354.42/147.61	MAYBE
354.42/147.61	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
354.42/147.61	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
354.42/147.61	
354.42/147.61	
354.42/147.61	Outermost Termination of the given OTRS could not be shown:
354.42/147.61	
354.42/147.61	(0) OTRS
354.42/147.61	(1) Thiemann-SpecialC-Transformation [EQUIVALENT, 0 ms]
354.42/147.61	(2) QTRS
354.42/147.61	    (3) DependencyPairsProof [EQUIVALENT, 0 ms]
354.42/147.61	    (4) QDP
354.42/147.61	    (5) DependencyGraphProof [EQUIVALENT, 0 ms]
354.42/147.61	    (6) AND
354.42/147.61	        (7) QDP
354.42/147.61	            (8) UsableRulesProof [EQUIVALENT, 0 ms]
354.42/147.61	            (9) QDP
354.42/147.61	            (10) QReductionProof [EQUIVALENT, 0 ms]
354.42/147.61	            (11) QDP
354.42/147.61	            (12) MRRProof [EQUIVALENT, 8 ms]
354.42/147.61	            (13) QDP
354.42/147.61	            (14) UsableRulesProof [EQUIVALENT, 0 ms]
354.42/147.61	            (15) QDP
354.42/147.61	            (16) QReductionProof [EQUIVALENT, 0 ms]
354.42/147.61	            (17) QDP
354.42/147.61	            (18) UsableRulesReductionPairsProof [EQUIVALENT, 7 ms]
354.42/147.61	            (19) QDP
354.42/147.61	            (20) DependencyGraphProof [EQUIVALENT, 0 ms]
354.42/147.61	            (21) QDP
354.42/147.61	            (22) QReductionProof [EQUIVALENT, 0 ms]
354.42/147.61	            (23) QDP
354.42/147.61	            (24) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
354.42/147.61	            (25) QDP
354.42/147.61	            (26) DependencyGraphProof [EQUIVALENT, 0 ms]
354.42/147.61	            (27) TRUE
354.42/147.61	        (28) QDP
354.42/147.61	            (29) UsableRulesProof [EQUIVALENT, 0 ms]
354.42/147.61	            (30) QDP
354.42/147.61	                (31) QReductionProof [EQUIVALENT, 0 ms]
354.42/147.61	                (32) QDP
354.42/147.61	                (33) TransformationProof [EQUIVALENT, 0 ms]
354.42/147.61	                (34) QDP
354.42/147.61	            (35) UsableRulesProof [EQUIVALENT, 0 ms]
354.42/147.61	            (36) QDP
354.42/147.61	                (37) QReductionProof [EQUIVALENT, 0 ms]
354.42/147.61	                (38) QDP
354.42/147.61	(39) Trivial-Transformation [SOUND, 0 ms]
354.42/147.61	(40) QTRS
354.42/147.61	    (41) DependencyPairsProof [EQUIVALENT, 0 ms]
354.42/147.61	    (42) QDP
354.42/147.61	    (43) DependencyGraphProof [EQUIVALENT, 5 ms]
354.42/147.61	    (44) AND
354.42/147.61	        (45) QDP
354.42/147.61	            (46) UsableRulesProof [EQUIVALENT, 0 ms]
354.42/147.61	            (47) QDP
354.42/147.61	            (48) QDPSizeChangeProof [EQUIVALENT, 0 ms]
354.42/147.61	            (49) YES
354.42/147.61	        (50) QDP
354.42/147.61	            (51) UsableRulesProof [EQUIVALENT, 0 ms]
354.42/147.61	            (52) QDP
354.42/147.61	            (53) QDPSizeChangeProof [EQUIVALENT, 0 ms]
354.42/147.61	            (54) YES
354.42/147.61	        (55) QDP
354.42/147.61	            (56) TransformationProof [EQUIVALENT, 1 ms]
354.42/147.61	            (57) QDP
354.42/147.61	            (58) QDPOrderProof [EQUIVALENT, 4 ms]
354.42/147.61	            (59) QDP
354.42/147.61	(60) Raffelsieper-Zantema-Transformation [SOUND, 0 ms]
354.42/147.61	(61) QTRS
354.42/147.61	    (62) AAECC Innermost [EQUIVALENT, 0 ms]
354.42/147.61	    (63) QTRS
354.42/147.61	    (64) DependencyPairsProof [EQUIVALENT, 0 ms]
354.42/147.61	    (65) QDP
354.42/147.61	    (66) DependencyGraphProof [EQUIVALENT, 0 ms]
354.42/147.61	    (67) AND
354.42/147.61	        (68) QDP
354.42/147.61	            (69) UsableRulesProof [EQUIVALENT, 0 ms]
354.42/147.61	            (70) QDP
354.42/147.61	            (71) QReductionProof [EQUIVALENT, 0 ms]
354.42/147.61	            (72) QDP
354.42/147.61	            (73) QDPSizeChangeProof [EQUIVALENT, 0 ms]
354.42/147.61	            (74) YES
354.42/147.61	        (75) QDP
354.42/147.61	            (76) UsableRulesProof [EQUIVALENT, 0 ms]
354.42/147.61	            (77) QDP
354.42/147.61	            (78) QReductionProof [EQUIVALENT, 0 ms]
354.42/147.61	            (79) QDP
354.42/147.61	            (80) TransformationProof [EQUIVALENT, 0 ms]
354.42/147.61	            (81) QDP
354.42/147.61	            (82) DependencyGraphProof [EQUIVALENT, 0 ms]
354.42/147.61	            (83) QDP
354.42/147.61	            (84) TransformationProof [EQUIVALENT, 0 ms]
354.42/147.61	            (85) QDP
354.42/147.61	            (86) QDPOrderProof [EQUIVALENT, 22 ms]
354.42/147.61	            (87) QDP
354.42/147.61	            (88) QDPOrderProof [EQUIVALENT, 15 ms]
354.42/147.61	            (89) QDP
354.42/147.61	            (90) MNOCProof [EQUIVALENT, 0 ms]
354.42/147.61	            (91) QDP
354.42/147.61	            (92) SplitQDPProof [EQUIVALENT, 0 ms]
354.42/147.61	            (93) AND
354.42/147.61	                (94) QDP
354.42/147.61	                    (95) SemLabProof [SOUND, 0 ms]
354.42/147.61	                    (96) QDP
