of the following rules:
26.18/8.15	
26.18/8.15	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.15	   active(U12(tt)) -> mark(tt)
26.18/8.15	   active(U21(tt)) -> mark(tt)
26.18/8.15	   active(U31(tt, N)) -> mark(N)
26.18/8.15	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.15	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.15	   active(isNat(0)) -> mark(tt)
26.18/8.15	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.15	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.15	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.15	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.15	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.15	   mark(tt) -> active(tt)
26.18/8.15	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.15	   mark(isNat(X)) -> active(isNat(X))
26.18/8.15	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.15	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.15	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.15	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.15	   mark(s(X)) -> active(s(mark(X)))
26.18/8.15	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.15	   mark(0) -> active(0)
26.18/8.15	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.15	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.15	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.15	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.15	   U12(mark(X)) -> U12(X)
26.18/8.15	   U12(active(X)) -> U12(X)
26.18/8.15	   isNat(mark(X)) -> isNat(X)
26.18/8.15	   isNat(active(X)) -> isNat(X)
26.18/8.15	   U21(mark(X)) -> U21(X)
26.18/8.15	   U21(active(X)) -> U21(X)
26.18/8.15	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.15	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.15	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.15	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.15	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.15	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.15	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.15	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.16	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.16	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.16	   s(mark(X)) -> s(X)
26.18/8.16	   s(active(X)) -> s(X)
26.18/8.16	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.16	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.16	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.16	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.16	
26.18/8.16	The set Q consists of the following terms:
26.18/8.16	
26.18/8.16	   active(U11(tt, x0))
26.18/8.16	   active(U12(tt))
26.18/8.16	   active(U21(tt))
26.18/8.16	   active(U31(tt, x0))
26.18/8.16	   active(U41(tt, x0, x1))
26.18/8.16	   active(U42(tt, x0, x1))
26.18/8.16	   active(isNat(0))
26.18/8.16	   active(isNat(plus(x0, x1)))
26.18/8.16	   active(isNat(s(x0)))
26.18/8.16	   active(plus(x0, 0))
26.18/8.16	   active(plus(x0, s(x1)))
26.18/8.16	   mark(U11(x0, x1))
26.18/8.16	   mark(tt)
26.18/8.16	   mark(U12(x0))
26.18/8.16	   mark(isNat(x0))
26.18/8.16	   mark(U21(x0))
26.18/8.16	   mark(U31(x0, x1))
26.18/8.16	   mark(U41(x0, x1, x2))
26.18/8.16	   mark(U42(x0, x1, x2))
26.18/8.16	   mark(s(x0))
26.18/8.16	   mark(plus(x0, x1))
26.18/8.16	   mark(0)
26.18/8.16	   U11(mark(x0), x1)
26.18/8.16	   U11(x0, mark(x1))
26.18/8.16	   U11(active(x0), x1)
26.18/8.16	   U11(x0, active(x1))
26.18/8.16	   U12(mark(x0))
26.18/8.16	   U12(active(x0))
26.18/8.16	   isNat(mark(x0))
26.18/8.16	   isNat(active(x0))
26.18/8.16	   U21(mark(x0))
26.18/8.16	   U21(active(x0))
26.18/8.16	   U31(mark(x0), x1)
26.18/8.16	   U31(x0, mark(x1))
26.18/8.16	   U31(active(x0), x1)
26.18/8.16	   U31(x0, active(x1))
26.18/8.16	   U41(mark(x0), x1, x2)
26.18/8.16	   U41(x0, mark(x1), x2)
26.18/8.16	   U41(x0, x1, mark(x2))
26.18/8.16	   U41(active(x0), x1, x2)
26.18/8.16	   U41(x0, active(x1), x2)
26.18/8.16	   U41(x0, x1, active(x2))
26.18/8.16	   U42(mark(x0), x1, x2)
26.18/8.16	   U42(x0, mark(x1), x2)
26.18/8.16	   U42(x0, x1, mark(x2))
26.18/8.16	   U42(active(x0), x1, x2)
26.18/8.16	   U42(x0, active(x1), x2)
26.18/8.16	   U42(x0, x1, active(x2))
26.18/8.16	   s(mark(x0))
26.18/8.16	   s(active(x0))
26.18/8.16	   plus(mark(x0), x1)
26.18/8.16	   plus(x0, mark(x1))
26.18/8.16	   plus(active(x0), x1)
26.18/8.16	   plus(x0, active(x1))
26.18/8.16	
26.18/8.16	We have to consider all minimal (P,Q,R)-chains.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(3) DependencyGraphProof (EQUIVALENT)
26.18/8.16	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 10 SCCs with 27 less nodes.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(4)
26.18/8.16	Complex Obligation (AND)
26.18/8.16	
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(5)
26.18/8.16	Obligation:
26.18/8.16	Q DP problem:
26.18/8.16	The TRS P consists of the following rules:
26.18/8.16	
26.18/8.16	   PLUS(X1, mark(X2)) -> PLUS(X1, X2)
26.18/8.16	   PLUS(mark(X1), X2) -> PLUS(X1, X2)
26.18/8.16	   PLUS(active(X1), X2) -> PLUS(X1, X2)
26.18/8.16	   PLUS(X1, active(X2)) -> PLUS(X1, X2)
26.18/8.16	
26.18/8.16	The TRS R consists of the following rules:
26.18/8.16	
26.18/8.16	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.16	   active(U12(tt)) -> mark(tt)
26.18/8.16	   active(U21(tt)) -> mark(tt)
26.18/8.16	   active(U31(tt, N)) -> mark(N)
26.18/8.16	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.16	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.16	   active(isNat(0)) -> mark(tt)
26.18/8.16	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.16	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.16	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.16	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.16	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.16	   mark(tt) -> active(tt)
26.18/8.16	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.16	   mark(isNat(X)) -> active(isNat(X))
26.18/8.16	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.16	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.16	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.16	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.16	   mark(s(X)) -> active(s(mark(X)))
26.18/8.16	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.16	   mark(0) -> active(0)
26.18/8.16	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.16	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.16	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.16	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.16	   U12(mark(X)) -> U12(X)
26.18/8.16	   U12(active(X)) -> U12(X)
26.18/8.16	   isNat(mark(X)) -> isNat(X)
26.18/8.16	   isNat(active(X)) -> isNat(X)
26.18/8.16	   U21(mark(X)) -> U21(X)
26.18/8.16	   U21(active(X)) -> U21(X)
26.18/8.16	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.16	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.16	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.16	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.16	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.16	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.16	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.16	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.16	   s(mark(X)) -> s(X)
26.18/8.16	   s(active(X)) -> s(X)
26.18/8.16	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.16	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.16	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.16	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.16	
26.18/8.16	The set Q consists of the following terms:
26.18/8.16	
26.18/8.16	   active(U11(tt, x0))
26.18/8.16	   active(U12(tt))
26.18/8.16	   active(U21(tt))
26.18/8.16	   active(U31(tt, x0))
26.18/8.16	   active(U41(tt, x0, x1))
26.18/8.16	   active(U42(tt, x0, x1))
26.18/8.16	   active(isNat(0))
26.18/8.16	   active(isNat(plus(x0, x1)))
26.18/8.16	   active(isNat(s(x0)))
26.18/8.16	   active(plus(x0, 0))
26.18/8.16	   active(plus(x0, s(x1)))
26.18/8.16	   mark(U11(x0, x1))
26.18/8.16	   mark(tt)
26.18/8.16	   mark(U12(x0))
26.18/8.16	   mark(isNat(x0))
26.18/8.16	   mark(U21(x0))
26.18/8.16	   mark(U31(x0, x1))
26.18/8.16	   mark(U41(x0, x1, x2))
26.18/8.16	   mark(U42(x0, x1, x2))
26.18/8.16	   mark(s(x0))
26.18/8.16	   mark(plus(x0, x1))
26.18/8.16	   mark(0)
26.18/8.16	   U11(mark(x0), x1)
26.18/8.16	   U11(x0, mark(x1))
26.18/8.16	   U11(active(x0), x1)
26.18/8.16	   U11(x0, active(x1))
26.18/8.16	   U12(mark(x0))
26.18/8.16	   U12(active(x0))
26.18/8.16	   isNat(mark(x0))
26.18/8.16	   isNat(active(x0))
26.18/8.16	   U21(mark(x0))
26.18/8.16	   U21(active(x0))
26.18/8.16	   U31(mark(x0), x1)
26.18/8.16	   U31(x0, mark(x1))
26.18/8.16	   U31(active(x0), x1)
26.18/8.16	   U31(x0, active(x1))
26.18/8.16	   U41(mark(x0), x1, x2)
26.18/8.16	   U41(x0, mark(x1), x2)
26.18/8.16	   U41(x0, x1, mark(x2))
26.18/8.16	   U41(active(x0), x1, x2)
26.18/8.16	   U41(x0, active(x1), x2)
26.18/8.16	   U41(x0, x1, active(x2))
26.18/8.16	   U42(mark(x0), x1, x2)
26.18/8.16	   U42(x0, mark(x1), x2)
26.18/8.16	   U42(x0, x1, mark(x2))
26.18/8.16	   U42(active(x0), x1, x2)
26.18/8.16	   U42(x0, active(x1), x2)
26.18/8.16	   U42(x0, x1, active(x2))
26.18/8.16	   s(mark(x0))
26.18/8.16	   s(active(x0))
26.18/8.16	   plus(mark(x0), x1)
26.18/8.16	   plus(x0, mark(x1))
26.18/8.16	   plus(active(x0), x1)
26.18/8.16	   plus(x0, active(x1))
26.18/8.16	
26.18/8.16	We have to consider all minimal (P,Q,R)-chains.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(6) UsableRulesProof (EQUIVALENT)
26.18/8.16	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.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(7)
26.18/8.16	Obligation:
26.18/8.16	Q DP problem:
26.18/8.16	The TRS P consists of the following rules:
26.18/8.16	
26.18/8.16	   PLUS(X1, mark(X2)) -> PLUS(X1, X2)
26.18/8.16	   PLUS(mark(X1), X2) -> PLUS(X1, X2)
26.18/8.16	   PLUS(active(X1), X2) -> PLUS(X1, X2)
26.18/8.16	   PLUS(X1, active(X2)) -> PLUS(X1, X2)
26.18/8.16	
26.18/8.16	R is empty.
26.18/8.16	The set Q consists of the following terms:
26.18/8.16	
26.18/8.16	   active(U11(tt, x0))
26.18/8.16	   active(U12(tt))
26.18/8.16	   active(U21(tt))
26.18/8.16	   active(U31(tt, x0))
26.18/8.16	   active(U41(tt, x0, x1))
26.18/8.16	   active(U42(tt, x0, x1))
26.18/8.16	   active(isNat(0))
26.18/8.16	   active(isNat(plus(x0, x1)))
26.18/8.16	   active(isNat(s(x0)))
26.18/8.16	   active(plus(x0, 0))
26.18/8.16	   active(plus(x0, s(x1)))
26.18/8.16	   mark(U11(x0, x1))
26.18/8.16	   mark(tt)
26.18/8.16	   mark(U12(x0))
26.18/8.16	   mark(isNat(x0))
26.18/8.16	   mark(U21(x0))
26.18/8.16	   mark(U31(x0, x1))
26.18/8.16	   mark(U41(x0, x1, x2))
26.18/8.16	   mark(U42(x0, x1, x2))
26.18/8.16	   mark(s(x0))
26.18/8.16	   mark(plus(x0, x1))
26.18/8.16	   mark(0)
26.18/8.16	   U11(mark(x0), x1)
26.18/8.16	   U11(x0, mark(x1))
26.18/8.16	   U11(active(x0), x1)
26.18/8.16	   U11(x0, active(x1))
26.18/8.16	   U12(mark(x0))
26.18/8.16	   U12(active(x0))
26.18/8.16	   isNat(mark(x0))
26.18/8.16	   isNat(active(x0))
26.18/8.16	   U21(mark(x0))
26.18/8.16	   U21(active(x0))
26.18/8.16	   U31(mark(x0), x1)
26.18/8.16	   U31(x0, mark(x1))
26.18/8.16	   U31(active(x0), x1)
26.18/8.16	   U31(x0, active(x1))
26.18/8.16	   U41(mark(x0), x1, x2)
26.18/8.16	   U41(x0, mark(x1), x2)
26.18/8.16	   U41(x0, x1, mark(x2))
26.18/8.16	   U41(active(x0), x1, x2)
26.18/8.16	   U41(x0, active(x1), x2)
26.18/8.16	   U41(x0, x1, active(x2))
26.18/8.16	   U42(mark(x0), x1, x2)
26.18/8.16	   U42(x0, mark(x1), x2)
26.18/8.16	   U42(x0, x1, mark(x2))
26.18/8.16	   U42(active(x0), x1, x2)
26.18/8.16	   U42(x0, active(x1), x2)
26.18/8.16	   U42(x0, x1, active(x2))
26.18/8.16	   s(mark(x0))
26.18/8.16	   s(active(x0))
26.18/8.16	   plus(mark(x0), x1)
26.18/8.16	   plus(x0, mark(x1))
26.18/8.16	   plus(active(x0), x1)
26.18/8.16	   plus(x0, active(x1))
26.18/8.16	
26.18/8.16	We have to consider all minimal (P,Q,R)-chains.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(8) QReductionProof (EQUIVALENT)
26.18/8.16	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
26.18/8.16	
26.18/8.16	   U11(mark(x0), x1)
26.18/8.16	   U11(x0, mark(x1))
26.18/8.16	   U11(active(x0), x1)
26.18/8.16	   U11(x0, active(x1))
26.18/8.16	   U12(mark(x0))
26.18/8.16	   U12(active(x0))
26.18/8.16	   isNat(mark(x0))
26.18/8.16	   isNat(active(x0))
26.18/8.16	   U21(mark(x0))
26.18/8.16	   U21(active(x0))
26.18/8.16	   U31(mark(x0), x1)
26.18/8.16	   U31(x0, mark(x1))
26.18/8.16	   U31(active(x0), x1)
26.18/8.16	   U31(x0, active(x1))
26.18/8.16	   U41(mark(x0), x1, x2)
26.18/8.16	   U41(x0, mark(x1), x2)
26.18/8.16	   U41(x0, x1, mark(x2))
26.18/8.16	   U41(active(x0), x1, x2)
26.18/8.16	   U41(x0, active(x1), x2)
26.18/8.16	   U41(x0, x1, active(x2))
26.18/8.16	   U42(mark(x0), x1, x2)
26.18/8.16	   U42(x0, mark(x1), x2)
26.18/8.16	   U42(x0, x1, mark(x2))
26.18/8.16	   U42(active(x0), x1, x2)
26.18/8.16	   U42(x0, active(x1), x2)
26.18/8.16	   U42(x0, x1, active(x2))
26.18/8.16	   s(mark(x0))
26.18/8.16	   s(active(x0))
26.18/8.16	   plus(mark(x0), x1)
26.18/8.16	   plus(x0, mark(x1))
26.18/8.16	   plus(active(x0), x1)
26.18/8.16	   plus(x0, active(x1))
26.18/8.16	
26.18/8.16	
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(9)
26.18/8.16	Obligation:
26.18/8.16	Q DP problem:
26.18/8.16	The TRS P consists of the following rules:
26.18/8.16	
26.18/8.16	   PLUS(X1, mark(X2)) -> PLUS(X1, X2)
26.18/8.16	   PLUS(mark(X1), X2) -> PLUS(X1, X2)
26.18/8.16	   PLUS(active(X1), X2) -> PLUS(X1, X2)
26.18/8.16	   PLUS(X1, active(X2)) -> PLUS(X1, X2)
26.18/8.16	
26.18/8.16	R is empty.
26.18/8.16	The set Q consists of the following terms:
26.18/8.16	
26.18/8.16	   active(U11(tt, x0))
26.18/8.16	   active(U12(tt))
26.18/8.16	   active(U21(tt))
26.18/8.16	   active(U31(tt, x0))
26.18/8.16	   active(U41(tt, x0, x1))
26.18/8.16	   active(U42(tt, x0, x1))
26.18/8.16	   active(isNat(0))
26.18/8.16	   active(isNat(plus(x0, x1)))
26.18/8.16	   active(isNat(s(x0)))
26.18/8.16	   active(plus(x0, 0))
26.18/8.16	   active(plus(x0, s(x1)))
26.18/8.16	   mark(U11(x0, x1))
26.18/8.16	   mark(tt)
26.18/8.16	   mark(U12(x0))
26.18/8.16	   mark(isNat(x0))
26.18/8.16	   mark(U21(x0))
26.18/8.16	   mark(U31(x0, x1))
26.18/8.16	   mark(U41(x0, x1, x2))
26.18/8.16	   mark(U42(x0, x1, x2))
26.18/8.16	   mark(s(x0))
26.18/8.16	   mark(plus(x0, x1))
26.18/8.16	   mark(0)
26.18/8.16	
26.18/8.16	We have to consider all minimal (P,Q,R)-chains.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(10) QDPSizeChangeProof (EQUIVALENT)
26.18/8.16	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. 
26.18/8.16	
26.18/8.16	From the DPs we obtained the following set of size-change graphs:
26.18/8.16	*PLUS(X1, mark(X2)) -> PLUS(X1, X2)
26.18/8.16	The graph contains the following edges 1 >= 1, 2 > 2
26.18/8.16	
26.18/8.16	
26.18/8.16	*PLUS(mark(X1), X2) -> PLUS(X1, X2)
26.18/8.16	The graph contains the following edges 1 > 1, 2 >= 2
26.18/8.16	
26.18/8.16	
26.18/8.16	*PLUS(active(X1), X2) -> PLUS(X1, X2)
26.18/8.16	The graph contains the following edges 1 > 1, 2 >= 2
26.18/8.16	
26.18/8.16	
26.18/8.16	*PLUS(X1, active(X2)) -> PLUS(X1, X2)
26.18/8.16	The graph contains the following edges 1 >= 1, 2 > 2
26.18/8.16	
26.18/8.16	
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(11)
26.18/8.16	YES
26.18/8.16	
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(12)
26.18/8.16	Obligation:
26.18/8.16	Q DP problem:
26.18/8.16	The TRS P consists of the following rules:
26.18/8.16	
26.18/8.16	   S(active(X)) -> S(X)
26.18/8.16	   S(mark(X)) -> S(X)
26.18/8.16	
26.18/8.16	The TRS R consists of the following rules:
26.18/8.16	
26.18/8.16	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.16	   active(U12(tt)) -> mark(tt)
26.18/8.16	   active(U21(tt)) -> mark(tt)
26.18/8.16	   active(U31(tt, N)) -> mark(N)
26.18/8.16	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.16	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.16	   active(isNat(0)) -> mark(tt)
26.18/8.16	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.16	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.16	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.16	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.16	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.16	   mark(tt) -> active(tt)
26.18/8.16	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.16	   mark(isNat(X)) -> active(isNat(X))
26.18/8.16	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.16	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.16	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.16	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.16	   mark(s(X)) -> active(s(mark(X)))
26.18/8.16	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.16	   mark(0) -> active(0)
26.18/8.16	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.16	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.16	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.16	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.16	   U12(mark(X)) -> U12(X)
26.18/8.16	   U12(active(X)) -> U12(X)
26.18/8.16	   isNat(mark(X)) -> isNat(X)
26.18/8.16	   isNat(active(X)) -> isNat(X)
26.18/8.16	   U21(mark(X)) -> U21(X)
26.18/8.16	   U21(active(X)) -> U21(X)
26.18/8.16	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.16	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.16	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.16	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.16	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.16	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.16	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.16	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.16	   s(mark(X)) -> s(X)
26.18/8.16	   s(active(X)) -> s(X)
26.18/8.16	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.16	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.16	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.16	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.16	
26.18/8.16	The set Q consists of the following terms:
26.18/8.16	
26.18/8.16	   active(U11(tt, x0))
26.18/8.16	   active(U12(tt))
26.18/8.16	   active(U21(tt))
26.18/8.16	   active(U31(tt, x0))
26.18/8.16	   active(U41(tt, x0, x1))
26.18/8.16	   active(U42(tt, x0, x1))
26.18/8.16	   active(isNat(0))
26.18/8.16	   active(isNat(plus(x0, x1)))
26.18/8.16	   active(isNat(s(x0)))
26.18/8.16	   active(plus(x0, 0))
26.18/8.16	   active(plus(x0, s(x1)))
26.18/8.16	   mark(U11(x0, x1))
26.18/8.16	   mark(tt)
26.18/8.16	   mark(U12(x0))
26.18/8.16	   mark(isNat(x0))
26.18/8.16	   mark(U21(x0))
26.18/8.16	   mark(U31(x0, x1))
26.18/8.16	   mark(U41(x0, x1, x2))
26.18/8.16	   mark(U42(x0, x1, x2))
26.18/8.16	   mark(s(x0))
26.18/8.16	   mark(plus(x0, x1))
26.18/8.16	   mark(0)
26.18/8.16	   U11(mark(x0), x1)
26.18/8.16	   U11(x0, mark(x1))
26.18/8.16	   U11(active(x0), x1)
26.18/8.16	   U11(x0, active(x1))
26.18/8.16	   U12(mark(x0))
26.18/8.16	   U12(active(x0))
26.18/8.16	   isNat(mark(x0))
26.18/8.16	   isNat(active(x0))
26.18/8.16	   U21(mark(x0))
26.18/8.16	   U21(active(x0))
26.18/8.16	   U31(mark(x0), x1)
26.18/8.16	   U31(x0, mark(x1))
26.18/8.16	   U31(active(x0), x1)
26.18/8.16	   U31(x0, active(x1))
26.18/8.16	   U41(mark(x0), x1, x2)
26.18/8.16	   U41(x0, mark(x1), x2)
26.18/8.16	   U41(x0, x1, mark(x2))
26.18/8.16	   U41(active(x0), x1, x2)
26.18/8.16	   U41(x0, active(x1), x2)
26.18/8.16	   U41(x0, x1, active(x2))
26.18/8.16	   U42(mark(x0), x1, x2)
26.18/8.16	   U42(x0, mark(x1), x2)
26.18/8.16	   U42(x0, x1, mark(x2))
26.18/8.16	   U42(active(x0), x1, x2)
26.18/8.16	   U42(x0, active(x1), x2)
26.18/8.16	   U42(x0, x1, active(x2))
26.18/8.16	   s(mark(x0))
26.18/8.16	   s(active(x0))
26.18/8.16	   plus(mark(x0), x1)
26.18/8.16	   plus(x0, mark(x1))
26.18/8.16	   plus(active(x0), x1)
26.18/8.16	   plus(x0, active(x1))
26.18/8.16	
26.18/8.16	We have to consider all minimal (P,Q,R)-chains.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(13) UsableRulesProof (EQUIVALENT)
26.18/8.16	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.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(14)
26.18/8.16	Obligation:
26.18/8.16	Q DP problem:
26.18/8.16	The TRS P consists of the following rules:
26.18/8.16	
26.18/8.16	   S(active(X)) -> S(X)
26.18/8.16	   S(mark(X)) -> S(X)
26.18/8.16	
26.18/8.16	R is empty.
26.18/8.16	The set Q consists of the following terms:
26.18/8.16	
26.18/8.16	   active(U11(tt, x0))
26.18/8.16	   active(U12(tt))
26.18/8.16	   active(U21(tt))
26.18/8.16	   active(U31(tt, x0))
26.18/8.16	   active(U41(tt, x0, x1))
26.18/8.16	   active(U42(tt, x0, x1))
26.18/8.16	   active(isNat(0))
26.18/8.16	   active(isNat(plus(x0, x1)))
26.18/8.16	   active(isNat(s(x0)))
26.18/8.16	   active(plus(x0, 0))
26.18/8.16	   active(plus(x0, s(x1)))
26.18/8.16	   mark(U11(x0, x1))
26.18/8.16	   mark(tt)
26.18/8.16	   mark(U12(x0))
26.18/8.16	   mark(isNat(x0))
26.18/8.16	   mark(U21(x0))
26.18/8.16	   mark(U31(x0, x1))
26.18/8.16	   mark(U41(x0, x1, x2))
26.18/8.16	   mark(U42(x0, x1, x2))
26.18/8.16	   mark(s(x0))
26.18/8.16	   mark(plus(x0, x1))
26.18/8.16	   mark(0)
26.18/8.16	   U11(mark(x0), x1)
26.18/8.16	   U11(x0, mark(x1))
26.18/8.16	   U11(active(x0), x1)
26.18/8.16	   U11(x0, active(x1))
26.18/8.16	   U12(mark(x0))
26.18/8.16	   U12(active(x0))
26.18/8.16	   isNat(mark(x0))
26.18/8.16	   isNat(active(x0))
26.18/8.16	   U21(mark(x0))
26.18/8.16	   U21(active(x0))
26.18/8.16	   U31(mark(x0), x1)
26.18/8.16	   U31(x0, mark(x1))
26.18/8.16	   U31(active(x0), x1)
26.18/8.16	   U31(x0, active(x1))
26.18/8.16	   U41(mark(x0), x1, x2)
26.18/8.16	   U41(x0, mark(x1), x2)
26.18/8.16	   U41(x0, x1, mark(x2))
26.18/8.16	   U41(active(x0), x1, x2)
26.18/8.16	   U41(x0, active(x1), x2)
26.18/8.16	   U41(x0, x1, active(x2))
26.18/8.16	   U42(mark(x0), x1, x2)
26.18/8.16	   U42(x0, mark(x1), x2)
26.18/8.16	   U42(x0, x1, mark(x2))
26.18/8.16	   U42(active(x0), x1, x2)
26.18/8.16	   U42(x0, active(x1), x2)
26.18/8.16	   U42(x0, x1, active(x2))
26.18/8.16	   s(mark(x0))
26.18/8.16	   s(active(x0))
26.18/8.16	   plus(mark(x0), x1)
26.18/8.16	   plus(x0, mark(x1))
26.18/8.16	   plus(active(x0), x1)
26.18/8.16	   plus(x0, active(x1))
26.18/8.16	
26.18/8.16	We have to consider all minimal (P,Q,R)-chains.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(15) QReductionProof (EQUIVALENT)
26.18/8.16	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
26.18/8.16	
26.18/8.16	   U11(mark(x0), x1)
26.18/8.16	   U11(x0, mark(x1))
26.18/8.16	   U11(active(x0), x1)
26.18/8.16	   U11(x0, active(x1))
26.18/8.16	   U12(mark(x0))
26.18/8.16	   U12(active(x0))
26.18/8.16	   isNat(mark(x0))
26.18/8.16	   isNat(active(x0))
26.18/8.16	   U21(mark(x0))
26.18/8.16	   U21(active(x0))
26.18/8.16	   U31(mark(x0), x1)
26.18/8.16	   U31(x0, mark(x1))
26.18/8.16	   U31(active(x0), x1)
26.18/8.16	   U31(x0, active(x1))
26.18/8.16	   U41(mark(x0), x1, x2)
26.18/8.16	   U41(x0, mark(x1), x2)
26.18/8.16	   U41(x0, x1, mark(x2))
26.18/8.16	   U41(active(x0), x1, x2)
26.18/8.16	   U41(x0, active(x1), x2)
26.18/8.16	   U41(x0, x1, active(x2))
26.18/8.16	   U42(mark(x0), x1, x2)
26.18/8.16	   U42(x0, mark(x1), x2)
26.18/8.16	   U42(x0, x1, mark(x2))
26.18/8.16	   U42(active(x0), x1, x2)
26.18/8.16	   U42(x0, active(x1), x2)
26.18/8.16	   U42(x0, x1, active(x2))
26.18/8.16	   s(mark(x0))
26.18/8.16	   s(active(x0))
26.18/8.16	   plus(mark(x0), x1)
26.18/8.16	   plus(x0, mark(x1))
26.18/8.16	   plus(active(x0), x1)
26.18/8.16	   plus(x0, active(x1))
26.18/8.16	
26.18/8.16	
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(16)
26.18/8.16	Obligation:
26.18/8.16	Q DP problem:
26.18/8.16	The TRS P consists of the following rules:
26.18/8.16	
26.18/8.16	   S(active(X)) -> S(X)
26.18/8.16	   S(mark(X)) -> S(X)
26.18/8.16	
26.18/8.16	R is empty.