354.42/147.61	                    (97) DependencyGraphProof [EQUIVALENT, 0 ms]
354.42/147.61	                    (98) QDP
354.42/147.61	                    (99) UsableRulesReductionPairsProof [EQUIVALENT, 8 ms]
354.42/147.61	                    (100) QDP
354.42/147.61	                    (101) MRRProof [EQUIVALENT, 0 ms]
354.42/147.61	                    (102) QDP
354.42/147.61	                    (103) DependencyGraphProof [EQUIVALENT, 0 ms]
354.42/147.61	                    (104) QDP
354.42/147.61	                    (105) MRRProof [EQUIVALENT, 5 ms]
354.42/147.61	                    (106) QDP
354.42/147.61	                    (107) MRRProof [EQUIVALENT, 0 ms]
354.42/147.61	                    (108) QDP
354.42/147.62	                    (109) PisEmptyProof [SOUND, 0 ms]
354.42/147.62	                    (110) TRUE
354.42/147.62	                (111) QDP
354.42/147.62	                    (112) QReductionProof [EQUIVALENT, 0 ms]
354.42/147.62	                    (113) QDP
354.42/147.62	                        (114) MNOCProof [EQUIVALENT, 0 ms]
354.42/147.62	                        (115) QDP
354.42/147.62	                    (116) SplitQDPProof [EQUIVALENT, 0 ms]
354.42/147.62	                    (117) AND
354.42/147.62	                        (118) QDP
354.42/147.62	                            (119) SemLabProof [SOUND, 0 ms]
354.42/147.62	                            (120) QDP
354.42/147.62	                            (121) DependencyGraphProof [EQUIVALENT, 0 ms]
354.42/147.62	                            (122) QDP
354.42/147.62	                            (123) MRRProof [EQUIVALENT, 16 ms]
354.42/147.62	                            (124) QDP
354.42/147.62	                            (125) DependencyGraphProof [EQUIVALENT, 0 ms]
354.42/147.62	                            (126) QDP
354.42/147.62	                            (127) MRRProof [EQUIVALENT, 0 ms]
354.42/147.62	                            (128) QDP
354.42/147.62	                            (129) MRRProof [EQUIVALENT, 0 ms]
354.42/147.62	                            (130) QDP
354.42/147.62	                            (131) PisEmptyProof [SOUND, 0 ms]
354.42/147.62	                            (132) TRUE
354.42/147.62	                        (133) QDP
354.42/147.62	                            (134) QReductionProof [EQUIVALENT, 0 ms]
354.42/147.62	                            (135) QDP
354.42/147.62	
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(0)
354.42/147.62	Obligation:
354.42/147.62	Term rewrite system R:
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   a(a(x)) -> c
354.42/147.62	   b(u(x)) -> b(d(x))
354.42/147.62	   d(a(x)) -> a(d(x))
354.42/147.62	   d(b(x)) -> u(a(b(x)))
354.42/147.62	   a(u(x)) -> u(a(x))
354.42/147.62	
354.42/147.62	
354.42/147.62	
354.42/147.62	Outermost Strategy.
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(1) Thiemann-SpecialC-Transformation (EQUIVALENT)
354.42/147.62	We applied the Thiemann-SpecialC transformation  to transform the outermost TRS to an innermost TRS.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(2)
354.42/147.62	Obligation:
354.42/147.62	Q restricted rewrite system:
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   top(go_up(x)) -> top(reduce(x))
354.42/147.62	   reduce(a(x_1)) -> check_a(redex_a(x_1))
354.42/147.62	   reduce(b(x_1)) -> check_b(redex_b(x_1))
354.42/147.62	   reduce(d(x_1)) -> check_d(redex_d(x_1))
354.42/147.62	   redex_a(a(x)) -> result_a(c)
354.42/147.62	   redex_b(u(x)) -> result_b(b(d(x)))
354.42/147.62	   redex_d(a(x)) -> result_d(a(d(x)))
354.42/147.62	   redex_d(b(x)) -> result_d(u(a(b(x))))
354.42/147.62	   redex_a(u(x)) -> result_a(u(a(x)))
354.42/147.62	   check_a(result_a(x)) -> go_up(x)
354.42/147.62	   check_b(result_b(x)) -> go_up(x)
354.42/147.62	   check_d(result_d(x)) -> go_up(x)
354.42/147.62	   check_a(redex_a(x_1)) -> in_a_1(reduce(x_1))
354.42/147.62	   check_b(redex_b(x_1)) -> in_b_1(reduce(x_1))
354.42/147.62	   check_d(redex_d(x_1)) -> in_d_1(reduce(x_1))
354.42/147.62	   reduce(u(x_1)) -> in_u_1(reduce(x_1))
354.42/147.62	   in_a_1(go_up(x_1)) -> go_up(a(x_1))
354.42/147.62	   in_b_1(go_up(x_1)) -> go_up(b(x_1))
354.42/147.62	   in_u_1(go_up(x_1)) -> go_up(u(x_1))
354.42/147.62	   in_d_1(go_up(x_1)) -> go_up(d(x_1))
354.42/147.62	
354.42/147.62	The set Q consists of the following terms:
354.42/147.62	
354.42/147.62	   top(go_up(x0))
354.42/147.62	   reduce(a(x0))
354.42/147.62	   reduce(b(x0))
354.42/147.62	   reduce(d(x0))
354.42/147.62	   redex_a(a(x0))
354.42/147.62	   redex_b(u(x0))
354.42/147.62	   redex_d(a(x0))
354.42/147.62	   redex_d(b(x0))
354.42/147.62	   redex_a(u(x0))
354.42/147.62	   check_a(result_a(x0))
354.42/147.62	   check_b(result_b(x0))
354.42/147.62	   check_d(result_d(x0))
354.42/147.62	   check_a(redex_a(x0))
354.42/147.62	   check_b(redex_b(x0))
354.42/147.62	   check_d(redex_d(x0))
354.42/147.62	   reduce(u(x0))
354.42/147.62	   in_a_1(go_up(x0))
354.42/147.62	   in_b_1(go_up(x0))
354.42/147.62	   in_u_1(go_up(x0))
354.42/147.62	   in_d_1(go_up(x0))
354.42/147.62	
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(3) DependencyPairsProof (EQUIVALENT)
354.42/147.62	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(4)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   TOP(go_up(x)) -> TOP(reduce(x))
354.42/147.62	   TOP(go_up(x)) -> REDUCE(x)
354.42/147.62	   REDUCE(a(x_1)) -> CHECK_A(redex_a(x_1))
354.42/147.62	   REDUCE(a(x_1)) -> REDEX_A(x_1)
354.42/147.62	   REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1))
354.42/147.62	   REDUCE(b(x_1)) -> REDEX_B(x_1)
354.42/147.62	   REDUCE(d(x_1)) -> CHECK_D(redex_d(x_1))
354.42/147.62	   REDUCE(d(x_1)) -> REDEX_D(x_1)
354.42/147.62	   CHECK_A(redex_a(x_1)) -> IN_A_1(reduce(x_1))
354.42/147.62	   CHECK_A(redex_a(x_1)) -> REDUCE(x_1)
354.42/147.62	   CHECK_B(redex_b(x_1)) -> IN_B_1(reduce(x_1))
354.42/147.62	   CHECK_B(redex_b(x_1)) -> REDUCE(x_1)
354.42/147.62	   CHECK_D(redex_d(x_1)) -> IN_D_1(reduce(x_1))
354.42/147.62	   CHECK_D(redex_d(x_1)) -> REDUCE(x_1)
354.42/147.62	   REDUCE(u(x_1)) -> IN_U_1(reduce(x_1))
354.42/147.62	   REDUCE(u(x_1)) -> REDUCE(x_1)
354.42/147.62	
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   top(go_up(x)) -> top(reduce(x))
354.42/147.62	   reduce(a(x_1)) -> check_a(redex_a(x_1))
354.42/147.62	   reduce(b(x_1)) -> check_b(redex_b(x_1))
354.42/147.62	   reduce(d(x_1)) -> check_d(redex_d(x_1))
354.42/147.62	   redex_a(a(x)) -> result_a(c)
354.42/147.62	   redex_b(u(x)) -> result_b(b(d(x)))
354.42/147.62	   redex_d(a(x)) -> result_d(a(d(x)))
354.42/147.62	   redex_d(b(x)) -> result_d(u(a(b(x))))
354.42/147.62	   redex_a(u(x)) -> result_a(u(a(x)))
354.42/147.62	   check_a(result_a(x)) -> go_up(x)
354.42/147.62	   check_b(result_b(x)) -> go_up(x)
354.42/147.62	   check_d(result_d(x)) -> go_up(x)
354.42/147.62	   check_a(redex_a(x_1)) -> in_a_1(reduce(x_1))
354.42/147.62	   check_b(redex_b(x_1)) -> in_b_1(reduce(x_1))
354.42/147.62	   check_d(redex_d(x_1)) -> in_d_1(reduce(x_1))
354.42/147.62	   reduce(u(x_1)) -> in_u_1(reduce(x_1))
354.42/147.62	   in_a_1(go_up(x_1)) -> go_up(a(x_1))
354.42/147.62	   in_b_1(go_up(x_1)) -> go_up(b(x_1))
354.42/147.62	   in_u_1(go_up(x_1)) -> go_up(u(x_1))
354.42/147.62	   in_d_1(go_up(x_1)) -> go_up(d(x_1))
354.42/147.62	
354.42/147.62	The set Q consists of the following terms:
354.42/147.62	
354.42/147.62	   top(go_up(x0))
354.42/147.62	   reduce(a(x0))
354.42/147.62	   reduce(b(x0))
354.42/147.62	   reduce(d(x0))
354.42/147.62	   redex_a(a(x0))
354.42/147.62	   redex_b(u(x0))
354.42/147.62	   redex_d(a(x0))
354.42/147.62	   redex_d(b(x0))
354.42/147.62	   redex_a(u(x0))
354.42/147.62	   check_a(result_a(x0))
354.42/147.62	   check_b(result_b(x0))
354.42/147.62	   check_d(result_d(x0))
354.42/147.62	   check_a(redex_a(x0))
354.42/147.62	   check_b(redex_b(x0))
354.42/147.62	   check_d(redex_d(x0))
354.42/147.62	   reduce(u(x0))
354.42/147.62	   in_a_1(go_up(x0))
354.42/147.62	   in_b_1(go_up(x0))
354.42/147.62	   in_u_1(go_up(x0))
354.42/147.62	   in_d_1(go_up(x0))
354.42/147.62	
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(5) DependencyGraphProof (EQUIVALENT)
354.42/147.62	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 8 less nodes.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(6)
354.42/147.62	Complex Obligation (AND)
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(7)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   CHECK_A(redex_a(x_1)) -> REDUCE(x_1)
354.42/147.62	   REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1))
354.42/147.62	   CHECK_B(redex_b(x_1)) -> REDUCE(x_1)
354.42/147.62	   REDUCE(a(x_1)) -> CHECK_A(redex_a(x_1))
354.42/147.62	   REDUCE(d(x_1)) -> CHECK_D(redex_d(x_1))
354.42/147.62	   CHECK_D(redex_d(x_1)) -> REDUCE(x_1)
354.42/147.62	   REDUCE(u(x_1)) -> REDUCE(x_1)
354.42/147.62	
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   top(go_up(x)) -> top(reduce(x))
354.42/147.62	   reduce(a(x_1)) -> check_a(redex_a(x_1))
354.42/147.62	   reduce(b(x_1)) -> check_b(redex_b(x_1))
354.42/147.62	   reduce(d(x_1)) -> check_d(redex_d(x_1))
354.42/147.62	   redex_a(a(x)) -> result_a(c)
354.42/147.62	   redex_b(u(x)) -> result_b(b(d(x)))
354.42/147.62	   redex_d(a(x)) -> result_d(a(d(x)))
354.42/147.62	   redex_d(b(x)) -> result_d(u(a(b(x))))
354.42/147.62	   redex_a(u(x)) -> result_a(u(a(x)))
354.42/147.62	   check_a(result_a(x)) -> go_up(x)
354.42/147.62	   check_b(result_b(x)) -> go_up(x)
354.42/147.62	   check_d(result_d(x)) -> go_up(x)
354.42/147.62	   check_a(redex_a(x_1)) -> in_a_1(reduce(x_1))
354.42/147.62	   check_b(redex_b(x_1)) -> in_b_1(reduce(x_1))
354.42/147.62	   check_d(redex_d(x_1)) -> in_d_1(reduce(x_1))
354.42/147.62	   reduce(u(x_1)) -> in_u_1(reduce(x_1))
354.42/147.62	   in_a_1(go_up(x_1)) -> go_up(a(x_1))
354.42/147.62	   in_b_1(go_up(x_1)) -> go_up(b(x_1))
354.42/147.62	   in_u_1(go_up(x_1)) -> go_up(u(x_1))
354.42/147.62	   in_d_1(go_up(x_1)) -> go_up(d(x_1))
354.42/147.62	