26.18/8.16	The set Q consists of the following terms:
26.18/8.16	
26.18/8.16	   active(U11(tt, x0))
26.18/8.16	   active(U12(tt))
26.18/8.16	   active(U21(tt))
26.18/8.16	   active(U31(tt, x0))
26.18/8.16	   active(U41(tt, x0, x1))
26.18/8.16	   active(U42(tt, x0, x1))
26.18/8.16	   active(isNat(0))
26.18/8.16	   active(isNat(plus(x0, x1)))
26.18/8.16	   active(isNat(s(x0)))
26.18/8.16	   active(plus(x0, 0))
26.18/8.16	   active(plus(x0, s(x1)))
26.18/8.16	   mark(U11(x0, x1))
26.18/8.16	   mark(tt)
26.18/8.16	   mark(U12(x0))
26.18/8.16	   mark(isNat(x0))
26.18/8.16	   mark(U21(x0))
26.18/8.16	   mark(U31(x0, x1))
26.18/8.16	   mark(U41(x0, x1, x2))
26.18/8.16	   mark(U42(x0, x1, x2))
26.18/8.16	   mark(s(x0))
26.18/8.16	   mark(plus(x0, x1))
26.18/8.16	   mark(0)
26.18/8.16	
26.18/8.16	We have to consider all minimal (P,Q,R)-chains.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(17) QDPSizeChangeProof (EQUIVALENT)
26.18/8.16	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. 
26.18/8.16	
26.18/8.16	From the DPs we obtained the following set of size-change graphs:
26.18/8.16	*S(active(X)) -> S(X)
26.18/8.16	The graph contains the following edges 1 > 1
26.18/8.16	
26.18/8.16	
26.18/8.16	*S(mark(X)) -> S(X)
26.18/8.16	The graph contains the following edges 1 > 1
26.18/8.16	
26.18/8.16	
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(18)
26.18/8.16	YES
26.18/8.16	
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(19)
26.18/8.16	Obligation:
26.18/8.16	Q DP problem:
26.18/8.16	The TRS P consists of the following rules:
26.18/8.16	
26.18/8.16	   U42^1(X1, mark(X2), X3) -> U42^1(X1, X2, X3)
26.18/8.16	   U42^1(mark(X1), X2, X3) -> U42^1(X1, X2, X3)
26.18/8.16	   U42^1(X1, X2, mark(X3)) -> U42^1(X1, X2, X3)
26.18/8.16	   U42^1(active(X1), X2, X3) -> U42^1(X1, X2, X3)
26.18/8.16	   U42^1(X1, active(X2), X3) -> U42^1(X1, X2, X3)
26.18/8.16	   U42^1(X1, X2, active(X3)) -> U42^1(X1, X2, X3)
26.18/8.16	
26.18/8.16	The TRS R consists of the following rules:
26.18/8.16	
26.18/8.16	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.16	   active(U12(tt)) -> mark(tt)
26.18/8.16	   active(U21(tt)) -> mark(tt)
26.18/8.16	   active(U31(tt, N)) -> mark(N)
26.18/8.16	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.16	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.16	   active(isNat(0)) -> mark(tt)
26.18/8.16	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.16	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.16	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.16	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.16	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.16	   mark(tt) -> active(tt)
26.18/8.16	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.16	   mark(isNat(X)) -> active(isNat(X))
26.18/8.16	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.16	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.16	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.16	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.16	   mark(s(X)) -> active(s(mark(X)))
26.18/8.16	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.16	   mark(0) -> active(0)
26.18/8.16	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.16	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.16	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.16	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.16	   U12(mark(X)) -> U12(X)
26.18/8.16	   U12(active(X)) -> U12(X)
26.18/8.16	   isNat(mark(X)) -> isNat(X)
26.18/8.16	   isNat(active(X)) -> isNat(X)
26.18/8.16	   U21(mark(X)) -> U21(X)
26.18/8.16	   U21(active(X)) -> U21(X)
26.18/8.16	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.16	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.16	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.16	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.16	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.16	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.16	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.16	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.16	   s(mark(X)) -> s(X)
26.18/8.16	   s(active(X)) -> s(X)
26.18/8.16	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.16	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.16	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.16	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.16	
26.18/8.16	The set Q consists of the following terms:
26.18/8.16	
26.18/8.16	   active(U11(tt, x0))
26.18/8.16	   active(U12(tt))
26.18/8.16	   active(U21(tt))
26.18/8.16	   active(U31(tt, x0))
26.18/8.16	   active(U41(tt, x0, x1))
26.18/8.16	   active(U42(tt, x0, x1))
26.18/8.16	   active(isNat(0))
26.18/8.16	   active(isNat(plus(x0, x1)))
26.18/8.16	   active(isNat(s(x0)))
26.18/8.16	   active(plus(x0, 0))
26.18/8.16	   active(plus(x0, s(x1)))
26.18/8.16	   mark(U11(x0, x1))
26.18/8.16	   mark(tt)
26.18/8.16	   mark(U12(x0))
26.18/8.16	   mark(isNat(x0))
26.18/8.16	   mark(U21(x0))
26.18/8.16	   mark(U31(x0, x1))
26.18/8.16	   mark(U41(x0, x1, x2))
26.18/8.16	   mark(U42(x0, x1, x2))
26.18/8.16	   mark(s(x0))
26.18/8.16	   mark(plus(x0, x1))
26.18/8.16	   mark(0)
26.18/8.16	   U11(mark(x0), x1)
26.18/8.16	   U11(x0, mark(x1))
26.18/8.16	   U11(active(x0), x1)
26.18/8.16	   U11(x0, active(x1))
26.18/8.16	   U12(mark(x0))
26.18/8.16	   U12(active(x0))
26.18/8.16	   isNat(mark(x0))
26.18/8.16	   isNat(active(x0))
26.18/8.16	   U21(mark(x0))
26.18/8.16	   U21(active(x0))
26.18/8.16	   U31(mark(x0), x1)
26.18/8.16	   U31(x0, mark(x1))
26.18/8.16	   U31(active(x0), x1)
26.18/8.16	   U31(x0, active(x1))
26.18/8.16	   U41(mark(x0), x1, x2)
26.18/8.16	   U41(x0, mark(x1), x2)
26.18/8.16	   U41(x0, x1, mark(x2))
26.18/8.16	   U41(active(x0), x1, x2)
26.18/8.16	   U41(x0, active(x1), x2)
26.18/8.16	   U41(x0, x1, active(x2))
26.18/8.16	   U42(mark(x0), x1, x2)
26.18/8.16	   U42(x0, mark(x1), x2)
26.18/8.16	   U42(x0, x1, mark(x2))
26.18/8.16	   U42(active(x0), x1, x2)
26.18/8.16	   U42(x0, active(x1), x2)
26.18/8.16	   U42(x0, x1, active(x2))
26.18/8.16	   s(mark(x0))
26.18/8.16	   s(active(x0))
26.18/8.16	   plus(mark(x0), x1)
26.18/8.16	   plus(x0, mark(x1))
26.18/8.16	   plus(active(x0), x1)
26.18/8.16	   plus(x0, active(x1))
26.18/8.16	
26.18/8.16	We have to consider all minimal (P,Q,R)-chains.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(20) UsableRulesProof (EQUIVALENT)
26.18/8.16	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.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(21)
26.18/8.16	Obligation:
26.18/8.16	Q DP problem:
26.18/8.16	The TRS P consists of the following rules:
26.18/8.16	
26.18/8.16	   U42^1(X1, mark(X2), X3) -> U42^1(X1, X2, X3)
26.18/8.16	   U42^1(mark(X1), X2, X3) -> U42^1(X1, X2, X3)
26.18/8.16	   U42^1(X1, X2, mark(X3)) -> U42^1(X1, X2, X3)
26.18/8.16	   U42^1(active(X1), X2, X3) -> U42^1(X1, X2, X3)
26.18/8.16	   U42^1(X1, active(X2), X3) -> U42^1(X1, X2, X3)
26.18/8.16	   U42^1(X1, X2, active(X3)) -> U42^1(X1, X2, X3)
26.18/8.16	
26.18/8.16	R is empty.
26.18/8.16	The set Q consists of the following terms:
26.18/8.16	
26.18/8.16	   active(U11(tt, x0))
26.18/8.16	   active(U12(tt))
26.18/8.16	   active(U21(tt))
26.18/8.16	   active(U31(tt, x0))
26.18/8.16	   active(U41(tt, x0, x1))
26.18/8.16	   active(U42(tt, x0, x1))
26.18/8.16	   active(isNat(0))
26.18/8.16	   active(isNat(plus(x0, x1)))
26.18/8.16	   active(isNat(s(x0)))
26.18/8.16	   active(plus(x0, 0))
26.18/8.16	   active(plus(x0, s(x1)))
26.18/8.16	   mark(U11(x0, x1))
26.18/8.16	   mark(tt)
26.18/8.16	   mark(U12(x0))
26.18/8.16	   mark(isNat(x0))
26.18/8.16	   mark(U21(x0))
26.18/8.16	   mark(U31(x0, x1))
26.18/8.16	   mark(U41(x0, x1, x2))
26.18/8.16	   mark(U42(x0, x1, x2))
26.18/8.16	   mark(s(x0))
26.18/8.16	   mark(plus(x0, x1))
26.18/8.16	   mark(0)
26.18/8.16	   U11(mark(x0), x1)
26.18/8.16	   U11(x0, mark(x1))
26.18/8.16	   U11(active(x0), x1)
26.18/8.16	   U11(x0, active(x1))
26.18/8.16	   U12(mark(x0))
26.18/8.16	   U12(active(x0))
26.18/8.16	   isNat(mark(x0))
26.18/8.16	   isNat(active(x0))
26.18/8.16	   U21(mark(x0))
26.18/8.16	   U21(active(x0))
26.18/8.16	   U31(mark(x0), x1)
26.18/8.16	   U31(x0, mark(x1))
26.18/8.16	   U31(active(x0), x1)
26.18/8.16	   U31(x0, active(x1))
26.18/8.16	   U41(mark(x0), x1, x2)
26.18/8.16	   U41(x0, mark(x1), x2)
26.18/8.16	   U41(x0, x1, mark(x2))
26.18/8.16	   U41(active(x0), x1, x2)
26.18/8.16	   U41(x0, active(x1), x2)
26.18/8.16	   U41(x0, x1, active(x2))
26.18/8.16	   U42(mark(x0), x1, x2)
26.18/8.16	   U42(x0, mark(x1), x2)
26.18/8.16	   U42(x0, x1, mark(x2))
26.18/8.16	   U42(active(x0), x1, x2)
26.18/8.16	   U42(x0, active(x1), x2)
26.18/8.16	   U42(x0, x1, active(x2))
26.18/8.16	   s(mark(x0))
26.18/8.16	   s(active(x0))
26.18/8.16	   plus(mark(x0), x1)
26.18/8.16	   plus(x0, mark(x1))
26.18/8.16	   plus(active(x0), x1)
26.18/8.16	   plus(x0, active(x1))
26.18/8.16	
26.18/8.16	We have to consider all minimal (P,Q,R)-chains.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(22) QReductionProof (EQUIVALENT)
26.18/8.16	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
26.18/8.16	
26.18/8.16	   U11(mark(x0), x1)
26.18/8.16	   U11(x0, mark(x1))
26.18/8.16	   U11(active(x0), x1)
26.18/8.16	   U11(x0, active(x1))
26.18/8.16	   U12(mark(x0))
26.18/8.16	   U12(active(x0))
26.18/8.16	   isNat(mark(x0))
26.18/8.16	   isNat(active(x0))
26.18/8.16	   U21(mark(x0))
26.18/8.16	   U21(active(x0))
26.18/8.16	   U31(mark(x0), x1)
26.18/8.16	   U31(x0, mark(x1))
26.18/8.16	   U31(active(x0), x1)
26.18/8.16	   U31(x0, active(x1))
26.18/8.16	   U41(mark(x0), x1, x2)
26.18/8.16	   U41(x0, mark(x1), x2)
26.18/8.16	   U41(x0, x1, mark(x2))
26.18/8.16	   U41(active(x0), x1, x2)
26.18/8.16	   U41(x0, active(x1), x2)
26.18/8.16	   U41(x0, x1, active(x2))
26.18/8.16	   U42(mark(x0), x1, x2)
26.18/8.16	   U42(x0, mark(x1), x2)
26.18/8.16	   U42(x0, x1, mark(x2))
26.18/8.16	   U42(active(x0), x1, x2)
26.18/8.16	   U42(x0, active(x1), x2)
26.18/8.16	   U42(x0, x1, active(x2))
26.18/8.16	   s(mark(x0))
26.18/8.16	   s(active(x0))
26.18/8.16	   plus(mark(x0), x1)
26.18/8.16	   plus(x0, mark(x1))
26.18/8.16	   plus(active(x0), x1)
26.18/8.16	   plus(x0, active(x1))
26.18/8.16	
26.18/8.16	
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(23)
26.18/8.16	Obligation:
26.18/8.16	Q DP problem:
26.18/8.16	The TRS P consists of the following rules:
26.18/8.16	
26.18/8.16	   U42^1(X1, mark(X2), X3) -> U42^1(X1, X2, X3)
26.18/8.16	   U42^1(mark(X1), X2, X3) -> U42^1(X1, X2, X3)
26.18/8.16	   U42^1(X1, X2, mark(X3)) -> U42^1(X1, X2, X3)
26.18/8.16	   U42^1(active(X1), X2, X3) -> U42^1(X1, X2, X3)
26.18/8.16	   U42^1(X1, active(X2), X3) -> U42^1(X1, X2, X3)
26.18/8.16	   U42^1(X1, X2, active(X3)) -> U42^1(X1, X2, X3)
26.18/8.16	
26.18/8.16	R is empty.
26.18/8.16	The set Q consists of the following terms:
26.18/8.16	
26.18/8.16	   active(U11(tt, x0))
26.18/8.16	   active(U12(tt))
26.18/8.16	   active(U21(tt))
26.18/8.16	   active(U31(tt, x0))
26.18/8.16	   active(U41(tt, x0, x1))
26.18/8.16	   active(U42(tt, x0, x1))
26.18/8.16	   active(isNat(0))
26.18/8.16	   active(isNat(plus(x0, x1)))
26.18/8.16	   active(isNat(s(x0)))
26.18/8.16	   active(plus(x0, 0))
26.18/8.16	   active(plus(x0, s(x1)))
26.18/8.16	   mark(U11(x0, x1))
26.18/8.16	   mark(tt)
26.18/8.16	   mark(U12(x0))
26.18/8.16	   mark(isNat(x0))
26.18/8.16	   mark(U21(x0))
26.18/8.16	   mark(U31(x0, x1))
26.18/8.16	   mark(U41(x0, x1, x2))
26.18/8.16	   mark(U42(x0, x1, x2))
26.18/8.16	   mark(s(x0))
26.18/8.16	   mark(plus(x0, x1))
26.18/8.16	   mark(0)
26.18/8.16	
26.18/8.16	We have to consider all minimal (P,Q,R)-chains.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(24) QDPSizeChangeProof (EQUIVALENT)
26.18/8.16	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. 
26.18/8.16	
26.18/8.16	From the DPs we obtained the following set of size-change graphs:
26.18/8.16	*U42^1(X1, mark(X2), X3) -> U42^1(X1, X2, X3)
26.18/8.16	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
26.18/8.16	
26.18/8.16	
26.18/8.16	*U42^1(mark(X1), X2, X3) -> U42^1(X1, X2, X3)
26.18/8.16	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
26.18/8.16	
26.18/8.16	
26.18/8.16	*U42^1(X1, X2, mark(X3)) -> U42^1(X1, X2, X3)
26.18/8.16	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
26.18/8.16	
26.18/8.16	
26.18/8.16	*U42^1(active(X1), X2, X3) -> U42^1(X1, X2, X3)
26.18/8.16	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
26.18/8.16	
26.18/8.16	
26.18/8.16	*U42^1(X1, active(X2), X3) -> U42^1(X1, X2, X3)
26.18/8.16	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
26.18/8.16	
26.18/8.16	
26.18/8.16	*U42^1(X1, X2, active(X3)) -> U42^1(X1, X2, X3)
26.18/8.16	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
26.18/8.16	
26.18/8.16	
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(25)
26.18/8.16	YES
26.18/8.16	
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(26)
26.18/8.16	Obligation:
26.18/8.16	Q DP problem:
26.18/8.16	The TRS P consists of the following rules:
26.18/8.16	
26.18/8.16	   U41^1(X1, mark(X2), X3) -> U41^1(X1, X2, X3)
26.18/8.16	   U41^1(mark(X1), X2, X3) -> U41^1(X1, X2, X3)
26.18/8.16	   U41^1(X1, X2, mark(X3)) -> U41^1(X1, X2, X3)
26.18/8.16	   U41^1(active(X1), X2, X3) -> U41^1(X1, X2, X3)
26.18/8.16	   U41^1(X1, active(X2), X3) -> U41^1(X1, X2, X3)
26.18/8.16	   U41^1(X1, X2, active(X3)) -> U41^1(X1, X2, X3)
26.18/8.16	
26.18/8.16	The TRS R consists of the following rules:
26.18/8.16	
26.18/8.16	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.16	   active(U12(tt)) -> mark(tt)
26.18/8.16	   active(U21(tt)) -> mark(tt)
26.18/8.16	   active(U31(tt, N)) -> mark(N)
26.18/8.16	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.16	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.16	   active(isNat(0)) -> mark(tt)
26.18/8.16	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.16	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.16	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.16	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.16	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.16	   mark(tt) -> active(tt)
26.18/8.16	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.16	   mark(isNat(X)) -> active(isNat(X))
26.18/8.16	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.16	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.16	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.16	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.16	   mark(s(X)) -> active(s(mark(X)))
26.18/8.16	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.16	   mark(0) -> active(0)
26.18/8.16	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.16	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.16	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.16	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.16	   U12(mark(X)) -> U12(X)
26.18/8.16	   U12(active(X)) -> U12(X)
26.18/8.16	   isNat(mark(X)) -> isNat(X)
26.18/8.16	   isNat(active(X)) -> isNat(X)
26.18/8.16	   U21(mark(X)) -> U21(X)
26.18/8.16	   U21(active(X)) -> U21(X)
26.18/8.16	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.16	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.16	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.16	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.16	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.16	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.16	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.16	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.16	   s(mark(X)) -> s(X)
26.18/8.16	   s(active(X)) -> s(X)
26.18/8.16	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.16	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.16	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.16	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.16	
26.18/8.16	The set Q consists of the following terms:
26.18/8.16	
26.18/8.16	   active(U11(tt, x0))
26.18/8.16	   active(U12(tt))
26.18/8.16	   active(U21(tt))
26.18/8.16	   active(U31(tt, x0))
26.18/8.16	   active(U41(tt, x0, x1))
26.18/8.16	   active(U42(tt, x0, x1))
26.18/8.16	   active(isNat(0))
26.18/8.16	   active(isNat(plus(x0, x1)))
26.18/8.16	   active(isNat(s(x0)))
26.18/8.16	   active(plus(x0, 0))
26.18/8.16	   active(plus(x0, s(x1)))
26.18/8.16	   mark(U11(x0, x1))
26.18/8.16	   mark(tt)
26.18/8.16	   mark(U12(x0))
26.18/8.16	   mark(isNat(x0))
26.18/8.16	   mark(U21(x0))
26.18/8.16	   mark(U31(x0, x1))
26.18/8.16	   mark(U41(x0, x1, x2))
26.18/8.16	   mark(U42(x0, x1, x2))
26.18/8.16	   mark(s(x0))
26.18/8.16	   mark(plus(x0, x1))
26.18/8.16	   mark(0)
26.18/8.16	   U11(mark(x0), x1)
26.18/8.16	   U11(x0, mark(x1))
26.18/8.16	   U11(active(x0), x1)
26.18/8.16	   U11(x0, active(x1))
26.18/8.16	   U12(mark(x0))
26.18/8.16	   U12(active(x0))
26.18/8.16	   isNat(mark(x0))
26.18/8.16	   isNat(active(x0))
26.18/8.16	   U21(mark(x0))
26.18/8.16	   U21(active(x0))
26.18/8.16	   U31(mark(x0), x1)
26.18/8.16	   U31(x0, mark(x1))
26.18/8.16	   U31(active(x0), x1)
26.18/8.16	   U31(x0, active(x1))
26.18/8.16	   U41(mark(x0), x1, x2)
26.18/8.16	   U41(x0, mark(x1), x2)
26.18/8.16	   U41(x0, x1, mark(x2))
26.18/8.16	   U41(active(x0), x1, x2)
26.18/8.16	   U41(x0, active(x1), x2)
26.18/8.16	   U41(x0, x1, active(x2))
26.18/8.16	   U42(mark(x0), x1, x2)
26.18/8.16	   U42(x0, mark(x1), x2)
26.18/8.16	   U42(x0, x1, mark(x2))
26.18/8.16	   U42(active(x0), x1, x2)
26.18/8.16	   U42(x0, active(x1), x2)
26.18/8.16	   U42(x0, x1, active(x2))
26.18/8.16	   s(mark(x0))
26.18/8.16	   s(active(x0))
26.18/8.16	   plus(mark(x0), x1)
26.18/8.16	   plus(x0, mark(x1))
26.18/8.16	   plus(active(x0), x1)
26.18/8.16	   plus(x0, active(x1))
26.18/8.16	
26.18/8.16	We have to consider all minimal (P,Q,R)-chains.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(27) UsableRulesProof (EQUIVALENT)
26.18/8.16	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.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(28)
26.18/8.16	Obligation:
26.18/8.16	Q DP problem:
26.18/8.16	The TRS P consists of the following rules:
26.18/8.16	
26.18/8.16	   U41^1(X1, mark(X2), X3) -> U41^1(X1, X2, X3)
26.18/8.16	   U41^1(mark(X1), X2, X3) -> U41^1(X1, X2, X3)
26.18/8.16	   U41^1(X1, X2, mark(X3)) -> U41^1(X1, X2, X3)
26.18/8.16	   U41^1(active(X1), X2, X3) -> U41^1(X1, X2, X3)
26.18/8.16	   U41^1(X1, active(X2), X3) -> U41^1(X1, X2, X3)
26.18/8.16	   U41^1(X1, X2, active(X3)) -> U41^1(X1, X2, X3)
26.18/8.16	
26.18/8.16	R is empty.
26.18/8.16	The set Q consists of the following terms:
26.18/8.16	
26.18/8.16	   active(U11(tt, x0))
26.18/8.16	   active(U12(tt))
26.18/8.16	   active(U21(tt))
26.18/8.16	   active(U31(tt, x0))
26.18/8.16	   active(U41(tt, x0, x1))
26.18/8.16	   active(U42(tt, x0, x1))
26.18/8.16	   active(isNat(0))
26.18/8.16	   active(isNat(plus(x0, x1)))
26.18/8.16	   active(isNat(s(x0)))
26.18/8.16	   active(plus(x0, 0))
26.18/8.16	   active(plus(x0, s(x1)))
26.18/8.16	   mark(U11(x0, x1))
26.18/8.16	   mark(tt)
26.18/8.16	   mark(U12(x0))
26.18/8.16	   mark(isNat(x0))
26.18/8.16	   mark(U21(x0))
26.18/8.16	   mark(U31(x0, x1))
26.18/8.16	   mark(U41(x0, x1, x2))
26.18/8.16	   mark(U42(x0, x1, x2))
26.18/8.16	   mark(s(x0))
26.18/8.16	   mark(plus(x0, x1))
26.18/8.16	   mark(0)
26.18/8.16	   U11(mark(x0), x1)
26.18/8.16	   U11(x0, mark(x1))
26.18/8.16	   U11(active(x0), x1)
26.18/8.16	   U11(x0, active(x1))
26.18/8.16	   U12(mark(x0))
26.18/8.16	   U12(active(x0))
26.18/8.16	   isNat(mark(x0))
26.18/8.16	   isNat(active(x0))
26.18/8.16	   U21(mark(x0))
26.18/8.16	   U21(active(x0))
26.18/8.16	   U31(mark(x0), x1)
26.18/8.16	   U31(x0, mark(x1))
26.18/8.16	   U31(active(x0), x1)
26.18/8.16	   U31(x0, active(x1))
26.18/8.16	   U41(mark(x0), x1, x2)
26.18/8.16	   U41(x0, mark(x1), x2)
26.18/8.16	   U41(x0, x1, mark(x2))
26.18/8.16	   U41(active(x0), x1, x2)
26.18/8.16	   U41(x0, active(x1), x2)
26.18/8.16	   U41(x0, x1, active(x2))
26.18/8.16	   U42(mark(x0), x1, x2)
26.18/8.16	   U42(x0, mark(x1), x2)
26.18/8.16	   U42(x0, x1, mark(x2))
26.18/8.16	   U42(active(x0), x1, x2)
26.18/8.16	   U42(x0, active(x1), x2)
26.18/8.16	   U42(x0, x1, active(x2))
26.18/8.16	   s(mark(x0))
26.18/8.16	   s(active(x0))
26.18/8.16	   plus(mark(x0), x1)
26.18/8.16	   plus(x0, mark(x1))
26.18/8.16	   plus(active(x0), x1)
26.18/8.16	   plus(x0, active(x1))
26.18/8.16	
26.18/8.16	We have to consider all minimal (P,Q,R)-chains.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(29) QReductionProof (EQUIVALENT)
26.18/8.16	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
26.18/8.16	
26.18/8.16	   U11(mark(x0), x1)
26.18/8.16	   U11(x0, mark(x1))
26.18/8.16	   U11(active(x0), x1)
26.18/8.16	   U11(x0, active(x1))
26.18/8.16	   U12(mark(x0))
26.18/8.16	   U12(active(x0))
26.18/8.16	   isNat(mark(x0))
26.18/8.16	   isNat(active(x0))
26.18/8.16	   U21(mark(x0))
26.18/8.16	   U21(active(x0))
26.18/8.16	   U31(mark(x0), x1)
26.18/8.16	   U31(x0, mark(x1))
26.18/8.16	   U31(active(x0), x1)
26.18/8.16	   U31(x0, active(x1))
26.18/8.16	   U41(mark(x0), x1, x2)
26.18/8.16	   U41(x0, mark(x1), x2)
26.18/8.16	   U41(x0, x1, mark(x2))
26.18/8.16	   U41(active(x0), x1, x2)
26.18/8.16	   U41(x0, active(x1), x2)
26.18/8.16	   U41(x0, x1, active(x2))
26.18/8.16	   U42(mark(x0), x1, x2)
26.18/8.16	   U42(x0, mark(x1), x2)
26.18/8.16	   U42(x0, x1, mark(x2))
26.18/8.16	   U42(active(x0), x1, x2)
26.18/8.16	   U42(x0, active(x1), x2)
26.18/8.16	   U42(x0, x1, active(x2))
26.18/8.16	   s(mark(x0))
26.18/8.16	   s(active(x0))
26.18/8.16	   plus(mark(x0), x1)
26.18/8.16	   plus(x0, mark(x1))
26.18/8.16	   plus(active(x0), x1)
26.18/8.16	   plus(x0, active(x1))
26.18/8.16	
26.18/8.16	
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(30)
26.18/8.16	Obligation:
26.18/8.16	Q DP problem:
26.18/8.16	The TRS P consists of the following rules:
26.18/8.16	
26.18/8.16	   U41^1(X1, mark(X2), X3) -> U41^1(X1, X2, X3)
26.18/8.16	   U41^1(mark(X1), X2, X3) -> U41^1(X1, X2, X3)
26.18/8.16	   U41^1(X1, X2, mark(X3)) -> U41^1(X1, X2, X3)
26.18/8.16	   U41^1(active(X1), X2, X3) -> U41^1(X1, X2, X3)
26.18/8.16	   U41^1(X1, active(X2), X3) -> U41^1(X1, X2, X3)
26.18/8.16	   U41^1(X1, X2, active(X3)) -> U41^1(X1, X2, X3)
26.18/8.16	
26.18/8.16	R is empty.