354.42/147.62	The set Q consists of the following terms:
354.42/147.62	
354.42/147.62	   top(go_up(x0))
354.42/147.62	   reduce(a(x0))
354.42/147.62	   reduce(b(x0))
354.42/147.62	   reduce(d(x0))
354.42/147.62	   redex_a(a(x0))
354.42/147.62	   redex_b(u(x0))
354.42/147.62	   redex_d(a(x0))
354.42/147.62	   redex_d(b(x0))
354.42/147.62	   redex_a(u(x0))
354.42/147.62	   check_a(result_a(x0))
354.42/147.62	   check_b(result_b(x0))
354.42/147.62	   check_d(result_d(x0))
354.42/147.62	   check_a(redex_a(x0))
354.42/147.62	   check_b(redex_b(x0))
354.42/147.62	   check_d(redex_d(x0))
354.42/147.62	   reduce(u(x0))
354.42/147.62	   in_a_1(go_up(x0))
354.42/147.62	   in_b_1(go_up(x0))
354.42/147.62	   in_u_1(go_up(x0))
354.42/147.62	   in_d_1(go_up(x0))
354.42/147.62	
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(8) UsableRulesProof (EQUIVALENT)
354.42/147.62	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.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(9)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   CHECK_A(redex_a(x_1)) -> REDUCE(x_1)
354.42/147.62	   REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1))
354.42/147.62	   CHECK_B(redex_b(x_1)) -> REDUCE(x_1)
354.42/147.62	   REDUCE(a(x_1)) -> CHECK_A(redex_a(x_1))
354.42/147.62	   REDUCE(d(x_1)) -> CHECK_D(redex_d(x_1))
354.42/147.62	   CHECK_D(redex_d(x_1)) -> REDUCE(x_1)
354.42/147.62	   REDUCE(u(x_1)) -> REDUCE(x_1)
354.42/147.62	
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   redex_d(a(x)) -> result_d(a(d(x)))
354.42/147.62	   redex_d(b(x)) -> result_d(u(a(b(x))))
354.42/147.62	   redex_a(a(x)) -> result_a(c)
354.42/147.62	   redex_a(u(x)) -> result_a(u(a(x)))
354.42/147.62	   redex_b(u(x)) -> result_b(b(d(x)))
354.42/147.62	
354.42/147.62	The set Q consists of the following terms:
354.42/147.62	
354.42/147.62	   top(go_up(x0))
354.42/147.62	   reduce(a(x0))
354.42/147.62	   reduce(b(x0))
354.42/147.62	   reduce(d(x0))
354.42/147.62	   redex_a(a(x0))
354.42/147.62	   redex_b(u(x0))
354.42/147.62	   redex_d(a(x0))
354.42/147.62	   redex_d(b(x0))
354.42/147.62	   redex_a(u(x0))
354.42/147.62	   check_a(result_a(x0))
354.42/147.62	   check_b(result_b(x0))
354.42/147.62	   check_d(result_d(x0))
354.42/147.62	   check_a(redex_a(x0))
354.42/147.62	   check_b(redex_b(x0))
354.42/147.62	   check_d(redex_d(x0))
354.42/147.62	   reduce(u(x0))
354.42/147.62	   in_a_1(go_up(x0))
354.42/147.62	   in_b_1(go_up(x0))
354.42/147.62	   in_u_1(go_up(x0))
354.42/147.62	   in_d_1(go_up(x0))
354.42/147.62	
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(10) QReductionProof (EQUIVALENT)
354.42/147.62	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
354.42/147.62	
354.42/147.62	   top(go_up(x0))
354.42/147.62	   reduce(a(x0))
354.42/147.62	   reduce(b(x0))
354.42/147.62	   reduce(d(x0))
354.42/147.62	   check_a(result_a(x0))
354.42/147.62	   check_b(result_b(x0))
354.42/147.62	   check_d(result_d(x0))
354.42/147.62	   check_a(redex_a(x0))
354.42/147.62	   check_b(redex_b(x0))
354.42/147.62	   check_d(redex_d(x0))
354.42/147.62	   reduce(u(x0))
354.42/147.62	   in_a_1(go_up(x0))
354.42/147.62	   in_b_1(go_up(x0))
354.42/147.62	   in_u_1(go_up(x0))
354.42/147.62	   in_d_1(go_up(x0))
354.42/147.62	
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(11)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   CHECK_A(redex_a(x_1)) -> REDUCE(x_1)
354.42/147.62	   REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1))
354.42/147.62	   CHECK_B(redex_b(x_1)) -> REDUCE(x_1)
354.42/147.62	   REDUCE(a(x_1)) -> CHECK_A(redex_a(x_1))
354.42/147.62	   REDUCE(d(x_1)) -> CHECK_D(redex_d(x_1))
354.42/147.62	   CHECK_D(redex_d(x_1)) -> REDUCE(x_1)
354.42/147.62	   REDUCE(u(x_1)) -> REDUCE(x_1)
354.42/147.62	
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   redex_d(a(x)) -> result_d(a(d(x)))
354.42/147.62	   redex_d(b(x)) -> result_d(u(a(b(x))))
354.42/147.62	   redex_a(a(x)) -> result_a(c)
354.42/147.62	   redex_a(u(x)) -> result_a(u(a(x)))
354.42/147.62	   redex_b(u(x)) -> result_b(b(d(x)))
354.42/147.62	
354.42/147.62	The set Q consists of the following terms:
354.42/147.62	
354.42/147.62	   redex_a(a(x0))
354.42/147.62	   redex_b(u(x0))
354.42/147.62	   redex_d(a(x0))
354.42/147.62	   redex_d(b(x0))
354.42/147.62	   redex_a(u(x0))
354.42/147.62	
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(12) MRRProof (EQUIVALENT)
354.42/147.62	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
354.42/147.62	
354.42/147.62	Strictly oriented dependency pairs:
354.42/147.62	
354.42/147.62	   REDUCE(d(x_1)) -> CHECK_D(redex_d(x_1))
354.42/147.62	   CHECK_D(redex_d(x_1)) -> REDUCE(x_1)
354.42/147.62	   REDUCE(u(x_1)) -> REDUCE(x_1)
354.42/147.62	
354.42/147.62	Strictly oriented rules of the TRS R:
354.42/147.62	
354.42/147.62	   redex_a(a(x)) -> result_a(c)
354.42/147.62	   redex_a(u(x)) -> result_a(u(a(x)))
354.42/147.62	
354.42/147.62	Used ordering: Polynomial interpretation [POLO]:
354.42/147.62	
354.42/147.62	   POL(CHECK_A(x_1)) = x_1