26.18/8.16	The set Q consists of the following terms:
26.18/8.16	
26.18/8.16	   active(U11(tt, x0))
26.18/8.16	   active(U12(tt))
26.18/8.16	   active(U21(tt))
26.18/8.16	   active(U31(tt, x0))
26.18/8.16	   active(U41(tt, x0, x1))
26.18/8.16	   active(U42(tt, x0, x1))
26.18/8.16	   active(isNat(0))
26.18/8.16	   active(isNat(plus(x0, x1)))
26.18/8.16	   active(isNat(s(x0)))
26.18/8.16	   active(plus(x0, 0))
26.18/8.16	   active(plus(x0, s(x1)))
26.18/8.16	   mark(U11(x0, x1))
26.18/8.16	   mark(tt)
26.18/8.16	   mark(U12(x0))
26.18/8.16	   mark(isNat(x0))
26.18/8.16	   mark(U21(x0))
26.18/8.16	   mark(U31(x0, x1))
26.18/8.16	   mark(U41(x0, x1, x2))
26.18/8.16	   mark(U42(x0, x1, x2))
26.18/8.16	   mark(s(x0))
26.18/8.16	   mark(plus(x0, x1))
26.18/8.16	   mark(0)
26.18/8.16	
26.18/8.16	We have to consider all minimal (P,Q,R)-chains.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(31) QDPSizeChangeProof (EQUIVALENT)
26.18/8.16	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. 
26.18/8.16	
26.18/8.16	From the DPs we obtained the following set of size-change graphs:
26.18/8.16	*U41^1(X1, mark(X2), X3) -> U41^1(X1, X2, X3)
26.18/8.16	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
26.18/8.16	
26.18/8.16	
26.18/8.16	*U41^1(mark(X1), X2, X3) -> U41^1(X1, X2, X3)
26.18/8.16	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
26.18/8.16	
26.18/8.16	
26.18/8.16	*U41^1(X1, X2, mark(X3)) -> U41^1(X1, X2, X3)
26.18/8.16	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
26.18/8.16	
26.18/8.16	
26.18/8.16	*U41^1(active(X1), X2, X3) -> U41^1(X1, X2, X3)
26.18/8.16	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
26.18/8.16	
26.18/8.16	
26.18/8.16	*U41^1(X1, active(X2), X3) -> U41^1(X1, X2, X3)
26.18/8.16	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
26.18/8.16	
26.18/8.16	
26.18/8.16	*U41^1(X1, X2, active(X3)) -> U41^1(X1, X2, X3)
26.18/8.16	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
26.18/8.16	
26.18/8.16	
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(32)
26.18/8.16	YES
26.18/8.16	
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(33)
26.18/8.16	Obligation:
26.18/8.16	Q DP problem:
26.18/8.16	The TRS P consists of the following rules:
26.18/8.16	
26.18/8.16	   U31^1(X1, mark(X2)) -> U31^1(X1, X2)
26.18/8.16	   U31^1(mark(X1), X2) -> U31^1(X1, X2)
26.18/8.16	   U31^1(active(X1), X2) -> U31^1(X1, X2)
26.18/8.16	   U31^1(X1, active(X2)) -> U31^1(X1, X2)
26.18/8.16	
26.18/8.16	The TRS R consists of the following rules:
26.18/8.16	
26.18/8.16	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.16	   active(U12(tt)) -> mark(tt)
26.18/8.16	   active(U21(tt)) -> mark(tt)
26.18/8.16	   active(U31(tt, N)) -> mark(N)
26.18/8.16	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.16	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.16	   active(isNat(0)) -> mark(tt)
26.18/8.16	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.16	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.16	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.16	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.16	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.16	   mark(tt) -> active(tt)
26.18/8.16	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.16	   mark(isNat(X)) -> active(isNat(X))
26.18/8.16	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.16	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.16	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.16	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.16	   mark(s(X)) -> active(s(mark(X)))
26.18/8.16	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.16	   mark(0) -> active(0)
26.18/8.16	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.16	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.16	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.16	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.16	   U12(mark(X)) -> U12(X)
26.18/8.16	   U12(active(X)) -> U12(X)
26.18/8.16	   isNat(mark(X)) -> isNat(X)
26.18/8.16	   isNat(active(X)) -> isNat(X)
26.18/8.16	   U21(mark(X)) -> U21(X)
26.18/8.16	   U21(active(X)) -> U21(X)
26.18/8.16	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.16	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.16	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.16	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.16	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.16	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.16	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.16	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.16	   s(mark(X)) -> s(X)
26.18/8.16	   s(active(X)) -> s(X)
26.18/8.16	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.16	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.16	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.16	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.16	
26.18/8.16	The set Q consists of the following terms:
26.18/8.16	
26.18/8.16	   active(U11(tt, x0))
26.18/8.16	   active(U12(tt))
26.18/8.16	   active(U21(tt))
26.18/8.16	   active(U31(tt, x0))
26.18/8.16	   active(U41(tt, x0, x1))
26.18/8.16	   active(U42(tt, x0, x1))
26.18/8.16	   active(isNat(0))
26.18/8.16	   active(isNat(plus(x0, x1)))
26.18/8.16	   active(isNat(s(x0)))
26.18/8.16	   active(plus(x0, 0))
26.18/8.16	   active(plus(x0, s(x1)))
26.18/8.16	   mark(U11(x0, x1))
26.18/8.16	   mark(tt)
26.18/8.16	   mark(U12(x0))
26.18/8.16	   mark(isNat(x0))
26.18/8.16	   mark(U21(x0))
26.18/8.16	   mark(U31(x0, x1))
26.18/8.16	   mark(U41(x0, x1, x2))
26.18/8.16	   mark(U42(x0, x1, x2))
26.18/8.16	   mark(s(x0))
26.18/8.16	   mark(plus(x0, x1))
26.18/8.16	   mark(0)
26.18/8.16	   U11(mark(x0), x1)
26.18/8.16	   U11(x0, mark(x1))
26.18/8.16	   U11(active(x0), x1)
26.18/8.16	   U11(x0, active(x1))
26.18/8.16	   U12(mark(x0))
26.18/8.16	   U12(active(x0))
26.18/8.16	   isNat(mark(x0))
26.18/8.16	   isNat(active(x0))
26.18/8.16	   U21(mark(x0))
26.18/8.16	   U21(active(x0))
26.18/8.16	   U31(mark(x0), x1)
26.18/8.16	   U31(x0, mark(x1))
26.18/8.16	   U31(active(x0), x1)
26.18/8.16	   U31(x0, active(x1))
26.18/8.16	   U41(mark(x0), x1, x2)
26.18/8.16	   U41(x0, mark(x1), x2)
26.18/8.16	   U41(x0, x1, mark(x2))
26.18/8.16	   U41(active(x0), x1, x2)
26.18/8.16	   U41(x0, active(x1), x2)
26.18/8.16	   U41(x0, x1, active(x2))
26.18/8.16	   U42(mark(x0), x1, x2)
26.18/8.16	   U42(x0, mark(x1), x2)
26.18/8.16	   U42(x0, x1, mark(x2))
26.18/8.16	   U42(active(x0), x1, x2)
26.18/8.16	   U42(x0, active(x1), x2)
26.18/8.16	   U42(x0, x1, active(x2))
26.18/8.16	   s(mark(x0))
26.18/8.16	   s(active(x0))
26.18/8.16	   plus(mark(x0), x1)
26.18/8.16	   plus(x0, mark(x1))
26.18/8.16	   plus(active(x0), x1)
26.18/8.16	   plus(x0, active(x1))
26.18/8.16	
26.18/8.16	We have to consider all minimal (P,Q,R)-chains.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(34) UsableRulesProof (EQUIVALENT)
26.18/8.16	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.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(35)
26.18/8.16	Obligation:
26.18/8.16	Q DP problem:
26.18/8.16	The TRS P consists of the following rules:
26.18/8.16	
26.18/8.16	   U31^1(X1, mark(X2)) -> U31^1(X1, X2)
26.18/8.16	   U31^1(mark(X1), X2) -> U31^1(X1, X2)
26.18/8.16	   U31^1(active(X1), X2) -> U31^1(X1, X2)
26.18/8.16	   U31^1(X1, active(X2)) -> U31^1(X1, X2)
26.18/8.16	
26.18/8.16	R is empty.
26.18/8.16	The set Q consists of the following terms:
26.18/8.16	
26.18/8.16	   active(U11(tt, x0))
26.18/8.16	   active(U12(tt))
26.18/8.16	   active(U21(tt))
26.18/8.16	   active(U31(tt, x0))
26.18/8.16	   active(U41(tt, x0, x1))
26.18/8.16	   active(U42(tt, x0, x1))
26.18/8.16	   active(isNat(0))
26.18/8.16	   active(isNat(plus(x0, x1)))
26.18/8.16	   active(isNat(s(x0)))
26.18/8.16	   active(plus(x0, 0))
26.18/8.16	   active(plus(x0, s(x1)))
26.18/8.16	   mark(U11(x0, x1))
26.18/8.16	   mark(tt)
26.18/8.16	   mark(U12(x0))
26.18/8.16	   mark(isNat(x0))
26.18/8.16	   mark(U21(x0))
26.18/8.16	   mark(U31(x0, x1))
26.18/8.16	   mark(U41(x0, x1, x2))
26.18/8.16	   mark(U42(x0, x1, x2))
26.18/8.16	   mark(s(x0))
26.18/8.16	   mark(plus(x0, x1))
26.18/8.16	   mark(0)
26.18/8.16	   U11(mark(x0), x1)
26.18/8.16	   U11(x0, mark(x1))
26.18/8.16	   U11(active(x0), x1)
26.18/8.16	   U11(x0, active(x1))
26.18/8.16	   U12(mark(x0))
26.18/8.16	   U12(active(x0))
26.18/8.16	   isNat(mark(x0))
26.18/8.16	   isNat(active(x0))
26.18/8.16	   U21(mark(x0))
26.18/8.16	   U21(active(x0))
26.18/8.16	   U31(mark(x0), x1)
26.18/8.16	   U31(x0, mark(x1))
26.18/8.16	   U31(active(x0), x1)
26.18/8.16	   U31(x0, active(x1))
26.18/8.16	   U41(mark(x0), x1, x2)
26.18/8.16	   U41(x0, mark(x1), x2)
26.18/8.16	   U41(x0, x1, mark(x2))
26.18/8.16	   U41(active(x0), x1, x2)
26.18/8.16	   U41(x0, active(x1), x2)
26.18/8.16	   U41(x0, x1, active(x2))
26.18/8.16	   U42(mark(x0), x1, x2)
26.18/8.16	   U42(x0, mark(x1), x2)
26.18/8.16	   U42(x0, x1, mark(x2))
26.18/8.16	   U42(active(x0), x1, x2)
26.18/8.16	   U42(x0, active(x1), x2)
26.18/8.16	   U42(x0, x1, active(x2))
26.18/8.16	   s(mark(x0))
26.18/8.16	   s(active(x0))
26.18/8.16	   plus(mark(x0), x1)
26.18/8.16	   plus(x0, mark(x1))
26.18/8.16	   plus(active(x0), x1)
26.18/8.16	   plus(x0, active(x1))
26.18/8.16	
26.18/8.16	We have to consider all minimal (P,Q,R)-chains.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(36) QReductionProof (EQUIVALENT)
26.18/8.16	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
26.18/8.16	
26.18/8.16	   U11(mark(x0), x1)
26.18/8.16	   U11(x0, mark(x1))
26.18/8.16	   U11(active(x0), x1)
26.18/8.16	   U11(x0, active(x1))
26.18/8.16	   U12(mark(x0))
26.18/8.16	   U12(active(x0))
26.18/8.16	   isNat(mark(x0))
26.18/8.16	   isNat(active(x0))
26.18/8.16	   U21(mark(x0))
26.18/8.16	   U21(active(x0))
26.18/8.16	   U31(mark(x0), x1)
26.18/8.16	   U31(x0, mark(x1))
26.18/8.16	   U31(active(x0), x1)
26.18/8.16	   U31(x0, active(x1))
26.18/8.16	   U41(mark(x0), x1, x2)
26.18/8.16	   U41(x0, mark(x1), x2)
26.18/8.16	   U41(x0, x1, mark(x2))
26.18/8.16	   U41(active(x0), x1, x2)
26.18/8.16	   U41(x0, active(x1), x2)
26.18/8.16	   U41(x0, x1, active(x2))
26.18/8.16	   U42(mark(x0), x1, x2)
26.18/8.16	   U42(x0, mark(x1), x2)
26.18/8.16	   U42(x0, x1, mark(x2))
26.18/8.16	   U42(active(x0), x1, x2)
26.18/8.16	   U42(x0, active(x1), x2)
26.18/8.16	   U42(x0, x1, active(x2))
26.18/8.16	   s(mark(x0))
26.18/8.16	   s(active(x0))
26.18/8.16	   plus(mark(x0), x1)
26.18/8.16	   plus(x0, mark(x1))
26.18/8.16	   plus(active(x0), x1)
26.18/8.16	   plus(x0, active(x1))
26.18/8.16	
26.18/8.16	
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(37)
26.18/8.16	Obligation:
26.18/8.16	Q DP problem:
26.18/8.16	The TRS P consists of the following rules:
26.18/8.16	
26.18/8.16	   U31^1(X1, mark(X2)) -> U31^1(X1, X2)
26.18/8.16	   U31^1(mark(X1), X2) -> U31^1(X1, X2)
26.18/8.16	   U31^1(active(X1), X2) -> U31^1(X1, X2)
26.18/8.16	   U31^1(X1, active(X2)) -> U31^1(X1, X2)
26.18/8.16	
26.18/8.16	R is empty.
26.18/8.16	The set Q consists of the following terms:
26.18/8.16	
26.18/8.16	   active(U11(tt, x0))
26.18/8.16	   active(U12(tt))
26.18/8.16	   active(U21(tt))
26.18/8.16	   active(U31(tt, x0))
26.18/8.16	   active(U41(tt, x0, x1))
26.18/8.16	   active(U42(tt, x0, x1))
26.18/8.16	   active(isNat(0))
26.18/8.16	   active(isNat(plus(x0, x1)))
26.18/8.16	   active(isNat(s(x0)))
26.18/8.16	   active(plus(x0, 0))
26.18/8.16	   active(plus(x0, s(x1)))
26.18/8.16	   mark(U11(x0, x1))
26.18/8.16	   mark(tt)
26.18/8.16	   mark(U12(x0))
26.18/8.16	   mark(isNat(x0))
26.18/8.16	   mark(U21(x0))
26.18/8.16	   mark(U31(x0, x1))
26.18/8.16	   mark(U41(x0, x1, x2))
26.18/8.16	   mark(U42(x0, x1, x2))
26.18/8.16	   mark(s(x0))
26.18/8.16	   mark(plus(x0, x1))
26.18/8.16	   mark(0)
26.18/8.16	
26.18/8.16	We have to consider all minimal (P,Q,R)-chains.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(38) QDPSizeChangeProof (EQUIVALENT)
26.18/8.16	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. 
26.18/8.16	
26.18/8.16	From the DPs we obtained the following set of size-change graphs:
26.18/8.16	*U31^1(X1, mark(X2)) -> U31^1(X1, X2)
26.18/8.16	The graph contains the following edges 1 >= 1, 2 > 2
26.18/8.16	
26.18/8.16	
26.18/8.16	*U31^1(mark(X1), X2) -> U31^1(X1, X2)
26.18/8.16	The graph contains the following edges 1 > 1, 2 >= 2
26.18/8.16	
26.18/8.16	
26.18/8.16	*U31^1(active(X1), X2) -> U31^1(X1, X2)
26.18/8.16	The graph contains the following edges 1 > 1, 2 >= 2
26.18/8.16	
26.18/8.16	
26.18/8.16	*U31^1(X1, active(X2)) -> U31^1(X1, X2)
26.18/8.16	The graph contains the following edges 1 >= 1, 2 > 2
26.18/8.16	
26.18/8.16	
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(39)
26.18/8.16	YES
26.18/8.16	
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(40)
26.18/8.16	Obligation:
26.18/8.16	Q DP problem:
26.18/8.16	The TRS P consists of the following rules:
26.18/8.16	
26.18/8.16	   U21^1(active(X)) -> U21^1(X)
26.18/8.16	   U21^1(mark(X)) -> U21^1(X)
26.18/8.16	
26.18/8.16	The TRS R consists of the following rules:
26.18/8.16	
26.18/8.16	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.16	   active(U12(tt)) -> mark(tt)
26.18/8.16	   active(U21(tt)) -> mark(tt)
26.18/8.16	   active(U31(tt, N)) -> mark(N)
26.18/8.16	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.16	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.16	   active(isNat(0)) -> mark(tt)
26.18/8.16	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.16	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.16	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.16	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.16	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.16	   mark(tt) -> active(tt)
26.18/8.16	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.16	   mark(isNat(X)) -> active(isNat(X))
26.18/8.16	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.16	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.16	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.16	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.16	   mark(s(X)) -> active(s(mark(X)))
26.18/8.16	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.16	   mark(0) -> active(0)
26.18/8.16	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.16	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.16	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.16	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.16	   U12(mark(X)) -> U12(X)
26.18/8.16	   U12(active(X)) -> U12(X)
26.18/8.16	   isNat(mark(X)) -> isNat(X)
26.18/8.16	   isNat(active(X)) -> isNat(X)
26.18/8.16	   U21(mark(X)) -> U21(X)
26.18/8.16	   U21(active(X)) -> U21(X)
26.18/8.16	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.16	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.16	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.16	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.16	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.16	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.16	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.16	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.16	   s(mark(X)) -> s(X)
26.18/8.16	   s(active(X)) -> s(X)
26.18/8.16	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.16	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.16	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.16	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.16	
26.18/8.16	The set Q consists of the following terms:
26.18/8.16	
26.18/8.16	   active(U11(tt, x0))
26.18/8.16	   active(U12(tt))
26.18/8.16	   active(U21(tt))
26.18/8.16	   active(U31(tt, x0))
26.18/8.16	   active(U41(tt, x0, x1))
26.18/8.16	   active(U42(tt, x0, x1))
26.18/8.16	   active(isNat(0))
26.18/8.16	   active(isNat(plus(x0, x1)))
26.18/8.16	   active(isNat(s(x0)))
26.18/8.16	   active(plus(x0, 0))
26.18/8.16	   active(plus(x0, s(x1)))
26.18/8.16	   mark(U11(x0, x1))
26.18/8.16	   mark(tt)
26.18/8.16	   mark(U12(x0))
26.18/8.16	   mark(isNat(x0))
26.18/8.16	   mark(U21(x0))
26.18/8.16	   mark(U31(x0, x1))
26.18/8.16	   mark(U41(x0, x1, x2))
26.18/8.16	   mark(U42(x0, x1, x2))
26.18/8.16	   mark(s(x0))
26.18/8.16	   mark(plus(x0, x1))
26.18/8.16	   mark(0)
26.18/8.16	   U11(mark(x0), x1)
26.18/8.16	   U11(x0, mark(x1))
26.18/8.16	   U11(active(x0), x1)
26.18/8.16	   U11(x0, active(x1))
26.18/8.16	   U12(mark(x0))
26.18/8.16	   U12(active(x0))
26.18/8.16	   isNat(mark(x0))
26.18/8.16	   isNat(active(x0))
26.18/8.16	   U21(mark(x0))
26.18/8.16	   U21(active(x0))
26.18/8.16	   U31(mark(x0), x1)
26.18/8.16	   U31(x0, mark(x1))
26.18/8.16	   U31(active(x0), x1)
26.18/8.16	   U31(x0, active(x1))
26.18/8.16	   U41(mark(x0), x1, x2)
26.18/8.16	   U41(x0, mark(x1), x2)
26.18/8.16	   U41(x0, x1, mark(x2))
26.18/8.16	   U41(active(x0), x1, x2)
26.18/8.16	   U41(x0, active(x1), x2)
26.18/8.16	   U41(x0, x1, active(x2))
26.18/8.16	   U42(mark(x0), x1, x2)
26.18/8.16	   U42(x0, mark(x1), x2)
26.18/8.16	   U42(x0, x1, mark(x2))
26.18/8.16	   U42(active(x0), x1, x2)
26.18/8.16	   U42(x0, active(x1), x2)
26.18/8.16	   U42(x0, x1, active(x2))
26.18/8.16	   s(mark(x0))
26.18/8.16	   s(active(x0))
26.18/8.16	   plus(mark(x0), x1)
26.18/8.16	   plus(x0, mark(x1))
26.18/8.16	   plus(active(x0), x1)
26.18/8.16	   plus(x0, active(x1))
26.18/8.16	
26.18/8.16	We have to consider all minimal (P,Q,R)-chains.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(41) UsableRulesProof (EQUIVALENT)
26.18/8.16	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.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(42)
26.18/8.16	Obligation:
26.18/8.16	Q DP problem:
26.18/8.16	The TRS P consists of the following rules:
26.18/8.16	
26.18/8.16	   U21^1(active(X)) -> U21^1(X)
26.18/8.16	   U21^1(mark(X)) -> U21^1(X)
26.18/8.16	
26.18/8.16	R is empty.
26.18/8.16	The set Q consists of the following terms:
26.18/8.16	
26.18/8.16	   active(U11(tt, x0))
26.18/8.16	   active(U12(tt))
26.18/8.16	   active(U21(tt))
26.18/8.16	   active(U31(tt, x0))
26.18/8.16	   active(U41(tt, x0, x1))
26.18/8.16	   active(U42(tt, x0, x1))
26.18/8.16	   active(isNat(0))
26.18/8.16	   active(isNat(plus(x0, x1)))
26.18/8.16	   active(isNat(s(x0)))
26.18/8.16	   active(plus(x0, 0))
26.18/8.16	   active(plus(x0, s(x1)))
26.18/8.16	   mark(U11(x0, x1))
26.18/8.16	   mark(tt)
26.18/8.16	   mark(U12(x0))
26.18/8.16	   mark(isNat(x0))
26.18/8.16	   mark(U21(x0))
26.18/8.16	   mark(U31(x0, x1))
26.18/8.16	   mark(U41(x0, x1, x2))
26.18/8.16	   mark(U42(x0, x1, x2))
26.18/8.16	   mark(s(x0))
26.18/8.16	   mark(plus(x0, x1))
26.18/8.16	   mark(0)
26.18/8.16	   U11(mark(x0), x1)
26.18/8.16	   U11(x0, mark(x1))
26.18/8.16	   U11(active(x0), x1)
26.18/8.16	   U11(x0, active(x1))
26.18/8.16	   U12(mark(x0))
26.18/8.16	   U12(active(x0))
26.18/8.16	   isNat(mark(x0))
26.18/8.16	   isNat(active(x0))
26.18/8.16	   U21(mark(x0))
26.18/8.16	   U21(active(x0))
26.18/8.16	   U31(mark(x0), x1)
26.18/8.16	   U31(x0, mark(x1))
26.18/8.16	   U31(active(x0), x1)
26.18/8.16	   U31(x0, active(x1))
26.18/8.16	   U41(mark(x0), x1, x2)
26.18/8.16	   U41(x0, mark(x1), x2)
26.18/8.16	   U41(x0, x1, mark(x2))
26.18/8.16	   U41(active(x0), x1, x2)
26.18/8.16	   U41(x0, active(x1), x2)
26.18/8.16	   U41(x0, x1, active(x2))
26.18/8.16	   U42(mark(x0), x1, x2)
26.18/8.16	   U42(x0, mark(x1), x2)
26.18/8.16	   U42(x0, x1, mark(x2))
26.18/8.16	   U42(active(x0), x1, x2)
26.18/8.16	   U42(x0, active(x1), x2)
26.18/8.16	   U42(x0, x1, active(x2))
26.18/8.16	   s(mark(x0))
26.18/8.16	   s(active(x0))
26.18/8.16	   plus(mark(x0), x1)
26.18/8.16	   plus(x0, mark(x1))
26.18/8.16	   plus(active(x0), x1)
26.18/8.16	   plus(x0, active(x1))
26.18/8.16	
26.18/8.16	We have to consider all minimal (P,Q,R)-chains.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(43) QReductionProof (EQUIVALENT)
26.18/8.16	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
26.18/8.16	
26.18/8.16	   U11(mark(x0), x1)
26.18/8.16	   U11(x0, mark(x1))
26.18/8.16	   U11(active(x0), x1)
26.18/8.16	   U11(x0, active(x1))
26.18/8.16	   U12(mark(x0))
26.18/8.16	   U12(active(x0))
26.18/8.16	   isNat(mark(x0))
26.18/8.16	   isNat(active(x0))
26.18/8.16	   U21(mark(x0))
26.18/8.16	   U21(active(x0))
26.18/8.16	   U31(mark(x0), x1)
26.18/8.16	   U31(x0, mark(x1))
26.18/8.16	   U31(active(x0), x1)
26.18/8.16	   U31(x0, active(x1))
26.18/8.16	   U41(mark(x0), x1, x2)
26.18/8.16	   U41(x0, mark(x1), x2)
26.18/8.16	   U41(x0, x1, mark(x2))
26.18/8.16	   U41(active(x0), x1, x2)
26.18/8.16	   U41(x0, active(x1), x2)
26.18/8.16	   U41(x0, x1, active(x2))
26.18/8.16	   U42(mark(x0), x1, x2)
26.18/8.16	   U42(x0, mark(x1), x2)
26.18/8.16	   U42(x0, x1, mark(x2))
26.18/8.16	   U42(active(x0), x1, x2)
26.18/8.16	   U42(x0, active(x1), x2)
26.18/8.16	   U42(x0, x1, active(x2))
26.18/8.16	   s(mark(x0))
26.18/8.16	   s(active(x0))
26.18/8.16	   plus(mark(x0), x1)
26.18/8.16	   plus(x0, mark(x1))
26.18/8.16	   plus(active(x0), x1)
26.18/8.16	   plus(x0, active(x1))
26.18/8.16	
26.18/8.16	
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(44)
26.18/8.16	Obligation:
26.18/8.16	Q DP problem:
26.18/8.16	The TRS P consists of the following rules:
26.18/8.16	
26.18/8.16	   U21^1(active(X)) -> U21^1(X)
26.18/8.16	   U21^1(mark(X)) -> U21^1(X)
26.18/8.16	
26.18/8.16	R is empty.
26.18/8.16	The set Q consists of the following terms:
26.18/8.16	
26.18/8.16	   active(U11(tt, x0))
26.18/8.16	   active(U12(tt))
26.18/8.16	   active(U21(tt))
26.18/8.16	   active(U31(tt, x0))
26.18/8.16	   active(U41(tt, x0, x1))
26.18/8.16	   active(U42(tt, x0, x1))
26.18/8.16	   active(isNat(0))
26.18/8.16	   active(isNat(plus(x0, x1)))
26.18/8.16	   active(isNat(s(x0)))
26.18/8.16	   active(plus(x0, 0))
26.18/8.16	   active(plus(x0, s(x1)))
26.18/8.16	   mark(U11(x0, x1))
26.18/8.16	   mark(tt)
26.18/8.16	   mark(U12(x0))
26.18/8.16	   mark(isNat(x0))
26.18/8.16	   mark(U21(x0))
26.18/8.16	   mark(U31(x0, x1))
26.18/8.16	   mark(U41(x0, x1, x2))
26.18/8.16	   mark(U42(x0, x1, x2))
26.18/8.16	   mark(s(x0))
26.18/8.16	   mark(plus(x0, x1))
26.18/8.16	   mark(0)
26.18/8.16	
26.18/8.16	We have to consider all minimal (P,Q,R)-chains.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(45) QDPSizeChangeProof (EQUIVALENT)
26.18/8.16	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. 
26.18/8.16	
26.18/8.16	From the DPs we obtained the following set of size-change graphs:
26.18/8.16	*U21^1(active(X)) -> U21^1(X)
26.18/8.16	The graph contains the following edges 1 > 1
26.18/8.16	
26.18/8.16	
26.18/8.16	*U21^1(mark(X)) -> U21^1(X)
26.18/8.16	The graph contains the following edges 1 > 1
26.18/8.16	
26.18/8.16	
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(46)
26.18/8.16	YES
26.18/8.16	
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(47)
26.18/8.16	Obligation:
26.18/8.16	Q DP problem:
26.18/8.16	The TRS P consists of the following rules:
26.18/8.16	
26.18/8.16	   ISNAT(active(X)) -> ISNAT(X)
26.18/8.16	   ISNAT(mark(X)) -> ISNAT(X)
26.18/8.16	
26.18/8.16	The TRS R consists of the following rules:
26.18/8.16	
26.18/8.16	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.16	   active(U12(tt)) -> mark(tt)
26.18/8.16	   active(U21(tt)) -> mark(tt)
26.18/8.16	   active(U31(tt, N)) -> mark(N)
26.18/8.16	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.16	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.16	   active(isNat(0)) -> mark(tt)
26.18/8.16	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.16	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.16	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.16	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.16	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.16	   mark(tt) -> active(tt)
26.18/8.16	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.16	   mark(isNat(X)) -> active(isNat(X))
26.18/8.16	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.16	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.16	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.16	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.16	   mark(s(X)) -> active(s(mark(X)))
26.18/8.16	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.16	   mark(0) -> active(0)
26.18/8.16	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.16	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.16	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.16	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.16	   U12(mark(X)) -> U12(X)
26.18/8.16	   U12(active(X)) -> U12(X)
26.18/8.16	   isNat(mark(X)) -> isNat(X)
26.18/8.16	   isNat(active(X)) -> isNat(X)
26.18/8.16	   U21(mark(X)) -> U21(X)
26.18/8.16	   U21(active(X)) -> U21(X)
26.18/8.16	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.16	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.16	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.16	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.16	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.16	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.16	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.16	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.16	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.16	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.16	   s(mark(X)) -> s(X)
26.18/8.16	   s(active(X)) -> s(X)
26.18/8.16	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.16	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.16	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.16	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.16	
26.18/8.16	The set Q consists of the following terms:
26.18/8.16	
26.18/8.16	   active(U11(tt, x0))
26.18/8.16	   active(U12(tt))
26.18/8.16	   active(U21(tt))
26.18/8.16	   active(U31(tt, x0))
26.18/8.16	   active(U41(tt, x0, x1))
26.18/8.16	   active(U42(tt, x0, x1))
26.18/8.16	   active(isNat(0))
26.18/8.16	   active(isNat(plus(x0, x1)))
26.18/8.16	   active(isNat(s(x0)))
26.18/8.16	   active(plus(x0, 0))
26.18/8.16	   active(plus(x0, s(x1)))
26.18/8.16	   mark(U11(x0, x1))
26.18/8.16	   mark(tt)
26.18/8.16	   mark(U12(x0))
26.18/8.16	   mark(isNat(x0))
26.18/8.16	   mark(U21(x0))
26.18/8.16	   mark(U31(x0, x1))
26.18/8.16	   mark(U41(x0, x1, x2))
26.18/8.16	   mark(U42(x0, x1, x2))
26.18/8.16	   mark(s(x0))
26.18/8.16	   mark(plus(x0, x1))
26.18/8.16	   mark(0)
26.18/8.16	   U11(mark(x0), x1)
26.18/8.16	   U11(x0, mark(x1))
26.18/8.16	   U11(active(x0), x1)
26.18/8.16	   U11(x0, active(x1))
26.18/8.16	   U12(mark(x0))
26.18/8.16	   U12(active(x0))
26.18/8.16	   isNat(mark(x0))
26.18/8.16	   isNat(active(x0))
26.18/8.16	   U21(mark(x0))
26.18/8.16	   U21(active(x0))
26.18/8.16	   U31(mark(x0), x1)
26.18/8.16	   U31(x0, mark(x1))
26.18/8.16	   U31(active(x0), x1)
26.18/8.16	   U31(x0, active(x1))
26.18/8.16	   U41(mark(x0), x1, x2)
26.18/8.16	   U41(x0, mark(x1), x2)
26.18/8.16	   U41(x0, x1, mark(x2))
26.18/8.16	   U41(active(x0), x1, x2)
26.18/8.16	   U41(x0, active(x1), x2)
26.18/8.16	   U41(x0, x1, active(x2))
26.18/8.16	   U42(mark(x0), x1, x2)
26.18/8.16	   U42(x0, mark(x1), x2)
26.18/8.16	   U42(x0, x1, mark(x2))
26.18/8.16	   U42(active(x0), x1, x2)
26.18/8.16	   U42(x0, active(x1), x2)
26.18/8.16	   U42(x0, x1, active(x2))
26.18/8.16	   s(mark(x0))
26.18/8.16	   s(active(x0))
26.18/8.16	   plus(mark(x0), x1)
26.18/8.16	   plus(x0, mark(x1))
26.18/8.16	   plus(active(x0), x1)
26.18/8.16	   plus(x0, active(x1))
26.18/8.16	
26.18/8.16	We have to consider all minimal (P,Q,R)-chains.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(48) UsableRulesProof (EQUIVALENT)
26.18/8.16	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.
26.18/8.16	----------------------------------------
26.18/8.16	
26.18/8.16	(49)
26.18/8.16	Obligation:
26.18/8.16	Q DP problem:
26.18/8.16	The TRS P consists of the following rules:
26.18/8.16	
26.18/8.16	   ISNAT(active(X)) -> ISNAT(X)
26.18/8.16	   ISNAT(mark(X)) -> ISNAT(X)
26.18/8.16	
26.18/8.16	R is empty.
26.18/8.16	The set Q consists of the following terms:
26.18/8.16	
26.18/8.16	   active(U11(tt, x0))
26.18/8.16	   active(U12(tt))
26.18/8.16	   active(U21(tt))
26.18/8.16	   active(U31(tt, x0))
26.18/8.16	   active(U41(tt, x0, x1))
26.18/8.16	   active(U42(tt, x0, x1))
26.18/8.16	   active(isNat(0))
26.18/8.16	   active(isNat(plus(x0, x1)))
26.18/8.16	   active(isNat(s(x0)))
26.18/8.16	   active(plus(x0, 0))
26.18/8.16	   active(plus(x0, s(x1)))
26.18/8.16	   mark(U11(x0, x1))
26.18/8.16	   mark(tt)
26.18/8.16	   mark(U12(x0))
26.18/8.16	   mark(isNat(x0))
26.18/8.16	   mark(U21(x0))
26.18/8.16	   mark(U31(x0, x1))
26.18/8.16	   mark(U41(x0, x1, x2))
26.18/8.16	   mark(U42(x0, x1, x2))
26.18/8.16	   mark(s(x0))
26.18/8.16	   mark(plus(x0, x1))
26.18/8.16	   mark(0)
26.18/8.16	   U11(mark(x0), x1)
26.18/8.16	   U11(x0, mark(x1))
26.18/8.16	   U11(active(x0), x1)
26.18/8.16	   U11(x0, active(x1))
26.18/8.16	   U12(mark(x0))
26.18/8.16	   U12(active(x0))
26.18/8.16	   isNat(mark(x0))
26.18/8.16	   isNat(active(x0))
26.18/8.16	   U21(mark(x0))
26.18/8.16	   U21(active(x0))
26.18/8.16	   U31(mark(x0), x1)
26.18/8.16	   U31(x0, mark(x1))
26.18/8.16	   U31(active(x0), x1)
26.18/8.16	   U31(x0, active(x1))
26.18/8.16	   U41(mark(x0), x1, x2)
26.18/8.16	   U41(x0, mark(x1), x2)
26.18/8.16	   U41(x0, x1, mark(x2))
26.18/8.16	   U41(active(x0), x1, x2)
26.18/8.16	   U41(x0, active(x1), x2)
26.18/8.16	   U41(x0, x1, active(x2))
26.18/8.16	   U42(mark(x0), x1, x2)
26.18/8.16	   U42(x0, mark(x1), x2)
26.18/8.16	   U42(x0, x1, mark(x2))
26.18/8.16	   U42(active(x0), x1, x2)
26.18/8.16	   U42(x0, active(x1), x2)
26.18/8.16	   U42(x0, x1, active(x2))
26.18/8.16	   s(mark(x0))
26.18/8.17	   s(active(x0))
26.18/8.17	   plus(mark(x0), x1)
26.18/8.17	   plus(x0, mark(x1))
26.18/8.17	   plus(active(x0), x1)
26.18/8.17	   plus(x0, active(x1))
26.18/8.17	
26.18/8.17	We have to consider all minimal (P,Q,R)-chains.
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(50) QReductionProof (EQUIVALENT)
26.18/8.17	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
26.18/8.17	
26.18/8.17	   U11(mark(x0), x1)
26.18/8.17	   U11(x0, mark(x1))
26.18/8.17	   U11(active(x0), x1)
26.18/8.17	   U11(x0, active(x1))
26.18/8.17	   U12(mark(x0))
26.18/8.17	   U12(active(x0))
26.18/8.17	   isNat(mark(x0))
26.18/8.17	   isNat(active(x0))
26.18/8.17	   U21(mark(x0))
26.18/8.17	   U21(active(x0))
26.18/8.17	   U31(mark(x0), x1)
26.18/8.17	   U31(x0, mark(x1))
26.18/8.17	   U31(active(x0), x1)
26.18/8.17	   U31(x0, active(x1))
26.18/8.17	   U41(mark(x0), x1, x2)
26.18/8.17	   U41(x0, mark(x1), x2)
26.18/8.17	   U41(x0, x1, mark(x2))
26.18/8.17	   U41(active(x0), x1, x2)
26.18/8.17	   U41(x0, active(x1), x2)
26.18/8.17	   U41(x0, x1, active(x2))
26.18/8.17	   U42(mark(x0), x1, x2)
26.18/8.17	   U42(x0, mark(x1), x2)
26.18/8.17	   U42(x0, x1, mark(x2))
26.18/8.17	   U42(active(x0), x1, x2)
26.18/8.17	   U42(x0, active(x1), x2)
26.18/8.17	   U42(x0, x1, active(x2))
26.18/8.17	   s(mark(x0))
26.18/8.17	   s(active(x0))
26.18/8.17	   plus(mark(x0), x1)
26.18/8.17	   plus(x0, mark(x1))
26.18/8.17	   plus(active(x0), x1)
26.18/8.17	   plus(x0, active(x1))
26.18/8.17	
26.18/8.17	
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(51)
26.18/8.17	Obligation:
26.18/8.17	Q DP problem:
26.18/8.17	The TRS P consists of the following rules:
26.18/8.17	
26.18/8.17	   ISNAT(active(X)) -> ISNAT(X)
26.18/8.17	   ISNAT(mark(X)) -> ISNAT(X)
26.18/8.17	
26.18/8.17	R is empty.
26.18/8.17	The set Q consists of the following terms:
26.18/8.17	
26.18/8.17	   active(U11(tt, x0))
26.18/8.17	   active(U12(tt))
26.18/8.17	   active(U21(tt))
26.18/8.17	   active(U31(tt, x0))
26.18/8.17	   active(U41(tt, x0, x1))
26.18/8.17	   active(U42(tt, x0, x1))
26.18/8.17	   active(isNat(0))
26.18/8.17	   active(isNat(plus(x0, x1)))
26.18/8.17	   active(isNat(s(x0)))
26.18/8.17	   active(plus(x0, 0))
26.18/8.17	   active(plus(x0, s(x1)))
26.18/8.17	   mark(U11(x0, x1))
26.18/8.17	   mark(tt)
26.18/8.17	   mark(U12(x0))
26.18/8.17	   mark(isNat(x0))
26.18/8.17	   mark(U21(x0))
26.18/8.17	   mark(U31(x0, x1))
26.18/8.17	   mark(U41(x0, x1, x2))
26.18/8.17	   mark(U42(x0, x1, x2))
26.18/8.17	   mark(s(x0))
26.18/8.17	   mark(plus(x0, x1))
26.18/8.17	   mark(0)
26.18/8.17	
26.18/8.17	We have to consider all minimal (P,Q,R)-chains.
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(52) QDPSizeChangeProof (EQUIVALENT)
26.18/8.17	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. 
26.18/8.17	
26.18/8.17	From the DPs we obtained the following set of size-change graphs:
26.18/8.17	*ISNAT(active(X)) -> ISNAT(X)
26.18/8.17	The graph contains the following edges 1 > 1
26.18/8.17	
26.18/8.17	
26.18/8.17	*ISNAT(mark(X)) -> ISNAT(X)
26.18/8.17	The graph contains the following edges 1 > 1
26.18/8.17	
26.18/8.17	
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(53)
26.18/8.17	YES
26.18/8.17	
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(54)
26.18/8.17	Obligation:
26.18/8.17	Q DP problem:
26.18/8.17	The TRS P consists of the following rules:
26.18/8.17	
26.18/8.17	   U12^1(active(X)) -> U12^1(X)
26.18/8.17	   U12^1(mark(X)) -> U12^1(X)
26.18/8.17	
26.18/8.17	The TRS R consists of the following rules:
26.18/8.17	
26.18/8.17	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.17	   active(U12(tt)) -> mark(tt)
26.18/8.17	   active(U21(tt)) -> mark(tt)
26.18/8.17	   active(U31(tt, N)) -> mark(N)
26.18/8.17	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.17	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.17	   active(isNat(0)) -> mark(tt)
26.18/8.17	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.17	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.17	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.17	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.17	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.17	   mark(tt) -> active(tt)
26.18/8.17	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.17	   mark(isNat(X)) -> active(isNat(X))
26.18/8.17	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.17	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.17	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.17	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.17	   mark(s(X)) -> active(s(mark(X)))
26.18/8.17	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.17	   mark(0) -> active(0)
26.18/8.17	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.17	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.17	   U12(mark(X)) -> U12(X)
26.18/8.17	   U12(active(X)) -> U12(X)
26.18/8.17	   isNat(mark(X)) -> isNat(X)
26.18/8.17	   isNat(active(X)) -> isNat(X)
26.18/8.17	   U21(mark(X)) -> U21(X)
26.18/8.17	   U21(active(X)) -> U21(X)
26.18/8.17	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.17	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.17	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.17	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.17	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.17	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.17	   s(mark(X)) -> s(X)
26.18/8.17	   s(active(X)) -> s(X)
26.18/8.17	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.17	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.17	
26.18/8.17	The set Q consists of the following terms:
26.18/8.17	
26.18/8.17	   active(U11(tt, x0))
26.18/8.17	   active(U12(tt))
26.18/8.17	   active(U21(tt))
26.18/8.17	   active(U31(tt, x0))
26.18/8.17	   active(U41(tt, x0, x1))
26.18/8.17	   active(U42(tt, x0, x1))
26.18/8.17	   active(isNat(0))
26.18/8.17	   active(isNat(plus(x0, x1)))
26.18/8.17	   active(isNat(s(x0)))
26.18/8.17	   active(plus(x0, 0))
26.18/8.17	   active(plus(x0, s(x1)))
26.18/8.17	   mark(U11(x0, x1))
26.18/8.17	   mark(tt)
26.18/8.17	   mark(U12(x0))
26.18/8.17	   mark(isNat(x0))
26.18/8.17	   mark(U21(x0))
26.18/8.17	   mark(U31(x0, x1))
26.18/8.17	   mark(U41(x0, x1, x2))
26.18/8.17	   mark(U42(x0, x1, x2))
26.18/8.17	   mark(s(x0))
26.18/8.17	   mark(plus(x0, x1))
26.18/8.17	   mark(0)
26.18/8.17	   U11(mark(x0), x1)
26.18/8.17	   U11(x0, mark(x1))
26.18/8.17	   U11(active(x0), x1)
26.18/8.17	   U11(x0, active(x1))
26.18/8.17	   U12(mark(x0))
26.18/8.17	   U12(active(x0))
26.18/8.17	   isNat(mark(x0))
26.18/8.17	   isNat(active(x0))
26.18/8.17	   U21(mark(x0))
26.18/8.17	   U21(active(x0))
26.18/8.17	   U31(mark(x0), x1)
26.18/8.17	   U31(x0, mark(x1))
26.18/8.17	   U31(active(x0), x1)
26.18/8.17	   U31(x0, active(x1))
26.18/8.17	   U41(mark(x0), x1, x2)
26.18/8.17	   U41(x0, mark(x1), x2)
26.18/8.17	   U41(x0, x1, mark(x2))
26.18/8.17	   U41(active(x0), x1, x2)
26.18/8.17	   U41(x0, active(x1), x2)
26.18/8.17	   U41(x0, x1, active(x2))
26.18/8.17	   U42(mark(x0), x1, x2)
26.18/8.17	   U42(x0, mark(x1), x2)
26.18/8.17	   U42(x0, x1, mark(x2))
26.18/8.17	   U42(active(x0), x1, x2)
26.18/8.17	   U42(x0, active(x1), x2)
26.18/8.17	   U42(x0, x1, active(x2))
26.18/8.17	   s(mark(x0))
26.18/8.17	   s(active(x0))
26.18/8.17	   plus(mark(x0), x1)
26.18/8.17	   plus(x0, mark(x1))
26.18/8.17	   plus(active(x0), x1)
26.18/8.17	   plus(x0, active(x1))
26.18/8.17	
26.18/8.17	We have to consider all minimal (P,Q,R)-chains.
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(55) UsableRulesProof (EQUIVALENT)
26.18/8.17	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.
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(56)
26.18/8.17	Obligation:
26.18/8.17	Q DP problem:
26.18/8.17	The TRS P consists of the following rules:
26.18/8.17	
26.18/8.17	   U12^1(active(X)) -> U12^1(X)
26.18/8.17	   U12^1(mark(X)) -> U12^1(X)
26.18/8.17	
26.18/8.17	R is empty.
26.18/8.17	The set Q consists of the following terms:
26.18/8.17	
26.18/8.17	   active(U11(tt, x0))
26.18/8.17	   active(U12(tt))
26.18/8.17	   active(U21(tt))
26.18/8.17	   active(U31(tt, x0))
26.18/8.17	   active(U41(tt, x0, x1))
26.18/8.17	   active(U42(tt, x0, x1))
26.18/8.17	   active(isNat(0))
26.18/8.17	   active(isNat(plus(x0, x1)))
26.18/8.17	   active(isNat(s(x0)))
26.18/8.17	   active(plus(x0, 0))
26.18/8.17	   active(plus(x0, s(x1)))
26.18/8.17	   mark(U11(x0, x1))
26.18/8.17	   mark(tt)
26.18/8.17	   mark(U12(x0))
26.18/8.17	   mark(isNat(x0))
26.18/8.17	   mark(U21(x0))
26.18/8.17	   mark(U31(x0, x1))
26.18/8.17	   mark(U41(x0, x1, x2))
26.18/8.17	   mark(U42(x0, x1, x2))
26.18/8.17	   mark(s(x0))
26.18/8.17	   mark(plus(x0, x1))
26.18/8.17	   mark(0)
26.18/8.17	   U11(mark(x0), x1)
26.18/8.17	   U11(x0, mark(x1))
26.18/8.17	   U11(active(x0), x1)
26.18/8.17	   U11(x0, active(x1))
26.18/8.17	   U12(mark(x0))
26.18/8.17	   U12(active(x0))
26.18/8.17	   isNat(mark(x0))
26.18/8.17	   isNat(active(x0))
26.18/8.17	   U21(mark(x0))
26.18/8.17	   U21(active(x0))
26.18/8.17	   U31(mark(x0), x1)
26.18/8.17	   U31(x0, mark(x1))
26.18/8.17	   U31(active(x0), x1)
26.18/8.17	   U31(x0, active(x1))
26.18/8.17	   U41(mark(x0), x1, x2)
26.18/8.17	   U41(x0, mark(x1), x2)
26.18/8.17	   U41(x0, x1, mark(x2))
26.18/8.17	   U41(active(x0), x1, x2)
26.18/8.17	   U41(x0, active(x1), x2)
26.18/8.17	   U41(x0, x1, active(x2))
26.18/8.17	   U42(mark(x0), x1, x2)
26.18/8.17	   U42(x0, mark(x1), x2)
26.18/8.17	   U42(x0, x1, mark(x2))
26.18/8.17	   U42(active(x0), x1, x2)
26.18/8.17	   U42(x0, active(x1), x2)
26.18/8.17	   U42(x0, x1, active(x2))
26.18/8.17	   s(mark(x0))
26.18/8.17	   s(active(x0))
26.18/8.17	   plus(mark(x0), x1)
26.18/8.17	   plus(x0, mark(x1))
26.18/8.17	   plus(active(x0), x1)
26.18/8.17	   plus(x0, active(x1))
26.18/8.17	
26.18/8.17	We have to consider all minimal (P,Q,R)-chains.
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(57) QReductionProof (EQUIVALENT)
26.18/8.17	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
26.18/8.17	
26.18/8.17	   U11(mark(x0), x1)
26.18/8.17	   U11(x0, mark(x1))
26.18/8.17	   U11(active(x0), x1)
26.18/8.17	   U11(x0, active(x1))
26.18/8.17	   U12(mark(x0))
26.18/8.17	   U12(active(x0))
26.18/8.17	   isNat(mark(x0))
26.18/8.17	   isNat(active(x0))
26.18/8.17	   U21(mark(x0))
26.18/8.17	   U21(active(x0))
26.18/8.17	   U31(mark(x0), x1)
26.18/8.17	   U31(x0, mark(x1))
26.18/8.17	   U31(active(x0), x1)
26.18/8.17	   U31(x0, active(x1))
26.18/8.17	   U41(mark(x0), x1, x2)
26.18/8.17	   U41(x0, mark(x1), x2)
26.18/8.17	   U41(x0, x1, mark(x2))
26.18/8.17	   U41(active(x0), x1, x2)
26.18/8.17	   U41(x0, active(x1), x2)
26.18/8.17	   U41(x0, x1, active(x2))
26.18/8.17	   U42(mark(x0), x1, x2)
26.18/8.17	   U42(x0, mark(x1), x2)
26.18/8.17	   U42(x0, x1, mark(x2))
26.18/8.17	   U42(active(x0), x1, x2)
26.18/8.17	   U42(x0, active(x1), x2)
26.18/8.17	   U42(x0, x1, active(x2))
26.18/8.17	   s(mark(x0))
26.18/8.17	   s(active(x0))
26.18/8.17	   plus(mark(x0), x1)
26.18/8.17	   plus(x0, mark(x1))
26.18/8.17	   plus(active(x0), x1)
26.18/8.17	   plus(x0, active(x1))
26.18/8.17	
26.18/8.17	
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(58)
26.18/8.17	Obligation:
26.18/8.17	Q DP problem:
26.18/8.17	The TRS P consists of the following rules:
26.18/8.17	
26.18/8.17	   U12^1(active(X)) -> U12^1(X)
26.18/8.17	   U12^1(mark(X)) -> U12^1(X)
26.18/8.17	
26.18/8.17	R is empty.
26.18/8.17	The set Q consists of the following terms:
26.18/8.17	
26.18/8.17	   active(U11(tt, x0))
26.18/8.17	   active(U12(tt))
26.18/8.17	   active(U21(tt))
26.18/8.17	   active(U31(tt, x0))
26.18/8.17	   active(U41(tt, x0, x1))
26.18/8.17	   active(U42(tt, x0, x1))
26.18/8.17	   active(isNat(0))
26.18/8.17	   active(isNat(plus(x0, x1)))
26.18/8.17	   active(isNat(s(x0)))
26.18/8.17	   active(plus(x0, 0))
26.18/8.17	   active(plus(x0, s(x1)))
26.18/8.17	   mark(U11(x0, x1))
26.18/8.17	   mark(tt)
26.18/8.17	   mark(U12(x0))
26.18/8.17	   mark(isNat(x0))
26.18/8.17	   mark(U21(x0))
26.18/8.17	   mark(U31(x0, x1))
26.18/8.17	   mark(U41(x0, x1, x2))
26.18/8.17	   mark(U42(x0, x1, x2))
26.18/8.17	   mark(s(x0))
26.18/8.17	   mark(plus(x0, x1))
26.18/8.17	   mark(0)
26.18/8.17	
26.18/8.17	We have to consider all minimal (P,Q,R)-chains.
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(59) QDPSizeChangeProof (EQUIVALENT)
26.18/8.17	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. 
26.18/8.17	
26.18/8.17	From the DPs we obtained the following set of size-change graphs:
26.18/8.17	*U12^1(active(X)) -> U12^1(X)
26.18/8.17	The graph contains the following edges 1 > 1
26.18/8.17	
26.18/8.17	
26.18/8.17	*U12^1(mark(X)) -> U12^1(X)
26.18/8.17	The graph contains the following edges 1 > 1
26.18/8.17	
26.18/8.17	
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(60)
26.18/8.17	YES
26.18/8.17	
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(61)
26.18/8.17	Obligation:
26.18/8.17	Q DP problem:
26.18/8.17	The TRS P consists of the following rules:
26.18/8.17	
26.18/8.17	   U11^1(X1, mark(X2)) -> U11^1(X1, X2)
26.18/8.17	   U11^1(mark(X1), X2) -> U11^1(X1, X2)
26.18/8.17	   U11^1(active(X1), X2) -> U11^1(X1, X2)
26.18/8.17	   U11^1(X1, active(X2)) -> U11^1(X1, X2)
26.18/8.17	
26.18/8.17	The TRS R consists of the following rules:
26.18/8.17	
26.18/8.17	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.17	   active(U12(tt)) -> mark(tt)
26.18/8.17	   active(U21(tt)) -> mark(tt)
26.18/8.17	   active(U31(tt, N)) -> mark(N)
26.18/8.17	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.17	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.17	   active(isNat(0)) -> mark(tt)
26.18/8.17	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.17	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.17	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.17	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.17	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.17	   mark(tt) -> active(tt)
26.18/8.17	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.17	   mark(isNat(X)) -> active(isNat(X))
26.18/8.17	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.17	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.17	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.17	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.17	   mark(s(X)) -> active(s(mark(X)))
26.18/8.17	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.17	   mark(0) -> active(0)
26.18/8.17	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.17	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.17	   U12(mark(X)) -> U12(X)
26.18/8.17	   U12(active(X)) -> U12(X)
26.18/8.17	   isNat(mark(X)) -> isNat(X)
26.18/8.17	   isNat(active(X)) -> isNat(X)
26.18/8.17	   U21(mark(X)) -> U21(X)
26.18/8.17	   U21(active(X)) -> U21(X)
26.18/8.17	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.17	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.17	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.17	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.17	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.17	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.17	   s(mark(X)) -> s(X)
26.18/8.17	   s(active(X)) -> s(X)
26.18/8.17	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.17	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.17	
26.18/8.17	The set Q consists of the following terms:
26.18/8.17	
26.18/8.17	   active(U11(tt, x0))
26.18/8.17	   active(U12(tt))
26.18/8.17	   active(U21(tt))
26.18/8.17	   active(U31(tt, x0))
26.18/8.17	   active(U41(tt, x0, x1))
26.18/8.17	   active(U42(tt, x0, x1))
26.18/8.17	   active(isNat(0))
26.18/8.17	   active(isNat(plus(x0, x1)))
26.18/8.17	   active(isNat(s(x0)))
26.18/8.17	   active(plus(x0, 0))
26.18/8.17	   active(plus(x0, s(x1)))
26.18/8.17	   mark(U11(x0, x1))
26.18/8.17	   mark(tt)
26.18/8.17	   mark(U12(x0))
26.18/8.17	   mark(isNat(x0))
26.18/8.17	   mark(U21(x0))
26.18/8.17	   mark(U31(x0, x1))
26.18/8.17	   mark(U41(x0, x1, x2))
26.18/8.17	   mark(U42(x0, x1, x2))
26.18/8.17	   mark(s(x0))
26.18/8.17	   mark(plus(x0, x1))
26.18/8.17	   mark(0)
26.18/8.17	   U11(mark(x0), x1)
26.18/8.17	   U11(x0, mark(x1))
26.18/8.17	   U11(active(x0), x1)
26.18/8.17	   U11(x0, active(x1))
26.18/8.17	   U12(mark(x0))
26.18/8.17	   U12(active(x0))
26.18/8.17	   isNat(mark(x0))
26.18/8.17	   isNat(active(x0))
26.18/8.17	   U21(mark(x0))
26.18/8.17	   U21(active(x0))
26.18/8.17	   U31(mark(x0), x1)
26.18/8.17	   U31(x0, mark(x1))
26.18/8.17	   U31(active(x0), x1)
26.18/8.17	   U31(x0, active(x1))
26.18/8.17	   U41(mark(x0), x1, x2)
26.18/8.17	   U41(x0, mark(x1), x2)
26.18/8.17	   U41(x0, x1, mark(x2))
26.18/8.17	   U41(active(x0), x1, x2)
26.18/8.17	   U41(x0, active(x1), x2)
26.18/8.17	   U41(x0, x1, active(x2))
26.18/8.17	   U42(mark(x0), x1, x2)
26.18/8.17	   U42(x0, mark(x1), x2)
26.18/8.17	   U42(x0, x1, mark(x2))
26.18/8.17	   U42(active(x0), x1, x2)
26.18/8.17	   U42(x0, active(x1), x2)
26.18/8.17	   U42(x0, x1, active(x2))
26.18/8.17	   s(mark(x0))
26.18/8.17	   s(active(x0))
26.18/8.17	   plus(mark(x0), x1)
26.18/8.17	   plus(x0, mark(x1))
26.18/8.17	   plus(active(x0), x1)
26.18/8.17	   plus(x0, active(x1))
26.18/8.17	
26.18/8.17	We have to consider all minimal (P,Q,R)-chains.