354.42/147.62	   POL(CHECK_B(x_1)) = 1 + 2*x_1
354.42/147.62	   POL(CHECK_D(x_1)) = x_1
354.42/147.62	   POL(REDUCE(x_1)) = 1 + 2*x_1
354.42/147.62	   POL(a(x_1)) = x_1
354.42/147.62	   POL(b(x_1)) = x_1
354.42/147.62	   POL(c) = 0
354.42/147.62	   POL(d(x_1)) = 2 + x_1
354.42/147.62	   POL(redex_a(x_1)) = 1 + 2*x_1
354.42/147.62	   POL(redex_b(x_1)) = x_1
354.42/147.62	   POL(redex_d(x_1)) = 2 + 2*x_1
354.42/147.62	   POL(result_a(x_1)) = 2*x_1
354.42/147.62	   POL(result_b(x_1)) = x_1
354.42/147.62	   POL(result_d(x_1)) = x_1
354.42/147.62	   POL(u(x_1)) = 2 + 2*x_1
354.42/147.62	
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(13)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   CHECK_A(redex_a(x_1)) -> REDUCE(x_1)
354.42/147.62	   REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1))
354.42/147.62	   CHECK_B(redex_b(x_1)) -> REDUCE(x_1)
354.42/147.62	   REDUCE(a(x_1)) -> CHECK_A(redex_a(x_1))
354.42/147.62	
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   redex_d(a(x)) -> result_d(a(d(x)))
354.42/147.62	   redex_d(b(x)) -> result_d(u(a(b(x))))
354.42/147.62	   redex_b(u(x)) -> result_b(b(d(x)))
354.42/147.62	
354.42/147.62	The set Q consists of the following terms:
354.42/147.62	
354.42/147.62	   redex_a(a(x0))
354.42/147.62	   redex_b(u(x0))
354.42/147.62	   redex_d(a(x0))
354.42/147.62	   redex_d(b(x0))
354.42/147.62	   redex_a(u(x0))
354.42/147.62	
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(14) UsableRulesProof (EQUIVALENT)
354.42/147.62	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.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(15)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   CHECK_A(redex_a(x_1)) -> REDUCE(x_1)
354.42/147.62	   REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1))
354.42/147.62	   CHECK_B(redex_b(x_1)) -> REDUCE(x_1)
354.42/147.62	   REDUCE(a(x_1)) -> CHECK_A(redex_a(x_1))
354.42/147.62	
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   redex_b(u(x)) -> result_b(b(d(x)))
354.42/147.62	
354.42/147.62	The set Q consists of the following terms:
354.42/147.62	
354.42/147.62	   redex_a(a(x0))
354.42/147.62	   redex_b(u(x0))
354.42/147.62	   redex_d(a(x0))
354.42/147.62	   redex_d(b(x0))
354.42/147.62	   redex_a(u(x0))
354.42/147.62	
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(16) QReductionProof (EQUIVALENT)
354.42/147.62	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
354.42/147.62	
354.42/147.62	   redex_d(a(x0))
354.42/147.62	   redex_d(b(x0))
354.42/147.62	
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(17)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   CHECK_A(redex_a(x_1)) -> REDUCE(x_1)
354.42/147.62	   REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1))
354.42/147.62	   CHECK_B(redex_b(x_1)) -> REDUCE(x_1)
354.42/147.62	   REDUCE(a(x_1)) -> CHECK_A(redex_a(x_1))
354.42/147.62	
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   redex_b(u(x)) -> result_b(b(d(x)))
354.42/147.62	
354.42/147.62	The set Q consists of the following terms:
354.42/147.62	
354.42/147.62	   redex_a(a(x0))
354.42/147.62	   redex_b(u(x0))
354.42/147.62	   redex_a(u(x0))
354.42/147.62	
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(18) UsableRulesReductionPairsProof (EQUIVALENT)
354.42/147.62	By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
354.42/147.62	
354.42/147.62	The following dependency pairs can be deleted:
354.42/147.62	
354.42/147.62	   REDUCE(a(x_1)) -> CHECK_A(redex_a(x_1))
354.42/147.62	The following rules are removed from R:
354.42/147.62	
354.42/147.62	   redex_b(u(x)) -> result_b(b(d(x)))
354.42/147.62	Used ordering: POLO with Polynomial interpretation [POLO]:
354.42/147.62	
354.42/147.62	   POL(CHECK_A(x_1)) = 2*x_1
354.42/147.62	   POL(CHECK_B(x_1)) = x_1
354.42/147.62	   POL(REDUCE(x_1)) = 2*x_1
354.42/147.62	   POL(a(x_1)) = 2*x_1
354.42/147.62	   POL(b(x_1)) = 2*x_1
354.42/147.62	   POL(d(x_1)) = 2*x_1
354.42/147.62	   POL(redex_a(x_1)) = x_1
354.42/147.62	   POL(redex_b(x_1)) = 2*x_1
354.42/147.62	   POL(result_b(x_1)) = x_1
354.42/147.62	   POL(u(x_1)) = 2*x_1
354.42/147.62	
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(19)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   CHECK_A(redex_a(x_1)) -> REDUCE(x_1)
354.42/147.62	   REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1))
354.42/147.62	   CHECK_B(redex_b(x_1)) -> REDUCE(x_1)
354.42/147.62	
354.42/147.62	R is empty.
354.42/147.62	The set Q consists of the following terms:
354.42/147.62	
354.42/147.62	   redex_a(a(x0))
354.42/147.62	   redex_b(u(x0))
354.42/147.62	   redex_a(u(x0))
354.42/147.62	
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(20) DependencyGraphProof (EQUIVALENT)
354.42/147.62	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(21)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   CHECK_B(redex_b(x_1)) -> REDUCE(x_1)
354.42/147.62	   REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1))
354.42/147.62	
354.42/147.62	R is empty.
354.42/147.62	The set Q consists of the following terms:
354.42/147.62	
354.42/147.62	   redex_a(a(x0))
354.42/147.62	   redex_b(u(x0))
354.42/147.62	   redex_a(u(x0))
354.42/147.62	
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(22) QReductionProof (EQUIVALENT)
354.42/147.62	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
354.42/147.62	
354.42/147.62	   redex_a(a(x0))
354.42/147.62	   redex_a(u(x0))
354.42/147.62	
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(23)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   CHECK_B(redex_b(x_1)) -> REDUCE(x_1)
354.42/147.62	   REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1))
354.42/147.62	
354.42/147.62	R is empty.
354.42/147.62	The set Q consists of the following terms:
354.42/147.62	
354.42/147.62	   redex_b(u(x0))
354.42/147.62	
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(24) UsableRulesReductionPairsProof (EQUIVALENT)
354.42/147.62	By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
354.42/147.62	
354.42/147.62	The following dependency pairs can be deleted:
354.42/147.62	
354.42/147.62	   REDUCE(b(x_1)) -> CHECK_B(redex_b(x_1))
354.42/147.62	No rules are removed from R.
354.42/147.62	
354.42/147.62	Used ordering: POLO with Polynomial interpretation [POLO]:
354.42/147.62	
354.42/147.62	   POL(CHECK_B(x_1)) = 2*x_1
354.42/147.62	   POL(REDUCE(x_1)) = 2*x_1
354.42/147.62	   POL(b(x_1)) = 2*x_1
354.42/147.62	   POL(redex_b(x_1)) = x_1
354.42/147.62	
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(25)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   CHECK_B(redex_b(x_1)) -> REDUCE(x_1)
354.42/147.62	
354.42/147.62	R is empty.
354.42/147.62	The set Q consists of the following terms:
354.42/147.62	
354.42/147.62	   redex_b(u(x0))
354.42/147.62	
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(26) DependencyGraphProof (EQUIVALENT)
354.42/147.62	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(27)
354.42/147.62	TRUE
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(28)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   TOP(go_up(x)) -> TOP(reduce(x))
354.42/147.62	
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   top(go_up(x)) -> top(reduce(x))
354.42/147.62	   reduce(a(x_1)) -> check_a(redex_a(x_1))
354.42/147.62	   reduce(b(x_1)) -> check_b(redex_b(x_1))
354.42/147.62	   reduce(d(x_1)) -> check_d(redex_d(x_1))
354.42/147.62	   redex_a(a(x)) -> result_a(c)
354.42/147.62	   redex_b(u(x)) -> result_b(b(d(x)))
354.42/147.62	   redex_d(a(x)) -> result_d(a(d(x)))
354.42/147.62	   redex_d(b(x)) -> result_d(u(a(b(x))))
354.42/147.62	   redex_a(u(x)) -> result_a(u(a(x)))
354.42/147.62	   check_a(result_a(x)) -> go_up(x)
354.42/147.62	   check_b(result_b(x)) -> go_up(x)
354.42/147.62	   check_d(result_d(x)) -> go_up(x)
354.42/147.62	   check_a(redex_a(x_1)) -> in_a_1(reduce(x_1))
354.42/147.62	   check_b(redex_b(x_1)) -> in_b_1(reduce(x_1))
354.42/147.62	   check_d(redex_d(x_1)) -> in_d_1(reduce(x_1))
354.42/147.62	   reduce(u(x_1)) -> in_u_1(reduce(x_1))
354.42/147.62	   in_a_1(go_up(x_1)) -> go_up(a(x_1))
354.42/147.62	   in_b_1(go_up(x_1)) -> go_up(b(x_1))
354.42/147.62	   in_u_1(go_up(x_1)) -> go_up(u(x_1))
354.42/147.62	   in_d_1(go_up(x_1)) -> go_up(d(x_1))
354.42/147.62	
354.42/147.62	The set Q consists of the following terms:
354.42/147.62	
354.42/147.62	   top(go_up(x0))
354.42/147.62	   reduce(a(x0))
354.42/147.62	   reduce(b(x0))
354.42/147.62	   reduce(d(x0))
354.42/147.62	   redex_a(a(x0))
354.42/147.62	   redex_b(u(x0))
354.42/147.62	   redex_d(a(x0))
354.42/147.62	   redex_d(b(x0))
354.42/147.62	   redex_a(u(x0))
354.42/147.62	   check_a(result_a(x0))
354.42/147.62	   check_b(result_b(x0))
354.42/147.62	   check_d(result_d(x0))
354.42/147.62	   check_a(redex_a(x0))
354.42/147.62	   check_b(redex_b(x0))
354.42/147.62	   check_d(redex_d(x0))
354.42/147.62	   reduce(u(x0))
354.42/147.62	   in_a_1(go_up(x0))
354.42/147.62	   in_b_1(go_up(x0))
354.42/147.62	   in_u_1(go_up(x0))
354.42/147.62	   in_d_1(go_up(x0))
354.42/147.62	
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(29) UsableRulesProof (EQUIVALENT)
354.42/147.62	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.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(30)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   TOP(go_up(x)) -> TOP(reduce(x))
354.42/147.62	
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   reduce(a(x_1)) -> check_a(redex_a(x_1))
354.42/147.62	   reduce(b(x_1)) -> check_b(redex_b(x_1))
354.42/147.62	   reduce(d(x_1)) -> check_d(redex_d(x_1))
354.42/147.62	   reduce(u(x_1)) -> in_u_1(reduce(x_1))
354.42/147.62	   in_u_1(go_up(x_1)) -> go_up(u(x_1))
354.42/147.62	   redex_d(a(x)) -> result_d(a(d(x)))
354.42/147.62	   redex_d(b(x)) -> result_d(u(a(b(x))))
354.42/147.62	   check_d(result_d(x)) -> go_up(x)
354.42/147.62	   check_d(redex_d(x_1)) -> in_d_1(reduce(x_1))