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(62) UsableRulesProof (EQUIVALENT)
26.18/8.17	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.
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(63)
26.18/8.17	Obligation:
26.18/8.17	Q DP problem:
26.18/8.17	The TRS P consists of the following rules:
26.18/8.17	
26.18/8.17	   U11^1(X1, mark(X2)) -> U11^1(X1, X2)
26.18/8.17	   U11^1(mark(X1), X2) -> U11^1(X1, X2)
26.18/8.17	   U11^1(active(X1), X2) -> U11^1(X1, X2)
26.18/8.17	   U11^1(X1, active(X2)) -> U11^1(X1, X2)
26.18/8.17	
26.18/8.17	R is empty.
26.18/8.17	The set Q consists of the following terms:
26.18/8.17	
26.18/8.17	   active(U11(tt, x0))
26.18/8.17	   active(U12(tt))
26.18/8.17	   active(U21(tt))
26.18/8.17	   active(U31(tt, x0))
26.18/8.17	   active(U41(tt, x0, x1))
26.18/8.17	   active(U42(tt, x0, x1))
26.18/8.17	   active(isNat(0))
26.18/8.17	   active(isNat(plus(x0, x1)))
26.18/8.17	   active(isNat(s(x0)))
26.18/8.17	   active(plus(x0, 0))
26.18/8.17	   active(plus(x0, s(x1)))
26.18/8.17	   mark(U11(x0, x1))
26.18/8.17	   mark(tt)
26.18/8.17	   mark(U12(x0))
26.18/8.17	   mark(isNat(x0))
26.18/8.17	   mark(U21(x0))
26.18/8.17	   mark(U31(x0, x1))
26.18/8.17	   mark(U41(x0, x1, x2))
26.18/8.17	   mark(U42(x0, x1, x2))
26.18/8.17	   mark(s(x0))
26.18/8.17	   mark(plus(x0, x1))
26.18/8.17	   mark(0)
26.18/8.17	   U11(mark(x0), x1)
26.18/8.17	   U11(x0, mark(x1))
26.18/8.17	   U11(active(x0), x1)
26.18/8.17	   U11(x0, active(x1))
26.18/8.17	   U12(mark(x0))
26.18/8.17	   U12(active(x0))
26.18/8.17	   isNat(mark(x0))
26.18/8.17	   isNat(active(x0))
26.18/8.17	   U21(mark(x0))
26.18/8.17	   U21(active(x0))
26.18/8.17	   U31(mark(x0), x1)
26.18/8.17	   U31(x0, mark(x1))
26.18/8.17	   U31(active(x0), x1)
26.18/8.17	   U31(x0, active(x1))
26.18/8.17	   U41(mark(x0), x1, x2)
26.18/8.17	   U41(x0, mark(x1), x2)
26.18/8.17	   U41(x0, x1, mark(x2))
26.18/8.17	   U41(active(x0), x1, x2)
26.18/8.17	   U41(x0, active(x1), x2)
26.18/8.17	   U41(x0, x1, active(x2))
26.18/8.17	   U42(mark(x0), x1, x2)
26.18/8.17	   U42(x0, mark(x1), x2)
26.18/8.17	   U42(x0, x1, mark(x2))
26.18/8.17	   U42(active(x0), x1, x2)
26.18/8.17	   U42(x0, active(x1), x2)
26.18/8.17	   U42(x0, x1, active(x2))
26.18/8.17	   s(mark(x0))
26.18/8.17	   s(active(x0))
26.18/8.17	   plus(mark(x0), x1)
26.18/8.17	   plus(x0, mark(x1))
26.18/8.17	   plus(active(x0), x1)
26.18/8.17	   plus(x0, active(x1))
26.18/8.17	
26.18/8.17	We have to consider all minimal (P,Q,R)-chains.
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(64) QReductionProof (EQUIVALENT)
26.18/8.17	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
26.18/8.17	
26.18/8.17	   U11(mark(x0), x1)
26.18/8.17	   U11(x0, mark(x1))
26.18/8.17	   U11(active(x0), x1)
26.18/8.17	   U11(x0, active(x1))
26.18/8.17	   U12(mark(x0))
26.18/8.17	   U12(active(x0))
26.18/8.17	   isNat(mark(x0))
26.18/8.17	   isNat(active(x0))
26.18/8.17	   U21(mark(x0))
26.18/8.17	   U21(active(x0))
26.18/8.17	   U31(mark(x0), x1)
26.18/8.17	   U31(x0, mark(x1))
26.18/8.17	   U31(active(x0), x1)
26.18/8.17	   U31(x0, active(x1))
26.18/8.17	   U41(mark(x0), x1, x2)
26.18/8.17	   U41(x0, mark(x1), x2)
26.18/8.17	   U41(x0, x1, mark(x2))
26.18/8.17	   U41(active(x0), x1, x2)
26.18/8.17	   U41(x0, active(x1), x2)
26.18/8.17	   U41(x0, x1, active(x2))
26.18/8.17	   U42(mark(x0), x1, x2)
26.18/8.17	   U42(x0, mark(x1), x2)
26.18/8.17	   U42(x0, x1, mark(x2))
26.18/8.17	   U42(active(x0), x1, x2)
26.18/8.17	   U42(x0, active(x1), x2)
26.18/8.17	   U42(x0, x1, active(x2))
26.18/8.17	   s(mark(x0))
26.18/8.17	   s(active(x0))
26.18/8.17	   plus(mark(x0), x1)
26.18/8.17	   plus(x0, mark(x1))
26.18/8.17	   plus(active(x0), x1)
26.18/8.17	   plus(x0, active(x1))
26.18/8.17	
26.18/8.17	
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(65)
26.18/8.17	Obligation:
26.18/8.17	Q DP problem:
26.18/8.17	The TRS P consists of the following rules:
26.18/8.17	
26.18/8.17	   U11^1(X1, mark(X2)) -> U11^1(X1, X2)
26.18/8.17	   U11^1(mark(X1), X2) -> U11^1(X1, X2)
26.18/8.17	   U11^1(active(X1), X2) -> U11^1(X1, X2)
26.18/8.17	   U11^1(X1, active(X2)) -> U11^1(X1, X2)
26.18/8.17	
26.18/8.17	R is empty.
26.18/8.17	The set Q consists of the following terms:
26.18/8.17	
26.18/8.17	   active(U11(tt, x0))
26.18/8.17	   active(U12(tt))
26.18/8.17	   active(U21(tt))
26.18/8.17	   active(U31(tt, x0))
26.18/8.17	   active(U41(tt, x0, x1))
26.18/8.17	   active(U42(tt, x0, x1))
26.18/8.17	   active(isNat(0))
26.18/8.17	   active(isNat(plus(x0, x1)))
26.18/8.17	   active(isNat(s(x0)))
26.18/8.17	   active(plus(x0, 0))
26.18/8.17	   active(plus(x0, s(x1)))
26.18/8.17	   mark(U11(x0, x1))
26.18/8.17	   mark(tt)
26.18/8.17	   mark(U12(x0))
26.18/8.17	   mark(isNat(x0))
26.18/8.17	   mark(U21(x0))
26.18/8.17	   mark(U31(x0, x1))
26.18/8.17	   mark(U41(x0, x1, x2))
26.18/8.17	   mark(U42(x0, x1, x2))
26.18/8.17	   mark(s(x0))
26.18/8.17	   mark(plus(x0, x1))
26.18/8.17	   mark(0)
26.18/8.17	
26.18/8.17	We have to consider all minimal (P,Q,R)-chains.
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(66) QDPSizeChangeProof (EQUIVALENT)
26.18/8.17	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. 
26.18/8.17	
26.18/8.17	From the DPs we obtained the following set of size-change graphs:
26.18/8.17	*U11^1(X1, mark(X2)) -> U11^1(X1, X2)
26.18/8.17	The graph contains the following edges 1 >= 1, 2 > 2
26.18/8.17	
26.18/8.17	
26.18/8.17	*U11^1(mark(X1), X2) -> U11^1(X1, X2)
26.18/8.17	The graph contains the following edges 1 > 1, 2 >= 2
26.18/8.17	
26.18/8.17	
26.18/8.17	*U11^1(active(X1), X2) -> U11^1(X1, X2)
26.18/8.17	The graph contains the following edges 1 > 1, 2 >= 2
26.18/8.17	
26.18/8.17	
26.18/8.17	*U11^1(X1, active(X2)) -> U11^1(X1, X2)
26.18/8.17	The graph contains the following edges 1 >= 1, 2 > 2
26.18/8.17	
26.18/8.17	
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(67)
26.18/8.17	YES
26.18/8.17	
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(68)
26.18/8.17	Obligation:
26.18/8.17	Q DP problem:
26.18/8.17	The TRS P consists of the following rules:
26.18/8.17	
26.18/8.17	   MARK(U12(X)) -> ACTIVE(U12(mark(X)))
26.18/8.17	   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
26.18/8.17	   MARK(U12(X)) -> MARK(X)
26.18/8.17	   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
26.18/8.17	   ACTIVE(U31(tt, N)) -> MARK(N)
26.18/8.17	   MARK(U11(X1, X2)) -> MARK(X1)
26.18/8.17	   MARK(isNat(X)) -> ACTIVE(isNat(X))
26.18/8.17	   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
26.18/8.17	   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
26.18/8.17	   MARK(U21(X)) -> ACTIVE(U21(mark(X)))
26.18/8.17	   ACTIVE(U41(tt, M, N)) -> MARK(U42(isNat(N), M, N))
26.18/8.17	   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
26.18/8.17	   ACTIVE(U42(tt, M, N)) -> MARK(s(plus(N, M)))
26.18/8.17	   MARK(s(X)) -> ACTIVE(s(mark(X)))
26.18/8.17	   ACTIVE(plus(N, 0)) -> MARK(U31(isNat(N), N))
26.18/8.17	   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
26.18/8.17	   ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N))
26.18/8.17	   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
26.18/8.17	   MARK(U41(X1, X2, X3)) -> MARK(X1)
26.18/8.17	   MARK(U21(X)) -> MARK(X)
26.18/8.17	   MARK(U31(X1, X2)) -> MARK(X1)
26.18/8.17	   MARK(U42(X1, X2, X3)) -> MARK(X1)
26.18/8.17	   MARK(s(X)) -> MARK(X)
26.18/8.17	   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
26.18/8.17	   MARK(plus(X1, X2)) -> MARK(X1)
26.18/8.17	   MARK(plus(X1, X2)) -> MARK(X2)
26.18/8.17	
26.18/8.17	The TRS R consists of the following rules:
26.18/8.17	
26.18/8.17	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.17	   active(U12(tt)) -> mark(tt)
26.18/8.17	   active(U21(tt)) -> mark(tt)
26.18/8.17	   active(U31(tt, N)) -> mark(N)
26.18/8.17	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.17	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.17	   active(isNat(0)) -> mark(tt)
26.18/8.17	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.17	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.17	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.17	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.17	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.17	   mark(tt) -> active(tt)
26.18/8.17	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.17	   mark(isNat(X)) -> active(isNat(X))
26.18/8.17	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.17	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.17	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.17	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.17	   mark(s(X)) -> active(s(mark(X)))
26.18/8.17	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.17	   mark(0) -> active(0)
26.18/8.17	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.17	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.17	   U12(mark(X)) -> U12(X)
26.18/8.17	   U12(active(X)) -> U12(X)
26.18/8.17	   isNat(mark(X)) -> isNat(X)
26.18/8.17	   isNat(active(X)) -> isNat(X)
26.18/8.17	   U21(mark(X)) -> U21(X)
26.18/8.17	   U21(active(X)) -> U21(X)
26.18/8.17	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.17	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.17	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.17	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.17	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.17	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.17	   s(mark(X)) -> s(X)
26.18/8.17	   s(active(X)) -> s(X)
26.18/8.17	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.17	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.17	
26.18/8.17	The set Q consists of the following terms:
26.18/8.17	
26.18/8.17	   active(U11(tt, x0))
26.18/8.17	   active(U12(tt))
26.18/8.17	   active(U21(tt))
26.18/8.17	   active(U31(tt, x0))
26.18/8.17	   active(U41(tt, x0, x1))
26.18/8.17	   active(U42(tt, x0, x1))
26.18/8.17	   active(isNat(0))
26.18/8.17	   active(isNat(plus(x0, x1)))
26.18/8.17	   active(isNat(s(x0)))
26.18/8.17	   active(plus(x0, 0))
26.18/8.17	   active(plus(x0, s(x1)))
26.18/8.17	   mark(U11(x0, x1))
26.18/8.17	   mark(tt)
26.18/8.17	   mark(U12(x0))
26.18/8.17	   mark(isNat(x0))
26.18/8.17	   mark(U21(x0))
26.18/8.17	   mark(U31(x0, x1))
26.18/8.17	   mark(U41(x0, x1, x2))
26.18/8.17	   mark(U42(x0, x1, x2))
26.18/8.17	   mark(s(x0))
26.18/8.17	   mark(plus(x0, x1))
26.18/8.17	   mark(0)
26.18/8.17	   U11(mark(x0), x1)
26.18/8.17	   U11(x0, mark(x1))
26.18/8.17	   U11(active(x0), x1)
26.18/8.17	   U11(x0, active(x1))
26.18/8.17	   U12(mark(x0))
26.18/8.17	   U12(active(x0))
26.18/8.17	   isNat(mark(x0))
26.18/8.17	   isNat(active(x0))
26.18/8.17	   U21(mark(x0))
26.18/8.17	   U21(active(x0))
26.18/8.17	   U31(mark(x0), x1)
26.18/8.17	   U31(x0, mark(x1))
26.18/8.17	   U31(active(x0), x1)
26.18/8.17	   U31(x0, active(x1))
26.18/8.17	   U41(mark(x0), x1, x2)
26.18/8.17	   U41(x0, mark(x1), x2)
26.18/8.17	   U41(x0, x1, mark(x2))
26.18/8.17	   U41(active(x0), x1, x2)
26.18/8.17	   U41(x0, active(x1), x2)
26.18/8.17	   U41(x0, x1, active(x2))
26.18/8.17	   U42(mark(x0), x1, x2)
26.18/8.17	   U42(x0, mark(x1), x2)
26.18/8.17	   U42(x0, x1, mark(x2))
26.18/8.17	   U42(active(x0), x1, x2)
26.18/8.17	   U42(x0, active(x1), x2)
26.18/8.17	   U42(x0, x1, active(x2))
26.18/8.17	   s(mark(x0))
26.18/8.17	   s(active(x0))
26.18/8.17	   plus(mark(x0), x1)
26.18/8.17	   plus(x0, mark(x1))
26.18/8.17	   plus(active(x0), x1)
26.18/8.17	   plus(x0, active(x1))
26.18/8.17	
26.18/8.17	We have to consider all minimal (P,Q,R)-chains.
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(69) UsableRulesProof (EQUIVALENT)
26.18/8.17	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.
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(70)
26.18/8.17	Obligation:
26.18/8.17	Q DP problem:
26.18/8.17	The TRS P consists of the following rules:
26.18/8.17	
26.18/8.17	   MARK(U12(X)) -> ACTIVE(U12(mark(X)))
26.18/8.17	   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
26.18/8.17	   MARK(U12(X)) -> MARK(X)
26.18/8.17	   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
26.18/8.17	   ACTIVE(U31(tt, N)) -> MARK(N)
26.18/8.17	   MARK(U11(X1, X2)) -> MARK(X1)
26.18/8.17	   MARK(isNat(X)) -> ACTIVE(isNat(X))
26.18/8.17	   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
26.18/8.17	   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
26.18/8.17	   MARK(U21(X)) -> ACTIVE(U21(mark(X)))
26.18/8.17	   ACTIVE(U41(tt, M, N)) -> MARK(U42(isNat(N), M, N))
26.18/8.17	   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
26.18/8.17	   ACTIVE(U42(tt, M, N)) -> MARK(s(plus(N, M)))
26.18/8.17	   MARK(s(X)) -> ACTIVE(s(mark(X)))
26.18/8.17	   ACTIVE(plus(N, 0)) -> MARK(U31(isNat(N), N))
26.18/8.17	   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
26.18/8.17	   ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N))
26.18/8.17	   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
26.18/8.17	   MARK(U41(X1, X2, X3)) -> MARK(X1)
26.18/8.17	   MARK(U21(X)) -> MARK(X)
26.18/8.17	   MARK(U31(X1, X2)) -> MARK(X1)
26.18/8.17	   MARK(U42(X1, X2, X3)) -> MARK(X1)
26.18/8.17	   MARK(s(X)) -> MARK(X)
26.18/8.17	   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
26.18/8.17	   MARK(plus(X1, X2)) -> MARK(X1)
26.18/8.17	   MARK(plus(X1, X2)) -> MARK(X2)
26.18/8.17	
26.18/8.17	The TRS R consists of the following rules:
26.18/8.17	
26.18/8.17	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.17	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.17	   active(U31(tt, N)) -> mark(N)
26.18/8.17	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.17	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.17	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.17	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.17	   mark(s(X)) -> active(s(mark(X)))
26.18/8.17	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.17	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.17	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.17	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.17	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.17	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.17	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.17	   mark(isNat(X)) -> active(isNat(X))
26.18/8.17	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.17	   mark(tt) -> active(tt)
26.18/8.17	   mark(0) -> active(0)
26.18/8.17	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.17	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.17	   U12(active(X)) -> U12(X)
26.18/8.17	   U12(mark(X)) -> U12(X)
26.18/8.17	   active(U12(tt)) -> mark(tt)
26.18/8.17	   active(U21(tt)) -> mark(tt)
26.18/8.17	   active(isNat(0)) -> mark(tt)
26.18/8.17	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.17	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.17	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.17	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.17	   s(active(X)) -> s(X)
26.18/8.17	   s(mark(X)) -> s(X)
26.18/8.17	   U21(active(X)) -> U21(X)
26.18/8.17	   U21(mark(X)) -> U21(X)
26.18/8.17	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.17	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.17	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.17	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.17	
26.18/8.17	The set Q consists of the following terms:
26.18/8.17	
26.18/8.17	   active(U11(tt, x0))
26.18/8.17	   active(U12(tt))
26.18/8.17	   active(U21(tt))
26.18/8.17	   active(U31(tt, x0))
26.18/8.17	   active(U41(tt, x0, x1))
26.18/8.17	   active(U42(tt, x0, x1))
26.18/8.17	   active(isNat(0))
26.18/8.17	   active(isNat(plus(x0, x1)))
26.18/8.17	   active(isNat(s(x0)))
26.18/8.17	   active(plus(x0, 0))
26.18/8.17	   active(plus(x0, s(x1)))
26.18/8.17	   mark(U11(x0, x1))
26.18/8.17	   mark(tt)
26.18/8.17	   mark(U12(x0))
26.18/8.17	   mark(isNat(x0))
26.18/8.17	   mark(U21(x0))
26.18/8.17	   mark(U31(x0, x1))
26.18/8.17	   mark(U41(x0, x1, x2))
26.18/8.17	   mark(U42(x0, x1, x2))
26.18/8.17	   mark(s(x0))
26.18/8.17	   mark(plus(x0, x1))
26.18/8.17	   mark(0)
26.18/8.17	   U11(mark(x0), x1)
26.18/8.17	   U11(x0, mark(x1))
26.18/8.17	   U11(active(x0), x1)
26.18/8.17	   U11(x0, active(x1))
26.18/8.17	   U12(mark(x0))
26.18/8.17	   U12(active(x0))
26.18/8.17	   isNat(mark(x0))
26.18/8.17	   isNat(active(x0))
26.18/8.17	   U21(mark(x0))
26.18/8.17	   U21(active(x0))
26.18/8.17	   U31(mark(x0), x1)
26.18/8.17	   U31(x0, mark(x1))
26.18/8.17	   U31(active(x0), x1)
26.18/8.17	   U31(x0, active(x1))
26.18/8.17	   U41(mark(x0), x1, x2)
26.18/8.17	   U41(x0, mark(x1), x2)
26.18/8.17	   U41(x0, x1, mark(x2))
26.18/8.17	   U41(active(x0), x1, x2)
26.18/8.17	   U41(x0, active(x1), x2)
26.18/8.17	   U41(x0, x1, active(x2))
26.18/8.17	   U42(mark(x0), x1, x2)
26.18/8.17	   U42(x0, mark(x1), x2)
26.18/8.17	   U42(x0, x1, mark(x2))
26.18/8.17	   U42(active(x0), x1, x2)
26.18/8.17	   U42(x0, active(x1), x2)
26.18/8.17	   U42(x0, x1, active(x2))
26.18/8.17	   s(mark(x0))
26.18/8.17	   s(active(x0))
26.18/8.17	   plus(mark(x0), x1)
26.18/8.17	   plus(x0, mark(x1))
26.18/8.17	   plus(active(x0), x1)
26.18/8.17	   plus(x0, active(x1))
26.18/8.17	
26.18/8.17	We have to consider all minimal (P,Q,R)-chains.
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(71) QDPOrderProof (EQUIVALENT)
26.18/8.17	We use the reduction pair processor [LPAR04,JAR06].
26.18/8.17	
26.18/8.17	
26.18/8.17	The following pairs can be oriented strictly and are deleted.
26.18/8.17	
26.18/8.17	   MARK(U12(X)) -> ACTIVE(U12(mark(X)))
26.18/8.17	   MARK(U21(X)) -> ACTIVE(U21(mark(X)))
26.18/8.17	   MARK(s(X)) -> ACTIVE(s(mark(X)))
26.18/8.17	The remaining pairs can at least be oriented weakly.
26.18/8.17	Used ordering:  Polynomial interpretation [POLO]:
26.18/8.17	
26.18/8.17	   POL(0) = 0
26.18/8.17	   POL(ACTIVE(x_1)) = x_1
26.18/8.17	   POL(MARK(x_1)) = 1
26.18/8.17	   POL(U11(x_1, x_2)) = 1
26.18/8.17	   POL(U12(x_1)) = 0
26.18/8.17	   POL(U21(x_1)) = 0
26.18/8.17	   POL(U31(x_1, x_2)) = 1
26.18/8.17	   POL(U41(x_1, x_2, x_3)) = 1
26.18/8.17	   POL(U42(x_1, x_2, x_3)) = 1
26.18/8.17	   POL(active(x_1)) = 0
26.18/8.17	   POL(isNat(x_1)) = 1
26.18/8.17	   POL(mark(x_1)) = 0
26.18/8.17	   POL(plus(x_1, x_2)) = 1
26.18/8.17	   POL(s(x_1)) = 0
26.18/8.17	   POL(tt) = 0
26.18/8.17	
26.18/8.17	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
26.18/8.17	
26.18/8.17	   U12(active(X)) -> U12(X)
26.18/8.17	   U12(mark(X)) -> U12(X)
26.18/8.17	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.17	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.17	   U21(active(X)) -> U21(X)
26.18/8.17	   U21(mark(X)) -> U21(X)
26.18/8.17	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.17	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.17	   s(active(X)) -> s(X)
26.18/8.17	   s(mark(X)) -> s(X)
26.18/8.17	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.17	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.17	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.17	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.17	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.17	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.17	
26.18/8.17	
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(72)
26.18/8.17	Obligation:
26.18/8.17	Q DP problem:
26.18/8.17	The TRS P consists of the following rules:
26.18/8.17	
26.18/8.17	   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
26.18/8.17	   MARK(U12(X)) -> MARK(X)
26.18/8.17	   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
26.18/8.17	   ACTIVE(U31(tt, N)) -> MARK(N)
26.18/8.17	   MARK(U11(X1, X2)) -> MARK(X1)
26.18/8.17	   MARK(isNat(X)) -> ACTIVE(isNat(X))
26.18/8.17	   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
26.18/8.17	   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
26.18/8.17	   ACTIVE(U41(tt, M, N)) -> MARK(U42(isNat(N), M, N))
26.18/8.17	   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
26.18/8.17	   ACTIVE(U42(tt, M, N)) -> MARK(s(plus(N, M)))
26.18/8.17	   ACTIVE(plus(N, 0)) -> MARK(U31(isNat(N), N))
26.18/8.17	   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
26.18/8.17	   ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N))
26.18/8.17	   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
26.18/8.17	   MARK(U41(X1, X2, X3)) -> MARK(X1)
26.18/8.17	   MARK(U21(X)) -> MARK(X)
26.18/8.17	   MARK(U31(X1, X2)) -> MARK(X1)
26.18/8.17	   MARK(U42(X1, X2, X3)) -> MARK(X1)
26.18/8.17	   MARK(s(X)) -> MARK(X)
26.18/8.17	   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
26.18/8.17	   MARK(plus(X1, X2)) -> MARK(X1)
26.18/8.17	   MARK(plus(X1, X2)) -> MARK(X2)
26.18/8.17	
26.18/8.17	The TRS R consists of the following rules:
26.18/8.17	
26.18/8.17	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.17	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.17	   active(U31(tt, N)) -> mark(N)
26.18/8.17	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.17	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.17	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.17	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.17	   mark(s(X)) -> active(s(mark(X)))
26.18/8.17	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.17	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.17	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.17	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.17	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.17	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.17	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.17	   mark(isNat(X)) -> active(isNat(X))
26.18/8.17	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.17	   mark(tt) -> active(tt)
26.18/8.17	   mark(0) -> active(0)
26.18/8.17	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.17	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.17	   U12(active(X)) -> U12(X)
26.18/8.17	   U12(mark(X)) -> U12(X)
26.18/8.17	   active(U12(tt)) -> mark(tt)
26.18/8.17	   active(U21(tt)) -> mark(tt)
26.18/8.17	   active(isNat(0)) -> mark(tt)
26.18/8.17	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.17	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.17	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.17	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.17	   s(active(X)) -> s(X)
26.18/8.17	   s(mark(X)) -> s(X)
26.18/8.17	   U21(active(X)) -> U21(X)
26.18/8.17	   U21(mark(X)) -> U21(X)
26.18/8.17	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.17	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.17	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.17	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.17	
26.18/8.17	The set Q consists of the following terms:
26.18/8.17	
26.18/8.17	   active(U11(tt, x0))
26.18/8.17	   active(U12(tt))
26.18/8.17	   active(U21(tt))
26.18/8.17	   active(U31(tt, x0))
26.18/8.17	   active(U41(tt, x0, x1))
26.18/8.17	   active(U42(tt, x0, x1))
26.18/8.17	   active(isNat(0))
26.18/8.17	   active(isNat(plus(x0, x1)))
26.18/8.17	   active(isNat(s(x0)))
26.18/8.17	   active(plus(x0, 0))
26.18/8.17	   active(plus(x0, s(x1)))
26.18/8.17	   mark(U11(x0, x1))
26.18/8.17	   mark(tt)
26.18/8.17	   mark(U12(x0))
26.18/8.17	   mark(isNat(x0))
26.18/8.17	   mark(U21(x0))
26.18/8.17	   mark(U31(x0, x1))
26.18/8.17	   mark(U41(x0, x1, x2))
26.18/8.17	   mark(U42(x0, x1, x2))
26.18/8.17	   mark(s(x0))
26.18/8.17	   mark(plus(x0, x1))
26.18/8.17	   mark(0)
26.18/8.17	   U11(mark(x0), x1)
26.18/8.17	   U11(x0, mark(x1))
26.18/8.17	   U11(active(x0), x1)
26.18/8.17	   U11(x0, active(x1))
26.18/8.17	   U12(mark(x0))
26.18/8.17	   U12(active(x0))
26.18/8.17	   isNat(mark(x0))
26.18/8.17	   isNat(active(x0))
26.18/8.17	   U21(mark(x0))
26.18/8.17	   U21(active(x0))
26.18/8.17	   U31(mark(x0), x1)
26.18/8.17	   U31(x0, mark(x1))
26.18/8.17	   U31(active(x0), x1)
26.18/8.17	   U31(x0, active(x1))
26.18/8.17	   U41(mark(x0), x1, x2)
26.18/8.17	   U41(x0, mark(x1), x2)
26.18/8.17	   U41(x0, x1, mark(x2))
26.18/8.17	   U41(active(x0), x1, x2)
26.18/8.17	   U41(x0, active(x1), x2)
26.18/8.17	   U41(x0, x1, active(x2))
26.18/8.17	   U42(mark(x0), x1, x2)
26.18/8.17	   U42(x0, mark(x1), x2)
26.18/8.17	   U42(x0, x1, mark(x2))
26.18/8.17	   U42(active(x0), x1, x2)
26.18/8.17	   U42(x0, active(x1), x2)
26.18/8.17	   U42(x0, x1, active(x2))
26.18/8.17	   s(mark(x0))
26.18/8.17	   s(active(x0))
26.18/8.17	   plus(mark(x0), x1)
26.18/8.17	   plus(x0, mark(x1))
26.18/8.17	   plus(active(x0), x1)
26.18/8.17	   plus(x0, active(x1))
26.18/8.17	
26.18/8.17	We have to consider all minimal (P,Q,R)-chains.