354.42/147.62	   in_d_1(go_up(x_1)) -> go_up(d(x_1))
354.42/147.62	   redex_b(u(x)) -> result_b(b(d(x)))
354.42/147.62	   check_b(result_b(x)) -> go_up(x)
354.42/147.62	   check_b(redex_b(x_1)) -> in_b_1(reduce(x_1))
354.42/147.62	   in_b_1(go_up(x_1)) -> go_up(b(x_1))
354.42/147.62	   redex_a(a(x)) -> result_a(c)
354.42/147.62	   redex_a(u(x)) -> result_a(u(a(x)))
354.42/147.62	   check_a(result_a(x)) -> go_up(x)
354.42/147.62	   check_a(redex_a(x_1)) -> in_a_1(reduce(x_1))
354.42/147.62	   in_a_1(go_up(x_1)) -> go_up(a(x_1))
354.42/147.62	
354.42/147.62	The set Q consists of the following terms:
354.42/147.62	
354.42/147.62	   top(go_up(x0))
354.42/147.62	   reduce(a(x0))
354.42/147.62	   reduce(b(x0))
354.42/147.62	   reduce(d(x0))
354.42/147.62	   redex_a(a(x0))
354.42/147.62	   redex_b(u(x0))
354.42/147.62	   redex_d(a(x0))
354.42/147.62	   redex_d(b(x0))
354.42/147.62	   redex_a(u(x0))
354.42/147.62	   check_a(result_a(x0))
354.42/147.62	   check_b(result_b(x0))
354.42/147.62	   check_d(result_d(x0))
354.42/147.62	   check_a(redex_a(x0))
354.42/147.62	   check_b(redex_b(x0))
354.42/147.62	   check_d(redex_d(x0))
354.42/147.62	   reduce(u(x0))
354.42/147.62	   in_a_1(go_up(x0))
354.42/147.62	   in_b_1(go_up(x0))
354.42/147.62	   in_u_1(go_up(x0))
354.42/147.62	   in_d_1(go_up(x0))
354.42/147.62	
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(31) QReductionProof (EQUIVALENT)
354.42/147.62	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
354.42/147.62	
354.42/147.62	   top(go_up(x0))
354.42/147.62	
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(32)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   TOP(go_up(x)) -> TOP(reduce(x))
354.42/147.62	
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   reduce(a(x_1)) -> check_a(redex_a(x_1))
354.42/147.62	   reduce(b(x_1)) -> check_b(redex_b(x_1))
354.42/147.62	   reduce(d(x_1)) -> check_d(redex_d(x_1))
354.42/147.62	   reduce(u(x_1)) -> in_u_1(reduce(x_1))
354.42/147.62	   in_u_1(go_up(x_1)) -> go_up(u(x_1))
354.42/147.62	   redex_d(a(x)) -> result_d(a(d(x)))
354.42/147.62	   redex_d(b(x)) -> result_d(u(a(b(x))))
354.42/147.62	   check_d(result_d(x)) -> go_up(x)
354.42/147.62	   check_d(redex_d(x_1)) -> in_d_1(reduce(x_1))
354.42/147.62	   in_d_1(go_up(x_1)) -> go_up(d(x_1))
354.42/147.62	   redex_b(u(x)) -> result_b(b(d(x)))
354.42/147.62	   check_b(result_b(x)) -> go_up(x)
354.42/147.62	   check_b(redex_b(x_1)) -> in_b_1(reduce(x_1))
354.42/147.62	   in_b_1(go_up(x_1)) -> go_up(b(x_1))
354.42/147.62	   redex_a(a(x)) -> result_a(c)
354.42/147.62	   redex_a(u(x)) -> result_a(u(a(x)))
354.42/147.62	   check_a(result_a(x)) -> go_up(x)
354.42/147.62	   check_a(redex_a(x_1)) -> in_a_1(reduce(x_1))
354.42/147.62	   in_a_1(go_up(x_1)) -> go_up(a(x_1))
354.42/147.62	
354.42/147.62	The set Q consists of the following terms:
354.42/147.62	
354.42/147.62	   reduce(a(x0))
354.42/147.62	   reduce(b(x0))
354.42/147.62	   reduce(d(x0))
354.42/147.62	   redex_a(a(x0))
354.42/147.62	   redex_b(u(x0))
354.42/147.62	   redex_d(a(x0))
354.42/147.62	   redex_d(b(x0))
354.42/147.62	   redex_a(u(x0))
354.42/147.62	   check_a(result_a(x0))
354.42/147.62	   check_b(result_b(x0))
354.42/147.62	   check_d(result_d(x0))
354.42/147.62	   check_a(redex_a(x0))
354.42/147.62	   check_b(redex_b(x0))
354.42/147.62	   check_d(redex_d(x0))
354.42/147.62	   reduce(u(x0))
354.42/147.62	   in_a_1(go_up(x0))
354.42/147.62	   in_b_1(go_up(x0))
354.42/147.62	   in_u_1(go_up(x0))
354.42/147.62	   in_d_1(go_up(x0))
354.42/147.62	
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(33) TransformationProof (EQUIVALENT)
354.42/147.62	By narrowing [LPAR04] the rule TOP(go_up(x)) -> TOP(reduce(x)) at position [0] we obtained the following new rules [LPAR04]:
354.42/147.62	
354.42/147.62	   (TOP(go_up(a(x0))) -> TOP(check_a(redex_a(x0))),TOP(go_up(a(x0))) -> TOP(check_a(redex_a(x0))))
354.42/147.62	   (TOP(go_up(b(x0))) -> TOP(check_b(redex_b(x0))),TOP(go_up(b(x0))) -> TOP(check_b(redex_b(x0))))
354.42/147.62	   (TOP(go_up(d(x0))) -> TOP(check_d(redex_d(x0))),TOP(go_up(d(x0))) -> TOP(check_d(redex_d(x0))))
354.42/147.62	   (TOP(go_up(u(x0))) -> TOP(in_u_1(reduce(x0))),TOP(go_up(u(x0))) -> TOP(in_u_1(reduce(x0))))
354.42/147.62	
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(34)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   TOP(go_up(a(x0))) -> TOP(check_a(redex_a(x0)))
354.42/147.62	   TOP(go_up(b(x0))) -> TOP(check_b(redex_b(x0)))
354.42/147.62	   TOP(go_up(d(x0))) -> TOP(check_d(redex_d(x0)))
354.42/147.62	   TOP(go_up(u(x0))) -> TOP(in_u_1(reduce(x0)))
354.42/147.62	
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   reduce(a(x_1)) -> check_a(redex_a(x_1))
354.42/147.62	   reduce(b(x_1)) -> check_b(redex_b(x_1))
354.42/147.62	   reduce(d(x_1)) -> check_d(redex_d(x_1))
354.42/147.62	   reduce(u(x_1)) -> in_u_1(reduce(x_1))