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(73) QDPOrderProof (EQUIVALENT)
26.18/8.17	We use the reduction pair processor [LPAR04,JAR06].
26.18/8.17	
26.18/8.17	
26.18/8.17	The following pairs can be oriented strictly and are deleted.
26.18/8.17	
26.18/8.17	   ACTIVE(plus(N, 0)) -> MARK(U31(isNat(N), N))
26.18/8.17	The remaining pairs can at least be oriented weakly.
26.18/8.17	Used ordering:  Polynomial interpretation [POLO]:
26.18/8.17	
26.18/8.17	   POL(0) = 1
26.18/8.17	   POL(ACTIVE(x_1)) = x_1
26.18/8.17	   POL(MARK(x_1)) = x_1
26.18/8.17	   POL(U11(x_1, x_2)) = x_1
26.18/8.17	   POL(U12(x_1)) = x_1
26.18/8.17	   POL(U21(x_1)) = x_1
26.18/8.17	   POL(U31(x_1, x_2)) = x_1 + x_2
26.18/8.17	   POL(U41(x_1, x_2, x_3)) = x_1 + x_2 + x_3
26.18/8.17	   POL(U42(x_1, x_2, x_3)) = x_1 + x_2 + x_3
26.18/8.17	   POL(active(x_1)) = x_1
26.18/8.17	   POL(isNat(x_1)) = 0
26.18/8.17	   POL(mark(x_1)) = x_1
26.18/8.17	   POL(plus(x_1, x_2)) = x_1 + x_2
26.18/8.17	   POL(s(x_1)) = x_1
26.18/8.17	   POL(tt) = 0
26.18/8.17	
26.18/8.17	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
26.18/8.17	
26.18/8.17	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.17	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.17	   active(U31(tt, N)) -> mark(N)
26.18/8.17	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.17	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.17	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.17	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.17	   mark(s(X)) -> active(s(mark(X)))
26.18/8.17	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.17	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.17	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.17	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.17	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.17	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.17	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.17	   mark(isNat(X)) -> active(isNat(X))
26.18/8.17	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.17	   mark(tt) -> active(tt)
26.18/8.17	   mark(0) -> active(0)
26.18/8.17	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.17	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.17	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.17	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.17	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.17	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.17	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.17	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.17	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.17	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.17	   U12(active(X)) -> U12(X)
26.18/8.17	   U12(mark(X)) -> U12(X)
26.18/8.17	   active(U12(tt)) -> mark(tt)
26.18/8.17	   active(U21(tt)) -> mark(tt)
26.18/8.17	   active(isNat(0)) -> mark(tt)
26.18/8.17	   s(active(X)) -> s(X)
26.18/8.17	   s(mark(X)) -> s(X)
26.18/8.17	   U21(active(X)) -> U21(X)
26.18/8.17	   U21(mark(X)) -> U21(X)
26.18/8.17	
26.18/8.17	
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(74)
26.18/8.17	Obligation:
26.18/8.17	Q DP problem:
26.18/8.17	The TRS P consists of the following rules:
26.18/8.17	
26.18/8.17	   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
26.18/8.17	   MARK(U12(X)) -> MARK(X)
26.18/8.17	   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
26.18/8.17	   ACTIVE(U31(tt, N)) -> MARK(N)
26.18/8.17	   MARK(U11(X1, X2)) -> MARK(X1)
26.18/8.17	   MARK(isNat(X)) -> ACTIVE(isNat(X))
26.18/8.17	   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
26.18/8.17	   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
26.18/8.17	   ACTIVE(U41(tt, M, N)) -> MARK(U42(isNat(N), M, N))
26.18/8.17	   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
26.18/8.17	   ACTIVE(U42(tt, M, N)) -> MARK(s(plus(N, M)))
26.18/8.17	   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
26.18/8.17	   ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N))
26.18/8.17	   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
26.18/8.17	   MARK(U41(X1, X2, X3)) -> MARK(X1)
26.18/8.17	   MARK(U21(X)) -> MARK(X)
26.18/8.17	   MARK(U31(X1, X2)) -> MARK(X1)
26.18/8.17	   MARK(U42(X1, X2, X3)) -> MARK(X1)
26.18/8.17	   MARK(s(X)) -> MARK(X)
26.18/8.17	   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
26.18/8.17	   MARK(plus(X1, X2)) -> MARK(X1)
26.18/8.17	   MARK(plus(X1, X2)) -> MARK(X2)
26.18/8.17	
26.18/8.17	The TRS R consists of the following rules:
26.18/8.17	
26.18/8.17	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.17	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.17	   active(U31(tt, N)) -> mark(N)
26.18/8.17	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.17	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.17	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.17	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.17	   mark(s(X)) -> active(s(mark(X)))
26.18/8.17	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.17	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.17	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.17	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.17	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.17	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.17	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.17	   mark(isNat(X)) -> active(isNat(X))
26.18/8.17	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.17	   mark(tt) -> active(tt)
26.18/8.17	   mark(0) -> active(0)
26.18/8.17	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.17	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.17	   U12(active(X)) -> U12(X)
26.18/8.17	   U12(mark(X)) -> U12(X)
26.18/8.17	   active(U12(tt)) -> mark(tt)
26.18/8.17	   active(U21(tt)) -> mark(tt)
26.18/8.17	   active(isNat(0)) -> mark(tt)
26.18/8.17	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.17	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.17	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.17	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.17	   s(active(X)) -> s(X)
26.18/8.17	   s(mark(X)) -> s(X)
26.18/8.17	   U21(active(X)) -> U21(X)
26.18/8.17	   U21(mark(X)) -> U21(X)
26.18/8.17	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.17	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.17	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.17	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.17	
26.18/8.17	The set Q consists of the following terms:
26.18/8.17	
26.18/8.17	   active(U11(tt, x0))
26.18/8.17	   active(U12(tt))
26.18/8.17	   active(U21(tt))
26.18/8.17	   active(U31(tt, x0))
26.18/8.17	   active(U41(tt, x0, x1))
26.18/8.17	   active(U42(tt, x0, x1))
26.18/8.17	   active(isNat(0))
26.18/8.17	   active(isNat(plus(x0, x1)))
26.18/8.17	   active(isNat(s(x0)))
26.18/8.17	   active(plus(x0, 0))
26.18/8.17	   active(plus(x0, s(x1)))
26.18/8.17	   mark(U11(x0, x1))
26.18/8.17	   mark(tt)
26.18/8.17	   mark(U12(x0))
26.18/8.17	   mark(isNat(x0))
26.18/8.17	   mark(U21(x0))
26.18/8.17	   mark(U31(x0, x1))
26.18/8.17	   mark(U41(x0, x1, x2))
26.18/8.17	   mark(U42(x0, x1, x2))
26.18/8.17	   mark(s(x0))
26.18/8.17	   mark(plus(x0, x1))
26.18/8.17	   mark(0)
26.18/8.17	   U11(mark(x0), x1)
26.18/8.17	   U11(x0, mark(x1))
26.18/8.17	   U11(active(x0), x1)
26.18/8.17	   U11(x0, active(x1))
26.18/8.17	   U12(mark(x0))
26.18/8.17	   U12(active(x0))
26.18/8.17	   isNat(mark(x0))
26.18/8.17	   isNat(active(x0))
26.18/8.17	   U21(mark(x0))
26.18/8.17	   U21(active(x0))
26.18/8.17	   U31(mark(x0), x1)
26.18/8.17	   U31(x0, mark(x1))
26.18/8.17	   U31(active(x0), x1)
26.18/8.17	   U31(x0, active(x1))
26.18/8.17	   U41(mark(x0), x1, x2)
26.18/8.17	   U41(x0, mark(x1), x2)
26.18/8.17	   U41(x0, x1, mark(x2))
26.18/8.17	   U41(active(x0), x1, x2)
26.18/8.17	   U41(x0, active(x1), x2)
26.18/8.17	   U41(x0, x1, active(x2))
26.18/8.17	   U42(mark(x0), x1, x2)
26.18/8.17	   U42(x0, mark(x1), x2)
26.18/8.17	   U42(x0, x1, mark(x2))
26.18/8.17	   U42(active(x0), x1, x2)
26.18/8.17	   U42(x0, active(x1), x2)
26.18/8.17	   U42(x0, x1, active(x2))
26.18/8.17	   s(mark(x0))
26.18/8.17	   s(active(x0))
26.18/8.17	   plus(mark(x0), x1)
26.18/8.17	   plus(x0, mark(x1))
26.18/8.17	   plus(active(x0), x1)
26.18/8.17	   plus(x0, active(x1))
26.18/8.17	
26.18/8.17	We have to consider all minimal (P,Q,R)-chains.
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(75) QDPOrderProof (EQUIVALENT)
26.18/8.17	We use the reduction pair processor [LPAR04,JAR06].
26.18/8.17	
26.18/8.17	
26.18/8.17	The following pairs can be oriented strictly and are deleted.
26.18/8.17	
26.18/8.17	   ACTIVE(U31(tt, N)) -> MARK(N)
26.18/8.17	   MARK(U31(X1, X2)) -> MARK(X1)
26.18/8.17	The remaining pairs can at least be oriented weakly.
26.18/8.17	Used ordering:  Polynomial interpretation [POLO]:
26.18/8.17	
26.18/8.17	   POL(0) = 1
26.18/8.17	   POL(ACTIVE(x_1)) = x_1
26.18/8.17	   POL(MARK(x_1)) = x_1
26.18/8.17	   POL(U11(x_1, x_2)) = x_1
26.18/8.17	   POL(U12(x_1)) = x_1
26.18/8.17	   POL(U21(x_1)) = x_1
26.18/8.17	   POL(U31(x_1, x_2)) = 1 + x_1 + x_2
26.18/8.17	   POL(U41(x_1, x_2, x_3)) = x_1 + x_2 + x_3
26.18/8.17	   POL(U42(x_1, x_2, x_3)) = x_1 + x_2 + x_3
26.18/8.17	   POL(active(x_1)) = x_1
26.18/8.17	   POL(isNat(x_1)) = 0
26.18/8.17	   POL(mark(x_1)) = x_1
26.18/8.17	   POL(plus(x_1, x_2)) = x_1 + x_2
26.18/8.17	   POL(s(x_1)) = x_1
26.18/8.17	   POL(tt) = 0
26.18/8.17	
26.18/8.17	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
26.18/8.17	
26.18/8.17	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.17	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.17	   active(U31(tt, N)) -> mark(N)
26.18/8.17	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.17	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.17	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.17	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.17	   mark(s(X)) -> active(s(mark(X)))
26.18/8.17	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.17	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.17	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.17	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.17	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.17	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.17	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.17	   mark(isNat(X)) -> active(isNat(X))
26.18/8.17	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.17	   mark(tt) -> active(tt)
26.18/8.17	   mark(0) -> active(0)
26.18/8.17	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.17	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.17	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.17	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.17	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.17	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.17	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.17	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.17	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.17	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.17	   U12(active(X)) -> U12(X)
26.18/8.17	   U12(mark(X)) -> U12(X)
26.18/8.17	   active(U12(tt)) -> mark(tt)
26.18/8.17	   active(U21(tt)) -> mark(tt)
26.18/8.17	   active(isNat(0)) -> mark(tt)
26.18/8.17	   s(active(X)) -> s(X)
26.18/8.17	   s(mark(X)) -> s(X)
26.18/8.17	   U21(active(X)) -> U21(X)
26.18/8.17	   U21(mark(X)) -> U21(X)
26.18/8.17	
26.18/8.17	
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(76)
26.18/8.17	Obligation:
26.18/8.17	Q DP problem:
26.18/8.17	The TRS P consists of the following rules:
26.18/8.17	
26.18/8.17	   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
26.18/8.17	   MARK(U12(X)) -> MARK(X)
26.18/8.17	   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
26.18/8.17	   MARK(U11(X1, X2)) -> MARK(X1)
26.18/8.17	   MARK(isNat(X)) -> ACTIVE(isNat(X))
26.18/8.17	   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
26.18/8.17	   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
26.18/8.17	   ACTIVE(U41(tt, M, N)) -> MARK(U42(isNat(N), M, N))
26.18/8.17	   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
26.18/8.17	   ACTIVE(U42(tt, M, N)) -> MARK(s(plus(N, M)))
26.18/8.17	   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
26.18/8.17	   ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N))
26.18/8.17	   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
26.18/8.17	   MARK(U41(X1, X2, X3)) -> MARK(X1)
26.18/8.17	   MARK(U21(X)) -> MARK(X)
26.18/8.17	   MARK(U42(X1, X2, X3)) -> MARK(X1)
26.18/8.17	   MARK(s(X)) -> MARK(X)
26.18/8.17	   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
26.18/8.17	   MARK(plus(X1, X2)) -> MARK(X1)
26.18/8.17	   MARK(plus(X1, X2)) -> MARK(X2)
26.18/8.17	
26.18/8.17	The TRS R consists of the following rules:
26.18/8.17	
26.18/8.17	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.17	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.17	   active(U31(tt, N)) -> mark(N)
26.18/8.17	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.17	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.17	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.17	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.17	   mark(s(X)) -> active(s(mark(X)))
26.18/8.17	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.17	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.17	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.17	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.17	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.17	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.17	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.17	   mark(isNat(X)) -> active(isNat(X))
26.18/8.17	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.17	   mark(tt) -> active(tt)
26.18/8.17	   mark(0) -> active(0)
26.18/8.17	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.17	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.17	   U12(active(X)) -> U12(X)
26.18/8.17	   U12(mark(X)) -> U12(X)
26.18/8.17	   active(U12(tt)) -> mark(tt)
26.18/8.17	   active(U21(tt)) -> mark(tt)
26.18/8.17	   active(isNat(0)) -> mark(tt)
26.18/8.17	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.17	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.17	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.17	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.17	   s(active(X)) -> s(X)
26.18/8.17	   s(mark(X)) -> s(X)
26.18/8.17	   U21(active(X)) -> U21(X)
26.18/8.17	   U21(mark(X)) -> U21(X)
26.18/8.17	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.17	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.17	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.17	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.17	
26.18/8.17	The set Q consists of the following terms:
26.18/8.17	
26.18/8.17	   active(U11(tt, x0))
26.18/8.17	   active(U12(tt))
26.18/8.17	   active(U21(tt))
26.18/8.17	   active(U31(tt, x0))
26.18/8.17	   active(U41(tt, x0, x1))
26.18/8.17	   active(U42(tt, x0, x1))
26.18/8.17	   active(isNat(0))
26.18/8.17	   active(isNat(plus(x0, x1)))
26.18/8.17	   active(isNat(s(x0)))
26.18/8.17	   active(plus(x0, 0))
26.18/8.17	   active(plus(x0, s(x1)))
26.18/8.17	   mark(U11(x0, x1))
26.18/8.17	   mark(tt)
26.18/8.17	   mark(U12(x0))
26.18/8.17	   mark(isNat(x0))
26.18/8.17	   mark(U21(x0))
26.18/8.17	   mark(U31(x0, x1))
26.18/8.17	   mark(U41(x0, x1, x2))
26.18/8.17	   mark(U42(x0, x1, x2))
26.18/8.17	   mark(s(x0))
26.18/8.17	   mark(plus(x0, x1))
26.18/8.17	   mark(0)
26.18/8.17	   U11(mark(x0), x1)
26.18/8.17	   U11(x0, mark(x1))
26.18/8.17	   U11(active(x0), x1)
26.18/8.17	   U11(x0, active(x1))
26.18/8.17	   U12(mark(x0))
26.18/8.17	   U12(active(x0))
26.18/8.17	   isNat(mark(x0))
26.18/8.17	   isNat(active(x0))
26.18/8.17	   U21(mark(x0))
26.18/8.17	   U21(active(x0))
26.18/8.17	   U31(mark(x0), x1)
26.18/8.17	   U31(x0, mark(x1))
26.18/8.17	   U31(active(x0), x1)
26.18/8.17	   U31(x0, active(x1))
26.18/8.17	   U41(mark(x0), x1, x2)
26.18/8.17	   U41(x0, mark(x1), x2)
26.18/8.17	   U41(x0, x1, mark(x2))
26.18/8.17	   U41(active(x0), x1, x2)
26.18/8.17	   U41(x0, active(x1), x2)
26.18/8.17	   U41(x0, x1, active(x2))
26.18/8.17	   U42(mark(x0), x1, x2)
26.18/8.17	   U42(x0, mark(x1), x2)
26.18/8.17	   U42(x0, x1, mark(x2))
26.18/8.17	   U42(active(x0), x1, x2)
26.18/8.17	   U42(x0, active(x1), x2)
26.18/8.17	   U42(x0, x1, active(x2))
26.18/8.17	   s(mark(x0))
26.18/8.17	   s(active(x0))
26.18/8.17	   plus(mark(x0), x1)
26.18/8.17	   plus(x0, mark(x1))
26.18/8.17	   plus(active(x0), x1)
26.18/8.17	   plus(x0, active(x1))
26.18/8.17	
26.18/8.17	We have to consider all minimal (P,Q,R)-chains.
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(77) QDPOrderProof (EQUIVALENT)
26.18/8.17	We use the reduction pair processor [LPAR04,JAR06].
26.18/8.17	
26.18/8.17	
26.18/8.17	The following pairs can be oriented strictly and are deleted.
26.18/8.17	
26.18/8.17	   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
26.18/8.17	The remaining pairs can at least be oriented weakly.
26.18/8.17	Used ordering:  Polynomial interpretation [POLO]:
26.18/8.17	
26.18/8.17	   POL(0) = 0
26.18/8.17	   POL(ACTIVE(x_1)) = x_1
26.18/8.17	   POL(MARK(x_1)) = 1
26.18/8.17	   POL(U11(x_1, x_2)) = 1
26.18/8.17	   POL(U12(x_1)) = 0
26.18/8.17	   POL(U21(x_1)) = 0
26.18/8.17	   POL(U31(x_1, x_2)) = 0
26.18/8.17	   POL(U41(x_1, x_2, x_3)) = 1
26.18/8.17	   POL(U42(x_1, x_2, x_3)) = 1
26.18/8.17	   POL(active(x_1)) = 0
26.18/8.17	   POL(isNat(x_1)) = 1
26.18/8.17	   POL(mark(x_1)) = 0
26.18/8.17	   POL(plus(x_1, x_2)) = 1
26.18/8.17	   POL(s(x_1)) = 0
26.18/8.17	   POL(tt) = 0
26.18/8.17	
26.18/8.17	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
26.18/8.17	
26.18/8.17	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.17	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.17	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.17	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.17	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.17	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.17	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.17	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.17	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.17	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.17	
26.18/8.17	
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(78)
26.18/8.17	Obligation:
26.18/8.17	Q DP problem:
26.18/8.17	The TRS P consists of the following rules:
26.18/8.17	
26.18/8.17	   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
26.18/8.17	   MARK(U12(X)) -> MARK(X)
26.18/8.17	   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
26.18/8.17	   MARK(U11(X1, X2)) -> MARK(X1)
26.18/8.17	   MARK(isNat(X)) -> ACTIVE(isNat(X))
26.18/8.17	   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
26.18/8.17	   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
26.18/8.17	   ACTIVE(U41(tt, M, N)) -> MARK(U42(isNat(N), M, N))
26.18/8.17	   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
26.18/8.17	   ACTIVE(U42(tt, M, N)) -> MARK(s(plus(N, M)))
26.18/8.17	   ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N))
26.18/8.17	   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
26.18/8.17	   MARK(U41(X1, X2, X3)) -> MARK(X1)
26.18/8.17	   MARK(U21(X)) -> MARK(X)
26.18/8.17	   MARK(U42(X1, X2, X3)) -> MARK(X1)
26.18/8.17	   MARK(s(X)) -> MARK(X)
26.18/8.17	   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
26.18/8.17	   MARK(plus(X1, X2)) -> MARK(X1)
26.18/8.17	   MARK(plus(X1, X2)) -> MARK(X2)
26.18/8.17	
26.18/8.17	The TRS R consists of the following rules:
26.18/8.17	
26.18/8.17	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.17	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.17	   active(U31(tt, N)) -> mark(N)
26.18/8.17	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.17	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.17	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.17	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.17	   mark(s(X)) -> active(s(mark(X)))
26.18/8.17	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.17	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.17	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.17	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.17	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.17	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.17	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.17	   mark(isNat(X)) -> active(isNat(X))
26.18/8.17	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.17	   mark(tt) -> active(tt)
26.18/8.17	   mark(0) -> active(0)
26.18/8.17	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.17	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.17	   U12(active(X)) -> U12(X)
26.18/8.17	   U12(mark(X)) -> U12(X)
26.18/8.17	   active(U12(tt)) -> mark(tt)
26.18/8.17	   active(U21(tt)) -> mark(tt)
26.18/8.17	   active(isNat(0)) -> mark(tt)
26.18/8.17	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.17	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.17	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.17	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.17	   s(active(X)) -> s(X)
26.18/8.17	   s(mark(X)) -> s(X)
26.18/8.17	   U21(active(X)) -> U21(X)
26.18/8.17	   U21(mark(X)) -> U21(X)
26.18/8.17	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.17	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.17	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.17	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.17	
26.18/8.17	The set Q consists of the following terms:
26.18/8.17	
26.18/8.17	   active(U11(tt, x0))
26.18/8.17	   active(U12(tt))
26.18/8.17	   active(U21(tt))
26.18/8.17	   active(U31(tt, x0))
26.18/8.17	   active(U41(tt, x0, x1))
26.18/8.17	   active(U42(tt, x0, x1))
26.18/8.17	   active(isNat(0))
26.18/8.17	   active(isNat(plus(x0, x1)))
26.18/8.17	   active(isNat(s(x0)))
26.18/8.17	   active(plus(x0, 0))
26.18/8.17	   active(plus(x0, s(x1)))
26.18/8.17	   mark(U11(x0, x1))
26.18/8.17	   mark(tt)
26.18/8.17	   mark(U12(x0))
26.18/8.17	   mark(isNat(x0))
26.18/8.17	   mark(U21(x0))
26.18/8.17	   mark(U31(x0, x1))
26.18/8.17	   mark(U41(x0, x1, x2))
26.18/8.17	   mark(U42(x0, x1, x2))
26.18/8.17	   mark(s(x0))
26.18/8.17	   mark(plus(x0, x1))
26.18/8.17	   mark(0)
26.18/8.17	   U11(mark(x0), x1)
26.18/8.17	   U11(x0, mark(x1))
26.18/8.17	   U11(active(x0), x1)
26.18/8.17	   U11(x0, active(x1))
26.18/8.17	   U12(mark(x0))
26.18/8.17	   U12(active(x0))
26.18/8.17	   isNat(mark(x0))
26.18/8.17	   isNat(active(x0))
26.18/8.17	   U21(mark(x0))
26.18/8.17	   U21(active(x0))
26.18/8.17	   U31(mark(x0), x1)
26.18/8.17	   U31(x0, mark(x1))
26.18/8.17	   U31(active(x0), x1)
26.18/8.17	   U31(x0, active(x1))
26.18/8.17	   U41(mark(x0), x1, x2)
26.18/8.17	   U41(x0, mark(x1), x2)
26.18/8.17	   U41(x0, x1, mark(x2))
26.18/8.17	   U41(active(x0), x1, x2)
26.18/8.17	   U41(x0, active(x1), x2)
26.18/8.17	   U41(x0, x1, active(x2))
26.18/8.17	   U42(mark(x0), x1, x2)
26.18/8.17	   U42(x0, mark(x1), x2)
26.18/8.17	   U42(x0, x1, mark(x2))
26.18/8.17	   U42(active(x0), x1, x2)
26.18/8.17	   U42(x0, active(x1), x2)
26.18/8.17	   U42(x0, x1, active(x2))
26.18/8.17	   s(mark(x0))
26.18/8.17	   s(active(x0))
26.18/8.17	   plus(mark(x0), x1)
26.18/8.17	   plus(x0, mark(x1))
26.18/8.17	   plus(active(x0), x1)
26.18/8.17	   plus(x0, active(x1))
26.18/8.17	
26.18/8.17	We have to consider all minimal (P,Q,R)-chains.
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(79) QDPOrderProof (EQUIVALENT)
26.18/8.17	We use the reduction pair processor [LPAR04,JAR06].
26.18/8.17	
26.18/8.17	
26.18/8.17	The following pairs can be oriented strictly and are deleted.
26.18/8.17	
26.18/8.17	   MARK(U41(X1, X2, X3)) -> MARK(X1)
26.18/8.17	   MARK(U42(X1, X2, X3)) -> MARK(X1)
26.18/8.17	   MARK(s(X)) -> MARK(X)
26.18/8.17	   MARK(plus(X1, X2)) -> MARK(X1)
26.18/8.17	   MARK(plus(X1, X2)) -> MARK(X2)
26.18/8.17	The remaining pairs can at least be oriented weakly.
26.18/8.17	Used ordering:  Polynomial interpretation [POLO]:
26.18/8.17	
26.18/8.17	   POL(0) = 0
26.18/8.17	   POL(ACTIVE(x_1)) = x_1
26.18/8.17	   POL(MARK(x_1)) = x_1
26.18/8.17	   POL(U11(x_1, x_2)) = x_1
26.18/8.17	   POL(U12(x_1)) = x_1
26.18/8.17	   POL(U21(x_1)) = x_1
26.18/8.17	   POL(U31(x_1, x_2)) = x_2
26.18/8.17	   POL(U41(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
26.18/8.17	   POL(U42(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
26.18/8.17	   POL(active(x_1)) = x_1
26.18/8.17	   POL(isNat(x_1)) = 1
26.18/8.17	   POL(mark(x_1)) = x_1
26.18/8.17	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
26.18/8.17	   POL(s(x_1)) = 1 + x_1
26.18/8.17	   POL(tt) = 1
26.18/8.17	
26.18/8.17	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
26.18/8.17	
26.18/8.17	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.17	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.17	   active(U31(tt, N)) -> mark(N)
26.18/8.17	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.17	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.17	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.17	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.17	   mark(s(X)) -> active(s(mark(X)))
26.18/8.17	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.17	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.17	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.17	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.17	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.17	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.17	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.17	   mark(isNat(X)) -> active(isNat(X))
26.18/8.17	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.17	   mark(tt) -> active(tt)
26.18/8.17	   mark(0) -> active(0)
26.18/8.17	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.17	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.17	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.17	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.17	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.17	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.17	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.17	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.17	   U12(active(X)) -> U12(X)
26.18/8.17	   U12(mark(X)) -> U12(X)
26.18/8.17	   active(U12(tt)) -> mark(tt)
26.18/8.17	   active(U21(tt)) -> mark(tt)
26.18/8.17	   active(isNat(0)) -> mark(tt)
26.18/8.17	   s(active(X)) -> s(X)
26.18/8.17	   s(mark(X)) -> s(X)
26.18/8.17	   U21(active(X)) -> U21(X)
26.18/8.17	   U21(mark(X)) -> U21(X)
26.18/8.17	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.17	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.17	
26.18/8.17	
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(80)
26.18/8.17	Obligation:
26.18/8.17	Q DP problem:
26.18/8.17	The TRS P consists of the following rules:
26.18/8.17	
26.18/8.17	   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
26.18/8.17	   MARK(U12(X)) -> MARK(X)
26.18/8.17	   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
26.18/8.17	   MARK(U11(X1, X2)) -> MARK(X1)
26.18/8.17	   MARK(isNat(X)) -> ACTIVE(isNat(X))
26.18/8.17	   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
26.18/8.17	   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
26.18/8.17	   ACTIVE(U41(tt, M, N)) -> MARK(U42(isNat(N), M, N))
26.18/8.17	   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
26.18/8.17	   ACTIVE(U42(tt, M, N)) -> MARK(s(plus(N, M)))
26.18/8.17	   ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N))
26.18/8.17	   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
26.18/8.17	   MARK(U21(X)) -> MARK(X)
26.18/8.17	   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
26.18/8.17	
26.18/8.17	The TRS R consists of the following rules:
26.18/8.17	
26.18/8.17	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.17	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.17	   active(U31(tt, N)) -> mark(N)
26.18/8.17	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.17	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.17	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.17	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.17	   mark(s(X)) -> active(s(mark(X)))
26.18/8.17	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.17	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.17	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.17	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.17	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.17	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.17	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.17	   mark(isNat(X)) -> active(isNat(X))
26.18/8.17	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.17	   mark(tt) -> active(tt)
26.18/8.17	   mark(0) -> active(0)
26.18/8.17	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.17	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.17	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.17	   U12(active(X)) -> U12(X)
26.18/8.17	   U12(mark(X)) -> U12(X)
26.18/8.17	   active(U12(tt)) -> mark(tt)
26.18/8.17	   active(U21(tt)) -> mark(tt)
26.18/8.17	   active(isNat(0)) -> mark(tt)
26.18/8.17	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.17	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.17	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.17	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.17	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.17	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.17	   s(active(X)) -> s(X)
26.18/8.17	   s(mark(X)) -> s(X)
26.18/8.17	   U21(active(X)) -> U21(X)
26.18/8.17	   U21(mark(X)) -> U21(X)
26.18/8.17	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.17	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.17	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.17	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.17	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.17	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.17	
26.18/8.17	The set Q consists of the following terms:
26.18/8.17	
26.18/8.17	   active(U11(tt, x0))
26.18/8.17	   active(U12(tt))
26.18/8.17	   active(U21(tt))
26.18/8.17	   active(U31(tt, x0))
26.18/8.17	   active(U41(tt, x0, x1))
26.18/8.17	   active(U42(tt, x0, x1))
26.18/8.17	   active(isNat(0))
26.18/8.17	   active(isNat(plus(x0, x1)))
26.18/8.17	   active(isNat(s(x0)))
26.18/8.17	   active(plus(x0, 0))
26.18/8.17	   active(plus(x0, s(x1)))
26.18/8.17	   mark(U11(x0, x1))
26.18/8.17	   mark(tt)
26.18/8.17	   mark(U12(x0))
26.18/8.17	   mark(isNat(x0))
26.18/8.17	   mark(U21(x0))
26.18/8.17	   mark(U31(x0, x1))
26.18/8.17	   mark(U41(x0, x1, x2))
26.18/8.17	   mark(U42(x0, x1, x2))
26.18/8.17	   mark(s(x0))
26.18/8.17	   mark(plus(x0, x1))
26.18/8.17	   mark(0)
26.18/8.17	   U11(mark(x0), x1)
26.18/8.17	   U11(x0, mark(x1))
26.18/8.17	   U11(active(x0), x1)
26.18/8.17	   U11(x0, active(x1))
26.18/8.17	   U12(mark(x0))
26.18/8.17	   U12(active(x0))
26.18/8.17	   isNat(mark(x0))
26.18/8.17	   isNat(active(x0))
26.18/8.17	   U21(mark(x0))
26.18/8.17	   U21(active(x0))
26.18/8.17	   U31(mark(x0), x1)
26.18/8.17	   U31(x0, mark(x1))
26.18/8.17	   U31(active(x0), x1)
26.18/8.17	   U31(x0, active(x1))
26.18/8.17	   U41(mark(x0), x1, x2)
26.18/8.17	   U41(x0, mark(x1), x2)
26.18/8.17	   U41(x0, x1, mark(x2))
26.18/8.17	   U41(active(x0), x1, x2)
26.18/8.17	   U41(x0, active(x1), x2)
26.18/8.17	   U41(x0, x1, active(x2))
26.18/8.17	   U42(mark(x0), x1, x2)
26.18/8.17	   U42(x0, mark(x1), x2)
26.18/8.17	   U42(x0, x1, mark(x2))
26.18/8.17	   U42(active(x0), x1, x2)
26.18/8.17	   U42(x0, active(x1), x2)
26.18/8.17	   U42(x0, x1, active(x2))
26.18/8.17	   s(mark(x0))
26.18/8.17	   s(active(x0))
26.18/8.17	   plus(mark(x0), x1)
26.18/8.17	   plus(x0, mark(x1))
26.18/8.17	   plus(active(x0), x1)
26.18/8.17	   plus(x0, active(x1))
26.18/8.17	
26.18/8.17	We have to consider all minimal (P,Q,R)-chains.
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(81) DependencyGraphProof (EQUIVALENT)
26.18/8.17	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
26.18/8.17	----------------------------------------
26.18/8.17	
26.18/8.17	(82)
26.18/8.17	Obligation:
26.18/8.17	Q DP problem:
26.18/8.17	The TRS P consists of the following rules:
26.18/8.17	
26.18/8.17	   MARK(U12(X)) -> MARK(X)
26.18/8.17	   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
26.18/8.17	   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
26.18/8.17	   ACTIVE(U41(tt, M, N)) -> MARK(U42(isNat(N), M, N))
26.18/8.17	   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
26.18/8.17	   ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N))
26.18/8.17	   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
26.18/8.17	   MARK(U11(X1, X2)) -> MARK(X1)
26.18/8.17	   MARK(isNat(X)) -> ACTIVE(isNat(X))
26.18/8.17	   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
26.18/8.18	   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
26.18/8.18	   MARK(U21(X)) -> MARK(X)
26.18/8.18	   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
26.18/8.18	
26.18/8.18	The TRS R consists of the following rules:
26.18/8.18	
26.18/8.18	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.18	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.18	   active(U31(tt, N)) -> mark(N)
26.18/8.18	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.18	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.18	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.18	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.18	   mark(s(X)) -> active(s(mark(X)))
26.18/8.18	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.18	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.18	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.18	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.18	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.18	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.18	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.18	   mark(isNat(X)) -> active(isNat(X))
26.18/8.18	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.18	   mark(tt) -> active(tt)
26.18/8.18	   mark(0) -> active(0)
26.18/8.18	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.18	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.18	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.18	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.18	   U12(active(X)) -> U12(X)
26.18/8.18	   U12(mark(X)) -> U12(X)
26.18/8.18	   active(U12(tt)) -> mark(tt)
26.18/8.18	   active(U21(tt)) -> mark(tt)
26.18/8.18	   active(isNat(0)) -> mark(tt)
26.18/8.18	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.18	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.18	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.18	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.18	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.18	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.18	   s(active(X)) -> s(X)
26.18/8.18	   s(mark(X)) -> s(X)
26.18/8.18	   U21(active(X)) -> U21(X)
26.18/8.18	   U21(mark(X)) -> U21(X)
26.18/8.18	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.18	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.18	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.18	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.18	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.18	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.18	
26.18/8.18	The set Q consists of the following terms:
26.18/8.18	
26.18/8.18	   active(U11(tt, x0))
26.18/8.18	   active(U12(tt))
26.18/8.18	   active(U21(tt))
26.18/8.18	   active(U31(tt, x0))
26.18/8.18	   active(U41(tt, x0, x1))
26.18/8.18	   active(U42(tt, x0, x1))
26.18/8.18	   active(isNat(0))
26.18/8.18	   active(isNat(plus(x0, x1)))
26.18/8.18	   active(isNat(s(x0)))
26.18/8.18	   active(plus(x0, 0))
26.18/8.18	   active(plus(x0, s(x1)))
26.18/8.18	   mark(U11(x0, x1))
26.18/8.18	   mark(tt)
26.18/8.18	   mark(U12(x0))
26.18/8.18	   mark(isNat(x0))
26.18/8.18	   mark(U21(x0))
26.18/8.18	   mark(U31(x0, x1))
26.18/8.18	   mark(U41(x0, x1, x2))
26.18/8.18	   mark(U42(x0, x1, x2))
26.18/8.18	   mark(s(x0))
26.18/8.18	   mark(plus(x0, x1))
26.18/8.18	   mark(0)
26.18/8.18	   U11(mark(x0), x1)
26.18/8.18	   U11(x0, mark(x1))
26.18/8.18	   U11(active(x0), x1)
26.18/8.18	   U11(x0, active(x1))
26.18/8.18	   U12(mark(x0))
26.18/8.18	   U12(active(x0))
26.18/8.18	   isNat(mark(x0))
26.18/8.18	   isNat(active(x0))
26.18/8.18	   U21(mark(x0))
26.18/8.18	   U21(active(x0))
26.18/8.18	   U31(mark(x0), x1)
26.18/8.18	   U31(x0, mark(x1))
26.18/8.18	   U31(active(x0), x1)
26.18/8.18	   U31(x0, active(x1))
26.18/8.18	   U41(mark(x0), x1, x2)
26.18/8.18	   U41(x0, mark(x1), x2)
26.18/8.18	   U41(x0, x1, mark(x2))
26.18/8.18	   U41(active(x0), x1, x2)
26.18/8.18	   U41(x0, active(x1), x2)
26.18/8.18	   U41(x0, x1, active(x2))
26.18/8.18	   U42(mark(x0), x1, x2)
26.18/8.18	   U42(x0, mark(x1), x2)
26.18/8.18	   U42(x0, x1, mark(x2))
26.18/8.18	   U42(active(x0), x1, x2)
26.18/8.18	   U42(x0, active(x1), x2)
26.18/8.18	   U42(x0, x1, active(x2))
26.18/8.18	   s(mark(x0))
26.18/8.18	   s(active(x0))
26.18/8.18	   plus(mark(x0), x1)
26.18/8.18	   plus(x0, mark(x1))
26.18/8.18	   plus(active(x0), x1)
26.18/8.18	   plus(x0, active(x1))
26.18/8.18	
26.18/8.18	We have to consider all minimal (P,Q,R)-chains.
26.18/8.18	----------------------------------------
26.18/8.18	
26.18/8.18	(83) QDPQMonotonicMRRProof (EQUIVALENT)
26.18/8.18	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.
26.18/8.18	
26.18/8.18	Strictly oriented dependency pairs:
26.18/8.18	
26.18/8.18	   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
26.18/8.18	
26.18/8.18	
26.18/8.18	Used ordering: Polynomial interpretation [POLO]:
26.18/8.18	
26.18/8.18	   POL(0) = 0
26.18/8.18	   POL(ACTIVE(x_1)) = x_1
26.18/8.18	   POL(MARK(x_1)) = 1
26.18/8.18	   POL(U11(x_1, x_2)) = 1
26.18/8.18	   POL(U12(x_1)) = 0
26.18/8.18	   POL(U21(x_1)) = 0
26.18/8.18	   POL(U31(x_1, x_2)) = 0
26.18/8.18	   POL(U41(x_1, x_2, x_3)) = 1
26.18/8.18	   POL(U42(x_1, x_2, x_3)) = 0
26.18/8.18	   POL(active(x_1)) = 0
26.18/8.18	   POL(isNat(x_1)) = 1
26.18/8.18	   POL(mark(x_1)) = 0
26.18/8.18	   POL(plus(x_1, x_2)) = 1
26.18/8.18	   POL(s(x_1)) = 0
26.18/8.18	   POL(tt) = 0
26.18/8.18	
26.18/8.18	
26.18/8.18	----------------------------------------
26.18/8.18	
26.18/8.18	(84)
26.18/8.18	Obligation:
26.18/8.18	Q DP problem:
26.18/8.18	The TRS P consists of the following rules:
26.18/8.18	
26.18/8.18	   MARK(U12(X)) -> MARK(X)
26.18/8.18	   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
26.18/8.18	   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
26.18/8.18	   ACTIVE(U41(tt, M, N)) -> MARK(U42(isNat(N), M, N))
26.18/8.18	   ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N))
26.18/8.18	   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
26.18/8.18	   MARK(U11(X1, X2)) -> MARK(X1)
26.18/8.18	   MARK(isNat(X)) -> ACTIVE(isNat(X))
26.18/8.18	   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
26.18/8.18	   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
26.18/8.18	   MARK(U21(X)) -> MARK(X)
26.18/8.18	   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
26.18/8.18	
26.18/8.18	The TRS R consists of the following rules:
26.18/8.18	
26.18/8.18	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.18	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.18	   active(U31(tt, N)) -> mark(N)
26.18/8.18	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.18	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.18	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.18	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.18	   mark(s(X)) -> active(s(mark(X)))
26.18/8.18	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.18	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.18	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.18	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.18	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.18	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.18	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.18	   mark(isNat(X)) -> active(isNat(X))
26.18/8.18	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.18	   mark(tt) -> active(tt)
26.18/8.18	   mark(0) -> active(0)
26.18/8.18	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.18	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.18	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.18	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.18	   U12(active(X)) -> U12(X)
26.18/8.18	   U12(mark(X)) -> U12(X)
26.18/8.18	   active(U12(tt)) -> mark(tt)
26.18/8.18	   active(U21(tt)) -> mark(tt)
26.18/8.18	   active(isNat(0)) -> mark(tt)
26.18/8.18	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.18	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.18	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.18	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.18	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.18	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.18	   s(active(X)) -> s(X)
26.18/8.18	   s(mark(X)) -> s(X)
26.18/8.18	   U21(active(X)) -> U21(X)
26.18/8.18	   U21(mark(X)) -> U21(X)
26.18/8.18	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.18	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.18	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.18	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.18	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.18	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.18	
26.18/8.18	The set Q consists of the following terms:
26.18/8.18	
26.18/8.18	   active(U11(tt, x0))
26.18/8.18	   active(U12(tt))
26.18/8.18	   active(U21(tt))
26.18/8.18	   active(U31(tt, x0))
26.18/8.18	   active(U41(tt, x0, x1))
26.18/8.18	   active(U42(tt, x0, x1))
26.18/8.18	   active(isNat(0))
26.18/8.18	   active(isNat(plus(x0, x1)))
26.18/8.18	   active(isNat(s(x0)))
26.18/8.18	   active(plus(x0, 0))
26.18/8.18	   active(plus(x0, s(x1)))
26.18/8.18	   mark(U11(x0, x1))
26.18/8.18	   mark(tt)
26.18/8.18	   mark(U12(x0))
26.18/8.18	   mark(isNat(x0))
26.18/8.18	   mark(U21(x0))
26.18/8.18	   mark(U31(x0, x1))
26.18/8.18	   mark(U41(x0, x1, x2))
26.18/8.18	   mark(U42(x0, x1, x2))
26.18/8.18	   mark(s(x0))
26.18/8.18	   mark(plus(x0, x1))
26.18/8.18	   mark(0)
26.18/8.18	   U11(mark(x0), x1)
26.18/8.18	   U11(x0, mark(x1))
26.18/8.18	   U11(active(x0), x1)
26.18/8.18	   U11(x0, active(x1))
26.18/8.18	   U12(mark(x0))
26.18/8.18	   U12(active(x0))
26.18/8.18	   isNat(mark(x0))
26.18/8.18	   isNat(active(x0))
26.18/8.18	   U21(mark(x0))
26.18/8.18	   U21(active(x0))
26.18/8.18	   U31(mark(x0), x1)
26.18/8.18	   U31(x0, mark(x1))
26.18/8.18	   U31(active(x0), x1)
26.18/8.18	   U31(x0, active(x1))
26.18/8.18	   U41(mark(x0), x1, x2)
26.18/8.18	   U41(x0, mark(x1), x2)
26.18/8.18	   U41(x0, x1, mark(x2))
26.18/8.18	   U41(active(x0), x1, x2)
26.18/8.18	   U41(x0, active(x1), x2)
26.18/8.18	   U41(x0, x1, active(x2))
26.18/8.18	   U42(mark(x0), x1, x2)
26.18/8.18	   U42(x0, mark(x1), x2)
26.18/8.18	   U42(x0, x1, mark(x2))
26.18/8.18	   U42(active(x0), x1, x2)
26.18/8.18	   U42(x0, active(x1), x2)
26.18/8.18	   U42(x0, x1, active(x2))
26.18/8.18	   s(mark(x0))
26.18/8.18	   s(active(x0))
26.18/8.18	   plus(mark(x0), x1)
26.18/8.18	   plus(x0, mark(x1))
26.18/8.18	   plus(active(x0), x1)
26.18/8.18	   plus(x0, active(x1))
26.18/8.18	
26.18/8.18	We have to consider all minimal (P,Q,R)-chains.
26.18/8.18	----------------------------------------
26.18/8.18	
26.18/8.18	(85) DependencyGraphProof (EQUIVALENT)
26.18/8.18	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
26.18/8.18	----------------------------------------
26.18/8.18	
26.18/8.18	(86)
26.18/8.18	Obligation:
26.18/8.18	Q DP problem:
26.18/8.18	The TRS P consists of the following rules:
26.18/8.18	
26.18/8.18	   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
26.18/8.18	   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
26.18/8.18	   MARK(U12(X)) -> MARK(X)
26.18/8.18	   MARK(U11(X1, X2)) -> MARK(X1)
26.18/8.18	   MARK(isNat(X)) -> ACTIVE(isNat(X))
26.18/8.18	   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
26.18/8.18	   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
26.18/8.18	   MARK(U21(X)) -> MARK(X)
26.18/8.18	   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
26.18/8.18	   ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N))
26.18/8.18	   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
26.18/8.18	
26.18/8.18	The TRS R consists of the following rules:
26.18/8.18	
26.18/8.18	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.18	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.18	   active(U31(tt, N)) -> mark(N)
26.18/8.18	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.18	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.18	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.18	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.18	   mark(s(X)) -> active(s(mark(X)))
26.18/8.18	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.18	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.18	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.18	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.18	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.18	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.18	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.18	   mark(isNat(X)) -> active(isNat(X))
26.18/8.18	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.18	   mark(tt) -> active(tt)
26.18/8.18	   mark(0) -> active(0)
26.18/8.18	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.18	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.18	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.18	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.18	   U12(active(X)) -> U12(X)
26.18/8.18	   U12(mark(X)) -> U12(X)
26.18/8.18	   active(U12(tt)) -> mark(tt)
26.18/8.18	   active(U21(tt)) -> mark(tt)
26.18/8.18	   active(isNat(0)) -> mark(tt)
26.18/8.18	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.18	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.18	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.18	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.18	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.18	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.18	   s(active(X)) -> s(X)
26.18/8.18	   s(mark(X)) -> s(X)
26.18/8.18	   U21(active(X)) -> U21(X)
26.18/8.18	   U21(mark(X)) -> U21(X)
26.18/8.18	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.18	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.18	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.18	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.18	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.18	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.18	
26.18/8.18	The set Q consists of the following terms:
26.18/8.18	
26.18/8.18	   active(U11(tt, x0))
26.18/8.18	   active(U12(tt))
26.18/8.18	   active(U21(tt))
26.18/8.18	   active(U31(tt, x0))
26.18/8.18	   active(U41(tt, x0, x1))
26.18/8.18	   active(U42(tt, x0, x1))
26.18/8.18	   active(isNat(0))
26.18/8.18	   active(isNat(plus(x0, x1)))
26.18/8.18	   active(isNat(s(x0)))
26.18/8.18	   active(plus(x0, 0))
26.18/8.18	   active(plus(x0, s(x1)))
26.18/8.18	   mark(U11(x0, x1))
26.18/8.18	   mark(tt)
26.18/8.18	   mark(U12(x0))
26.18/8.18	   mark(isNat(x0))
26.18/8.18	   mark(U21(x0))
26.18/8.18	   mark(U31(x0, x1))
26.18/8.18	   mark(U41(x0, x1, x2))
26.18/8.18	   mark(U42(x0, x1, x2))
26.18/8.18	   mark(s(x0))
26.18/8.18	   mark(plus(x0, x1))
26.18/8.18	   mark(0)
26.18/8.18	   U11(mark(x0), x1)
26.18/8.18	   U11(x0, mark(x1))
26.18/8.18	   U11(active(x0), x1)
26.18/8.18	   U11(x0, active(x1))
26.18/8.18	   U12(mark(x0))
26.18/8.18	   U12(active(x0))
26.18/8.18	   isNat(mark(x0))
26.18/8.18	   isNat(active(x0))
26.18/8.18	   U21(mark(x0))
26.18/8.18	   U21(active(x0))
26.18/8.18	   U31(mark(x0), x1)
26.18/8.18	   U31(x0, mark(x1))
26.18/8.18	   U31(active(x0), x1)
26.18/8.18	   U31(x0, active(x1))
26.18/8.18	   U41(mark(x0), x1, x2)
26.18/8.18	   U41(x0, mark(x1), x2)
26.18/8.18	   U41(x0, x1, mark(x2))
26.18/8.18	   U41(active(x0), x1, x2)
26.18/8.18	   U41(x0, active(x1), x2)
26.18/8.18	   U41(x0, x1, active(x2))
26.18/8.18	   U42(mark(x0), x1, x2)
26.18/8.18	   U42(x0, mark(x1), x2)
26.18/8.18	   U42(x0, x1, mark(x2))
26.18/8.18	   U42(active(x0), x1, x2)
26.18/8.18	   U42(x0, active(x1), x2)
26.18/8.18	   U42(x0, x1, active(x2))
26.18/8.18	   s(mark(x0))
26.18/8.18	   s(active(x0))
26.18/8.18	   plus(mark(x0), x1)
26.18/8.18	   plus(x0, mark(x1))
26.18/8.18	   plus(active(x0), x1)
26.18/8.18	   plus(x0, active(x1))
26.18/8.18	
26.18/8.18	We have to consider all minimal (P,Q,R)-chains.
26.18/8.18	----------------------------------------
26.18/8.18	
26.18/8.18	(87) QDPQMonotonicMRRProof (EQUIVALENT)
26.18/8.18	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.
26.18/8.18	
26.18/8.18	Strictly oriented dependency pairs:
26.18/8.18	
26.18/8.18	   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
26.18/8.18	
26.18/8.18	
26.18/8.18	Used ordering: Polynomial interpretation [POLO]:
26.18/8.18	
26.18/8.18	   POL(0) = 0
26.18/8.18	   POL(ACTIVE(x_1)) = 2*x_1
26.18/8.18	   POL(MARK(x_1)) = 2
26.18/8.18	   POL(U11(x_1, x_2)) = 1
26.18/8.18	   POL(U12(x_1)) = 0
26.18/8.18	   POL(U21(x_1)) = 0
26.18/8.18	   POL(U31(x_1, x_2)) = 0
26.18/8.18	   POL(U41(x_1, x_2, x_3)) = 0
26.18/8.18	   POL(U42(x_1, x_2, x_3)) = 0
26.18/8.18	   POL(active(x_1)) = 0
26.18/8.18	   POL(isNat(x_1)) = 1
26.18/8.18	   POL(mark(x_1)) = 0
26.18/8.18	   POL(plus(x_1, x_2)) = 1
26.18/8.18	   POL(s(x_1)) = 0
26.18/8.18	   POL(tt) = 0
26.18/8.18	
26.18/8.18	
26.18/8.18	----------------------------------------
26.18/8.18	
26.18/8.18	(88)
26.18/8.18	Obligation:
26.18/8.18	Q DP problem:
26.18/8.18	The TRS P consists of the following rules:
26.18/8.18	
26.18/8.18	   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
26.18/8.18	   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
26.18/8.18	   MARK(U12(X)) -> MARK(X)
26.18/8.18	   MARK(U11(X1, X2)) -> MARK(X1)
26.18/8.18	   MARK(isNat(X)) -> ACTIVE(isNat(X))
26.18/8.18	   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
26.18/8.18	   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
26.18/8.18	   MARK(U21(X)) -> MARK(X)
26.18/8.18	   ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N))
26.18/8.18	   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
26.18/8.18	
26.18/8.18	The TRS R consists of the following rules:
26.18/8.18	
26.18/8.18	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.18	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.18	   active(U31(tt, N)) -> mark(N)
26.18/8.18	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.18	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.18	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.18	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.18	   mark(s(X)) -> active(s(mark(X)))
26.18/8.18	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.18	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.18	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.18	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.18	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.18	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.18	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.18	   mark(isNat(X)) -> active(isNat(X))
26.18/8.18	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.18	   mark(tt) -> active(tt)
26.18/8.18	   mark(0) -> active(0)
26.18/8.18	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.18	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.18	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.18	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.18	   U12(active(X)) -> U12(X)
26.18/8.18	   U12(mark(X)) -> U12(X)
26.18/8.18	   active(U12(tt)) -> mark(tt)
26.18/8.18	   active(U21(tt)) -> mark(tt)
26.18/8.18	   active(isNat(0)) -> mark(tt)
26.18/8.18	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.18	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.18	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.18	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.18	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.18	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.18	   s(active(X)) -> s(X)
26.18/8.18	   s(mark(X)) -> s(X)
26.18/8.18	   U21(active(X)) -> U21(X)
26.18/8.18	   U21(mark(X)) -> U21(X)
26.18/8.18	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.18	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.18	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.18	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.18	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.18	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.18	
26.18/8.18	The set Q consists of the following terms:
26.18/8.18	
26.18/8.18	   active(U11(tt, x0))
26.18/8.18	   active(U12(tt))
26.18/8.18	   active(U21(tt))
26.18/8.18	   active(U31(tt, x0))
26.18/8.18	   active(U41(tt, x0, x1))
26.18/8.18	   active(U42(tt, x0, x1))
26.18/8.18	   active(isNat(0))
26.18/8.18	   active(isNat(plus(x0, x1)))
26.18/8.18	   active(isNat(s(x0)))
26.18/8.18	   active(plus(x0, 0))
26.18/8.18	   active(plus(x0, s(x1)))
26.18/8.18	   mark(U11(x0, x1))
26.18/8.18	   mark(tt)
26.18/8.18	   mark(U12(x0))
26.18/8.18	   mark(isNat(x0))
26.18/8.18	   mark(U21(x0))
26.18/8.18	   mark(U31(x0, x1))
26.18/8.18	   mark(U41(x0, x1, x2))
26.18/8.18	   mark(U42(x0, x1, x2))
26.18/8.18	   mark(s(x0))
26.18/8.18	   mark(plus(x0, x1))
26.18/8.18	   mark(0)
26.18/8.18	   U11(mark(x0), x1)
26.18/8.18	   U11(x0, mark(x1))
26.18/8.18	   U11(active(x0), x1)
26.18/8.18	   U11(x0, active(x1))
26.18/8.18	   U12(mark(x0))
26.18/8.18	   U12(active(x0))
26.18/8.18	   isNat(mark(x0))
26.18/8.18	   isNat(active(x0))
26.18/8.18	   U21(mark(x0))
26.18/8.18	   U21(active(x0))
26.18/8.18	   U31(mark(x0), x1)
26.18/8.18	   U31(x0, mark(x1))
26.18/8.18	   U31(active(x0), x1)
26.18/8.18	   U31(x0, active(x1))
26.18/8.18	   U41(mark(x0), x1, x2)
26.18/8.18	   U41(x0, mark(x1), x2)
26.18/8.18	   U41(x0, x1, mark(x2))
26.18/8.18	   U41(active(x0), x1, x2)
26.18/8.18	   U41(x0, active(x1), x2)
26.18/8.18	   U41(x0, x1, active(x2))
26.18/8.18	   U42(mark(x0), x1, x2)
26.18/8.18	   U42(x0, mark(x1), x2)
26.18/8.18	   U42(x0, x1, mark(x2))
26.18/8.18	   U42(active(x0), x1, x2)
26.18/8.18	   U42(x0, active(x1), x2)
26.18/8.18	   U42(x0, x1, active(x2))
26.18/8.18	   s(mark(x0))
26.18/8.18	   s(active(x0))
26.18/8.18	   plus(mark(x0), x1)
26.18/8.18	   plus(x0, mark(x1))
26.18/8.18	   plus(active(x0), x1)
26.18/8.18	   plus(x0, active(x1))
26.18/8.18	
26.18/8.18	We have to consider all minimal (P,Q,R)-chains.
26.18/8.18	----------------------------------------
26.18/8.18	
26.18/8.18	(89) DependencyGraphProof (EQUIVALENT)
26.18/8.18	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
26.18/8.18	----------------------------------------
26.18/8.18	
26.18/8.18	(90)
26.18/8.18	Obligation:
26.18/8.18	Q DP problem:
26.18/8.18	The TRS P consists of the following rules:
26.18/8.18	
26.18/8.18	   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
26.18/8.18	   MARK(U12(X)) -> MARK(X)
26.18/8.18	   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
26.18/8.18	   MARK(U11(X1, X2)) -> MARK(X1)
26.18/8.18	   MARK(isNat(X)) -> ACTIVE(isNat(X))
26.18/8.18	   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
26.18/8.18	   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
26.18/8.18	   MARK(U21(X)) -> MARK(X)
26.18/8.18	   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
26.18/8.18	
26.18/8.18	The TRS R consists of the following rules:
26.18/8.18	
26.18/8.18	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.18	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.18	   active(U31(tt, N)) -> mark(N)
26.18/8.18	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.18	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.18	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.18	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.18	   mark(s(X)) -> active(s(mark(X)))
26.18/8.18	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.18	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.18	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.18	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.18	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.18	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.18	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.18	   mark(isNat(X)) -> active(isNat(X))
26.18/8.18	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.18	   mark(tt) -> active(tt)
26.18/8.18	   mark(0) -> active(0)
26.18/8.18	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.18	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.18	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.18	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.18	   U12(active(X)) -> U12(X)
26.18/8.18	   U12(mark(X)) -> U12(X)
26.18/8.18	   active(U12(tt)) -> mark(tt)
26.18/8.18	   active(U21(tt)) -> mark(tt)
26.18/8.18	   active(isNat(0)) -> mark(tt)
26.18/8.18	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.18	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.18	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.18	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.18	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.18	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.18	   s(active(X)) -> s(X)
26.18/8.18	   s(mark(X)) -> s(X)
26.18/8.18	   U21(active(X)) -> U21(X)
26.18/8.18	   U21(mark(X)) -> U21(X)
26.18/8.18	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.18	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.18	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.18	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.18	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.18	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.18	
26.18/8.18	The set Q consists of the following terms:
26.18/8.18	
26.18/8.18	   active(U11(tt, x0))
26.18/8.18	   active(U12(tt))
26.18/8.18	   active(U21(tt))
26.18/8.18	   active(U31(tt, x0))
26.18/8.18	   active(U41(tt, x0, x1))
26.18/8.18	   active(U42(tt, x0, x1))
26.18/8.18	   active(isNat(0))
26.18/8.18	   active(isNat(plus(x0, x1)))
26.18/8.18	   active(isNat(s(x0)))
26.18/8.18	   active(plus(x0, 0))
26.18/8.18	   active(plus(x0, s(x1)))
26.18/8.18	   mark(U11(x0, x1))
26.18/8.18	   mark(tt)
26.18/8.18	   mark(U12(x0))
26.18/8.18	   mark(isNat(x0))
26.18/8.18	   mark(U21(x0))
26.18/8.18	   mark(U31(x0, x1))
26.18/8.18	   mark(U41(x0, x1, x2))
26.18/8.18	   mark(U42(x0, x1, x2))
26.18/8.18	   mark(s(x0))
26.18/8.18	   mark(plus(x0, x1))
26.18/8.18	   mark(0)
26.18/8.18	   U11(mark(x0), x1)
26.18/8.18	   U11(x0, mark(x1))
26.18/8.18	   U11(active(x0), x1)
26.18/8.18	   U11(x0, active(x1))
26.18/8.18	   U12(mark(x0))
26.18/8.18	   U12(active(x0))
26.18/8.18	   isNat(mark(x0))
26.18/8.18	   isNat(active(x0))
26.18/8.18	   U21(mark(x0))
26.18/8.18	   U21(active(x0))
26.18/8.18	   U31(mark(x0), x1)
26.18/8.18	   U31(x0, mark(x1))
26.18/8.18	   U31(active(x0), x1)
26.18/8.18	   U31(x0, active(x1))
26.18/8.18	   U41(mark(x0), x1, x2)
26.18/8.18	   U41(x0, mark(x1), x2)
26.18/8.18	   U41(x0, x1, mark(x2))
26.18/8.18	   U41(active(x0), x1, x2)
26.18/8.18	   U41(x0, active(x1), x2)
26.18/8.18	   U41(x0, x1, active(x2))
26.18/8.18	   U42(mark(x0), x1, x2)
26.18/8.18	   U42(x0, mark(x1), x2)
26.18/8.18	   U42(x0, x1, mark(x2))
26.18/8.18	   U42(active(x0), x1, x2)
26.18/8.18	   U42(x0, active(x1), x2)
26.18/8.18	   U42(x0, x1, active(x2))
26.18/8.18	   s(mark(x0))
26.18/8.18	   s(active(x0))
26.18/8.18	   plus(mark(x0), x1)
26.18/8.18	   plus(x0, mark(x1))
26.18/8.18	   plus(active(x0), x1)
26.18/8.18	   plus(x0, active(x1))
26.18/8.18	
26.18/8.18	We have to consider all minimal (P,Q,R)-chains.
26.18/8.18	----------------------------------------
26.18/8.18	
26.18/8.18	(91) QDPQMonotonicMRRProof (EQUIVALENT)
26.18/8.18	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.
26.18/8.18	
26.18/8.18	Strictly oriented dependency pairs:
26.18/8.18	
26.18/8.18	   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
26.18/8.18	
26.18/8.18	
26.18/8.18	Used ordering: Polynomial interpretation [POLO]:
26.18/8.18	
26.18/8.18	   POL(0) = 0
26.18/8.18	   POL(ACTIVE(x_1)) = 2*x_1
26.18/8.18	   POL(MARK(x_1)) = 2
26.18/8.18	   POL(U11(x_1, x_2)) = 1
26.18/8.18	   POL(U12(x_1)) = 0
26.18/8.18	   POL(U21(x_1)) = 0
26.18/8.18	   POL(U31(x_1, x_2)) = 0
26.18/8.18	   POL(U41(x_1, x_2, x_3)) = 0
26.18/8.18	   POL(U42(x_1, x_2, x_3)) = 0
26.18/8.18	   POL(active(x_1)) = 0
26.18/8.18	   POL(isNat(x_1)) = 1
26.18/8.18	   POL(mark(x_1)) = 0
26.18/8.18	   POL(plus(x_1, x_2)) = 0
26.18/8.18	   POL(s(x_1)) = 0
26.18/8.18	   POL(tt) = 0
26.18/8.18	
26.18/8.18	
26.18/8.18	----------------------------------------
26.18/8.18	
26.18/8.18	(92)
26.18/8.18	Obligation:
26.18/8.18	Q DP problem:
26.18/8.18	The TRS P consists of the following rules:
26.18/8.18	
26.18/8.18	   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
26.18/8.18	   MARK(U12(X)) -> MARK(X)
26.18/8.18	   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
26.18/8.18	   MARK(U11(X1, X2)) -> MARK(X1)
26.18/8.18	   MARK(isNat(X)) -> ACTIVE(isNat(X))
26.18/8.18	   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
26.18/8.18	   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
26.18/8.18	   MARK(U21(X)) -> MARK(X)
26.18/8.18	
26.18/8.18	The TRS R consists of the following rules:
26.18/8.18	
26.18/8.18	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.18	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.18	   active(U31(tt, N)) -> mark(N)
26.18/8.18	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.18	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.18	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.18	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.18	   mark(s(X)) -> active(s(mark(X)))
26.18/8.18	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.18	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.18	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.18	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.18	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.18	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.18	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.18	   mark(isNat(X)) -> active(isNat(X))
26.18/8.18	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.18	   mark(tt) -> active(tt)
26.18/8.18	   mark(0) -> active(0)
26.18/8.18	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.18	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.18	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.18	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.18	   U12(active(X)) -> U12(X)
26.18/8.18	   U12(mark(X)) -> U12(X)
26.18/8.18	   active(U12(tt)) -> mark(tt)
26.18/8.18	   active(U21(tt)) -> mark(tt)
26.18/8.18	   active(isNat(0)) -> mark(tt)
26.18/8.18	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.18	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.18	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.18	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.18	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.18	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.18	   s(active(X)) -> s(X)
26.18/8.18	   s(mark(X)) -> s(X)
26.18/8.18	   U21(active(X)) -> U21(X)
26.18/8.18	   U21(mark(X)) -> U21(X)
26.18/8.18	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.18	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.18	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.18	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.18	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.18	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.18	
26.18/8.18	The set Q consists of the following terms:
26.18/8.18	
26.18/8.18	   active(U11(tt, x0))
26.18/8.18	   active(U12(tt))
26.18/8.18	   active(U21(tt))
26.18/8.18	   active(U31(tt, x0))
26.18/8.18	   active(U41(tt, x0, x1))
26.18/8.18	   active(U42(tt, x0, x1))
26.18/8.18	   active(isNat(0))
26.18/8.18	   active(isNat(plus(x0, x1)))
26.18/8.18	   active(isNat(s(x0)))
26.18/8.18	   active(plus(x0, 0))
26.18/8.18	   active(plus(x0, s(x1)))
26.18/8.18	   mark(U11(x0, x1))
26.18/8.18	   mark(tt)
26.18/8.18	   mark(U12(x0))
26.18/8.18	   mark(isNat(x0))
26.18/8.18	   mark(U21(x0))
26.18/8.18	   mark(U31(x0, x1))
26.18/8.18	   mark(U41(x0, x1, x2))
26.18/8.18	   mark(U42(x0, x1, x2))
26.18/8.18	   mark(s(x0))
26.18/8.18	   mark(plus(x0, x1))
26.18/8.18	   mark(0)
26.18/8.18	   U11(mark(x0), x1)
26.18/8.18	   U11(x0, mark(x1))
26.18/8.18	   U11(active(x0), x1)
26.18/8.18	   U11(x0, active(x1))
26.18/8.18	   U12(mark(x0))
26.18/8.18	   U12(active(x0))
26.18/8.18	   isNat(mark(x0))
26.18/8.18	   isNat(active(x0))
26.18/8.18	   U21(mark(x0))
26.18/8.18	   U21(active(x0))
26.18/8.18	   U31(mark(x0), x1)
26.18/8.18	   U31(x0, mark(x1))
26.18/8.18	   U31(active(x0), x1)
26.18/8.18	   U31(x0, active(x1))
26.18/8.18	   U41(mark(x0), x1, x2)
26.18/8.18	   U41(x0, mark(x1), x2)
26.18/8.18	   U41(x0, x1, mark(x2))
26.18/8.18	   U41(active(x0), x1, x2)
26.18/8.18	   U41(x0, active(x1), x2)
26.18/8.18	   U41(x0, x1, active(x2))
26.18/8.18	   U42(mark(x0), x1, x2)
26.18/8.18	   U42(x0, mark(x1), x2)
26.18/8.18	   U42(x0, x1, mark(x2))
26.18/8.18	   U42(active(x0), x1, x2)
26.18/8.18	   U42(x0, active(x1), x2)
26.18/8.18	   U42(x0, x1, active(x2))
26.18/8.18	   s(mark(x0))
26.18/8.18	   s(active(x0))
26.18/8.18	   plus(mark(x0), x1)
26.18/8.18	   plus(x0, mark(x1))
26.18/8.18	   plus(active(x0), x1)
26.18/8.18	   plus(x0, active(x1))
26.18/8.18	
26.18/8.18	We have to consider all minimal (P,Q,R)-chains.
26.18/8.18	----------------------------------------
26.18/8.18	
26.18/8.18	(93) QDPQMonotonicMRRProof (EQUIVALENT)
26.18/8.18	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.
26.18/8.18	
26.18/8.18	Strictly oriented dependency pairs:
26.18/8.18	
26.18/8.18	   ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2)))
26.18/8.18	   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
26.18/8.18	   MARK(isNat(X)) -> ACTIVE(isNat(X))
26.18/8.18	   ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2))
26.18/8.18	   ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1)))
26.18/8.18	
26.18/8.18	
26.18/8.18	Used ordering: Polynomial interpretation [POLO]:
26.18/8.18	
26.18/8.18	   POL(0) = 2
26.18/8.18	   POL(ACTIVE(x_1)) = 2*x_1
26.18/8.18	   POL(MARK(x_1)) = 2 + 2*x_1
26.18/8.18	   POL(U11(x_1, x_2)) = 2*x_1 + 2*x_2
26.18/8.18	   POL(U12(x_1)) = x_1
26.18/8.18	   POL(U21(x_1)) = x_1
26.18/8.18	   POL(U31(x_1, x_2)) = 2 + 2*x_2
26.18/8.18	   POL(U41(x_1, x_2, x_3)) = 2 + 2*x_2 + 2*x_3
26.18/8.18	   POL(U42(x_1, x_2, x_3)) = 2 + 2*x_2 + 2*x_3
26.18/8.18	   POL(active(x_1)) = x_1
26.18/8.18	   POL(isNat(x_1)) = 2*x_1
26.18/8.18	   POL(mark(x_1)) = x_1
26.18/8.18	   POL(plus(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
26.18/8.18	   POL(s(x_1)) = 1 + x_1
26.18/8.18	   POL(tt) = 1
26.18/8.18	
26.18/8.18	
26.18/8.18	----------------------------------------
26.18/8.18	
26.18/8.18	(94)
26.18/8.18	Obligation:
26.18/8.18	Q DP problem:
26.18/8.18	The TRS P consists of the following rules:
26.18/8.18	
26.18/8.18	   MARK(U12(X)) -> MARK(X)
26.18/8.18	   MARK(U11(X1, X2)) -> MARK(X1)
26.18/8.18	   MARK(U21(X)) -> MARK(X)
26.18/8.18	
26.18/8.18	The TRS R consists of the following rules:
26.18/8.18	
26.18/8.18	   mark(U12(X)) -> active(U12(mark(X)))
26.18/8.18	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
26.18/8.18	   active(U31(tt, N)) -> mark(N)
26.18/8.18	   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
26.18/8.18	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
26.18/8.18	   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
26.18/8.18	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
26.18/8.18	   mark(s(X)) -> active(s(mark(X)))
26.18/8.18	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
26.18/8.18	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
26.18/8.18	   mark(U21(X)) -> active(U21(mark(X)))
26.18/8.18	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
26.18/8.18	   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
26.18/8.18	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
26.18/8.18	   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
26.18/8.18	   mark(isNat(X)) -> active(isNat(X))
26.18/8.18	   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
26.18/8.18	   mark(tt) -> active(tt)
26.18/8.18	   mark(0) -> active(0)
26.18/8.18	   plus(X1, mark(X2)) -> plus(X1, X2)
26.18/8.18	   plus(mark(X1), X2) -> plus(X1, X2)
26.18/8.18	   plus(active(X1), X2) -> plus(X1, X2)
26.18/8.18	   plus(X1, active(X2)) -> plus(X1, X2)
26.18/8.18	   U12(active(X)) -> U12(X)
26.18/8.18	   U12(mark(X)) -> U12(X)
26.18/8.18	   active(U12(tt)) -> mark(tt)
26.18/8.18	   active(U21(tt)) -> mark(tt)
26.18/8.18	   active(isNat(0)) -> mark(tt)
26.18/8.18	   U11(X1, mark(X2)) -> U11(X1, X2)
26.18/8.18	   U11(mark(X1), X2) -> U11(X1, X2)
26.18/8.18	   U11(active(X1), X2) -> U11(X1, X2)
26.18/8.18	   U11(X1, active(X2)) -> U11(X1, X2)
26.18/8.18	   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
26.18/8.18	   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
26.18/8.18	   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
26.18/8.18	   s(active(X)) -> s(X)
26.18/8.18	   s(mark(X)) -> s(X)
26.18/8.18	   U21(active(X)) -> U21(X)
26.18/8.18	   U21(mark(X)) -> U21(X)
26.18/8.18	   U31(X1, mark(X2)) -> U31(X1, X2)
26.18/8.18	   U31(mark(X1), X2) -> U31(X1, X2)
26.18/8.18	   U31(active(X1), X2) -> U31(X1, X2)
26.18/8.18	   U31(X1, active(X2)) -> U31(X1, X2)
26.18/8.18	   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
26.18/8.18	   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
26.18/8.18	   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
26.18/8.18	
26.18/8.18	The set Q consists of the following terms:
26.18/8.18	
26.18/8.18	   active(U11(tt, x0))
26.18/8.18	   active(U12(tt))
26.18/8.18	   active(U21(tt))
26.18/8.18	   active(U31(tt, x0))
26.18/8.18	   active(U41(tt, x0, x1))
26.18/8.18	   active(U42(tt, x0, x1))
26.18/8.18	   active(isNat(0))
26.18/8.18	   active(isNat(plus(x0, x1)))
26.18/8.18	   active(isNat(s(x0)))
26.18/8.18	   active(plus(x0, 0))
26.18/8.18	   active(plus(x0, s(x1)))
26.18/8.18	   mark(U11(x0, x1))
26.18/8.18	   mark(tt)
26.18/8.18	   mark(U12(x0))
26.18/8.18	   mark(isNat(x0))
26.18/8.18	   mark(U21(x0))
26.18/8.18	   mark(U31(x0, x1))
26.18/8.18	   mark(U41(x0, x1, x2))
26.18/8.18	   mark(U42(x0, x1, x2))
26.18/8.18	   mark(s(x0))
26.18/8.18	   mark(plus(x0, x1))
26.18/8.18	   mark(0)
26.18/8.18	   U11(mark(x0), x1)
26.18/8.18	   U11(x0, mark(x1))
26.18/8.18	   U11(active(x0), x1)
26.18/8.18	   U11(x0, active(x1))
26.18/8.18	   U12(mark(x0))
26.18/8.18	   U12(active(x0))
26.18/8.18	   isNat(mark(x0))
26.18/8.18	   isNat(active(x0))
26.18/8.18	   U21(mark(x0))
26.18/8.18	   U21(active(x0))
26.18/8.18	   U31(mark(x0), x1)
26.18/8.18	   U31(x0, mark(x1))
26.18/8.18	   U31(active(x0), x1)
26.18/8.18	   U31(x0, active(x1))
26.18/8.18	   U41(mark(x0), x1, x2)
26.18/8.18	   U41(x0, mark(x1), x2)
26.18/8.18	   U41(x0, x1, mark(x2))
26.18/8.18	   U41(active(x0), x1, x2)
26.18/8.18	   U41(x0, active(x1), x2)
26.18/8.18	   U41(x0, x1, active(x2))
26.18/8.18	   U42(mark(x0), x1, x2)
26.18/8.18	   U42(x0, mark(x1), x2)
26.18/8.18	   U42(x0, x1, mark(x2))
26.18/8.18	   U42(active(x0), x1, x2)
26.18/8.18	   U42(x0, active(x1), x2)
26.18/8.18	   U42(x0, x1, active(x2))
26.18/8.18	   s(mark(x0))
26.18/8.18	   s(active(x0))
26.18/8.18	   plus(mark(x0), x1)
26.18/8.18	   plus(x0, mark(x1))
26.18/8.18	   plus(active(x0), x1)
26.18/8.18	   plus(x0, active(x1))
26.18/8.18	
26.18/8.18	We have to consider all minimal (P,Q,R)-chains.
26.18/8.18	----------------------------------------
26.18/8.18	
26.18/8.18	(95) UsableRulesProof (EQUIVALENT)
26.18/8.18	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.
26.18/8.18	----------------------------------------
26.18/8.18	
26.18/8.18	(96)
26.18/8.18	Obligation:
26.18/8.18	Q DP problem:
26.18/8.18	The TRS P consists of the following rules:
26.18/8.18	
26.18/8.18	   MARK(U12(X)) -> MARK(X)
26.18/8.18	   MARK(U11(X1, X2)) -> MARK(X1)
26.18/8.18	   MARK(U21(X)) -> MARK(X)
26.18/8.18	
26.18/8.18	R is empty.
26.18/8.18	The set Q consists of the following terms:
26.18/8.18	
26.18/8.18	   active(U11(tt, x0))
26.18/8.18	   active(U12(tt))
26.18/8.18	   active(U21(tt))
26.18/8.18	   active(U31(tt, x0))
26.18/8.18	   active(U41(tt, x0, x1))
26.18/8.18	   active(U42(tt, x0, x1))
26.18/8.18	   active(isNat(0))
26.18/8.18	   active(isNat(plus(x0, x1)))
26.18/8.18	   active(isNat(s(x0)))
26.18/8.18	   active(plus(x0, 0))
26.18/8.18	   active(plus(x0, s(x1)))
26.18/8.18	   mark(U11(x0, x1))
26.18/8.18	   mark(tt)
26.18/8.18	   mark(U12(x0))
26.18/8.18	   mark(isNat(x0))
26.18/8.18	   mark(U21(x0))
26.18/8.18	   mark(U31(x0, x1))
26.18/8.18	   mark(U41(x0, x1, x2))
26.18/8.18	   mark(U42(x0, x1, x2))
26.18/8.18	   mark(s(x0))
26.18/8.18	   mark(plus(x0, x1))
26.18/8.18	   mark(0)
26.18/8.18	   U11(mark(x0), x1)
26.18/8.18	   U11(x0, mark(x1))
26.18/8.18	   U11(active(x0), x1)
26.18/8.18	   U11(x0, active(x1))
26.18/8.18	   U12(mark(x0))
26.18/8.18	   U12(active(x0))
26.18/8.18	   isNat(mark(x0))
26.18/8.18	   isNat(active(x0))
26.18/8.18	   U21(mark(x0))
26.18/8.18	   U21(active(x0))
26.18/8.18	   U31(mark(x0), x1)
26.18/8.18	   U31(x0, mark(x1))
26.18/8.18	   U31(active(x0), x1)
26.18/8.18	   U31(x0, active(x1))
26.18/8.18	   U41(mark(x0), x1, x2)
26.18/8.18	   U41(x0, mark(x1), x2)
26.18/8.18	   U41(x0, x1, mark(x2))
26.18/8.18	   U41(active(x0), x1, x2)
26.18/8.18	   U41(x0, active(x1), x2)
26.18/8.18	   U41(x0, x1, active(x2))
26.18/8.18	   U42(mark(x0), x1, x2)
26.18/8.18	   U42(x0, mark(x1), x2)
26.18/8.18	   U42(x0, x1, mark(x2))
26.18/8.18	   U42(active(x0), x1, x2)
26.18/8.18	   U42(x0, active(x1), x2)
26.18/8.18	   U42(x0, x1, active(x2))
26.18/8.18	   s(mark(x0))
26.18/8.18	   s(active(x0))
26.18/8.18	   plus(mark(x0), x1)
26.18/8.18	   plus(x0, mark(x1))
26.18/8.18	   plus(active(x0), x1)
26.18/8.18	   plus(x0, active(x1))
26.18/8.18	
26.18/8.18	We have to consider all minimal (P,Q,R)-chains.
26.18/8.18	----------------------------------------
26.18/8.18	
26.18/8.18	(97) QReductionProof (EQUIVALENT)
26.18/8.18	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
26.18/8.18	
26.18/8.18	   active(U11(tt, x0))
26.18/8.18	   active(U12(tt))
26.18/8.18	   active(U21(tt))
26.18/8.18	   active(U31(tt, x0))
26.18/8.18	   active(U41(tt, x0, x1))
26.18/8.18	   active(U42(tt, x0, x1))
26.18/8.18	   active(isNat(0))
26.18/8.18	   active(isNat(plus(x0, x1)))
26.18/8.18	   active(isNat(s(x0)))
26.18/8.18	   active(plus(x0, 0))
26.18/8.18	   active(plus(x0, s(x1)))
26.18/8.18	   mark(U11(x0, x1))
26.18/8.18	   mark(tt)
26.18/8.18	   mark(U12(x0))
26.18/8.18	   mark(isNat(x0))
26.18/8.18	   mark(U21(x0))
26.18/8.18	   mark(U31(x0, x1))
26.18/8.18	   mark(U41(x0, x1, x2))
26.18/8.18	   mark(U42(x0, x1, x2))
26.18/8.18	   mark(s(x0))
26.18/8.18	   mark(plus(x0, x1))
26.18/8.18	   mark(0)
26.18/8.18	   isNat(mark(x0))
26.18/8.18	   isNat(active(x0))
26.18/8.18	   U31(mark(x0), x1)
26.18/8.18	   U31(x0, mark(x1))
26.18/8.18	   U31(active(x0), x1)
26.18/8.18	   U31(x0, active(x1))
26.18/8.18	   U41(mark(x0), x1, x2)
26.18/8.18	   U41(x0, mark(x1), x2)
26.18/8.18	   U41(x0, x1, mark(x2))
26.18/8.18	   U41(active(x0), x1, x2)
26.18/8.18	   U41(x0, active(x1), x2)
26.18/8.18	   U41(x0, x1, active(x2))
26.18/8.18	   U42(mark(x0), x1, x2)
26.18/8.18	   U42(x0, mark(x1), x2)
26.18/8.18	   U42(x0, x1, mark(x2))
26.18/8.18	   U42(active(x0), x1, x2)
26.18/8.18	   U42(x0, active(x1), x2)
26.18/8.18	   U42(x0, x1, active(x2))
26.18/8.18	   s(mark(x0))
26.18/8.18	   s(active(x0))
26.18/8.18	   plus(mark(x0), x1)
26.18/8.18	   plus(x0, mark(x1))
26.18/8.18	   plus(active(x0), x1)
26.18/8.18	   plus(x0, active(x1))
26.18/8.18	
26.18/8.18	
26.18/8.18	----------------------------------------
26.18/8.18	
26.18/8.18	(98)
26.18/8.18	Obligation:
26.18/8.18	Q DP problem:
26.18/8.18	The TRS P consists of the following rules:
26.18/8.18	
26.18/8.18	   MARK(U12(X)) -> MARK(X)
26.18/8.18	   MARK(U11(X1, X2)) -> MARK(X1)
26.18/8.18	   MARK(U21(X)) -> MARK(X)
26.18/8.18	
26.18/8.18	R is empty.
26.18/8.18	The set Q consists of the following terms:
26.18/8.18	
26.18/8.18	   U11(mark(x0), x1)
26.18/8.18	   U11(x0, mark(x1))
26.18/8.18	   U11(active(x0), x1)
26.18/8.18	   U11(x0, active(x1))
26.18/8.18	   U12(mark(x0))
26.18/8.18	   U12(active(x0))
26.18/8.18	   U21(mark(x0))
26.18/8.18	   U21(active(x0))
26.18/8.18	
26.18/8.18	We have to consider all minimal (P,Q,R)-chains.
26.18/8.18	----------------------------------------
26.18/8.18	
26.18/8.18	(99) QDPSizeChangeProof (EQUIVALENT)
26.18/8.18	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. 
26.18/8.18	
26.18/8.18	From the DPs we obtained the following set of size-change graphs:
26.18/8.18	*MARK(U12(X)) -> MARK(X)
26.18/8.18	The graph contains the following edges 1 > 1
26.18/8.18	
26.18/8.18	
26.18/8.18	*MARK(U11(X1, X2)) -> MARK(X1)
26.18/8.18	The graph contains the following edges 1 > 1
26.18/8.18	
26.18/8.18	
26.18/8.18	*MARK(U21(X)) -> MARK(X)
26.18/8.18	The graph contains the following edges 1 > 1
26.18/8.18	
26.18/8.18	
26.18/8.18	----------------------------------------
26.18/8.18	
26.18/8.18	(100)
26.18/8.18	YES
26.18/8.21	EOF