354.42/147.62	   in_u_1(go_up(x_1)) -> go_up(u(x_1))
354.42/147.62	   redex_d(a(x)) -> result_d(a(d(x)))
354.42/147.62	   redex_d(b(x)) -> result_d(u(a(b(x))))
354.42/147.62	   check_d(result_d(x)) -> go_up(x)
354.42/147.62	   check_d(redex_d(x_1)) -> in_d_1(reduce(x_1))
354.42/147.62	   in_d_1(go_up(x_1)) -> go_up(d(x_1))
354.42/147.62	   redex_b(u(x)) -> result_b(b(d(x)))
354.42/147.62	   check_b(result_b(x)) -> go_up(x)
354.42/147.62	   check_b(redex_b(x_1)) -> in_b_1(reduce(x_1))
354.42/147.62	   in_b_1(go_up(x_1)) -> go_up(b(x_1))
354.42/147.62	   redex_a(a(x)) -> result_a(c)
354.42/147.62	   redex_a(u(x)) -> result_a(u(a(x)))
354.42/147.62	   check_a(result_a(x)) -> go_up(x)
354.42/147.62	   check_a(redex_a(x_1)) -> in_a_1(reduce(x_1))
354.42/147.62	   in_a_1(go_up(x_1)) -> go_up(a(x_1))
354.42/147.62	
354.42/147.62	The set Q consists of the following terms:
354.42/147.62	
354.42/147.62	   reduce(a(x0))
354.42/147.62	   reduce(b(x0))
354.42/147.62	   reduce(d(x0))
354.42/147.62	   redex_a(a(x0))
354.42/147.62	   redex_b(u(x0))
354.42/147.62	   redex_d(a(x0))
354.42/147.62	   redex_d(b(x0))
354.42/147.62	   redex_a(u(x0))
354.42/147.62	   check_a(result_a(x0))
354.42/147.62	   check_b(result_b(x0))
354.42/147.62	   check_d(result_d(x0))
354.42/147.62	   check_a(redex_a(x0))
354.42/147.62	   check_b(redex_b(x0))
354.42/147.62	   check_d(redex_d(x0))
354.42/147.62	   reduce(u(x0))
354.42/147.62	   in_a_1(go_up(x0))
354.42/147.62	   in_b_1(go_up(x0))
354.42/147.62	   in_u_1(go_up(x0))
354.42/147.62	   in_d_1(go_up(x0))
354.42/147.62	
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(35) UsableRulesProof (EQUIVALENT)
354.42/147.62	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.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(36)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   TOP(go_up(x)) -> TOP(reduce(x))
354.42/147.62	
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   reduce(a(x_1)) -> check_a(redex_a(x_1))
354.42/147.62	   reduce(b(x_1)) -> check_b(redex_b(x_1))
354.42/147.62	   reduce(d(x_1)) -> check_d(redex_d(x_1))
354.42/147.62	   reduce(u(x_1)) -> in_u_1(reduce(x_1))
354.42/147.62	   in_u_1(go_up(x_1)) -> go_up(u(x_1))
354.42/147.62	   redex_d(a(x)) -> result_d(a(d(x)))
354.42/147.62	   redex_d(b(x)) -> result_d(u(a(b(x))))
354.42/147.62	   check_d(result_d(x)) -> go_up(x)
354.42/147.62	   check_d(redex_d(x_1)) -> in_d_1(reduce(x_1))
354.42/147.62	   in_d_1(go_up(x_1)) -> go_up(d(x_1))
354.42/147.62	   redex_b(u(x)) -> result_b(b(d(x)))
354.42/147.62	   check_b(result_b(x)) -> go_up(x)
354.42/147.62	   check_b(redex_b(x_1)) -> in_b_1(reduce(x_1))
354.42/147.62	   in_b_1(go_up(x_1)) -> go_up(b(x_1))
354.42/147.62	   redex_a(a(x)) -> result_a(c)
354.42/147.62	   redex_a(u(x)) -> result_a(u(a(x)))
354.42/147.62	   check_a(result_a(x)) -> go_up(x)
354.42/147.62	   check_a(redex_a(x_1)) -> in_a_1(reduce(x_1))
354.42/147.62	   in_a_1(go_up(x_1)) -> go_up(a(x_1))
354.42/147.62	
354.42/147.62	The set Q consists of the following terms:
354.42/147.62	
354.42/147.62	   top(go_up(x0))
354.42/147.62	   reduce(a(x0))
354.42/147.62	   reduce(b(x0))
354.42/147.62	   reduce(d(x0))
354.42/147.62	   redex_a(a(x0))
354.42/147.62	   redex_b(u(x0))
354.42/147.62	   redex_d(a(x0))
354.42/147.62	   redex_d(b(x0))
354.42/147.62	   redex_a(u(x0))
354.42/147.62	   check_a(result_a(x0))
354.42/147.62	   check_b(result_b(x0))
354.42/147.62	   check_d(result_d(x0))
354.42/147.62	   check_a(redex_a(x0))
354.42/147.62	   check_b(redex_b(x0))
354.42/147.62	   check_d(redex_d(x0))
354.42/147.62	   reduce(u(x0))
354.42/147.62	   in_a_1(go_up(x0))
354.42/147.62	   in_b_1(go_up(x0))
354.42/147.62	   in_u_1(go_up(x0))
354.42/147.62	   in_d_1(go_up(x0))
354.42/147.62	
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(37) QReductionProof (EQUIVALENT)
354.42/147.62	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
354.42/147.62	
354.42/147.62	   top(go_up(x0))
354.42/147.62	
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(38)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   TOP(go_up(x)) -> TOP(reduce(x))
354.42/147.62	
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   reduce(a(x_1)) -> check_a(redex_a(x_1))
354.42/147.62	   reduce(b(x_1)) -> check_b(redex_b(x_1))
354.42/147.62	   reduce(d(x_1)) -> check_d(redex_d(x_1))
354.42/147.62	   reduce(u(x_1)) -> in_u_1(reduce(x_1))
354.42/147.62	   in_u_1(go_up(x_1)) -> go_up(u(x_1))
354.42/147.62	   redex_d(a(x)) -> result_d(a(d(x)))
354.42/147.62	   redex_d(b(x)) -> result_d(u(a(b(x))))
354.42/147.62	   check_d(result_d(x)) -> go_up(x)
354.42/147.62	   check_d(redex_d(x_1)) -> in_d_1(reduce(x_1))
354.42/147.62	   in_d_1(go_up(x_1)) -> go_up(d(x_1))
354.42/147.62	   redex_b(u(x)) -> result_b(b(d(x)))
354.42/147.62	   check_b(result_b(x)) -> go_up(x)
354.42/147.62	   check_b(redex_b(x_1)) -> in_b_1(reduce(x_1))
354.42/147.62	   in_b_1(go_up(x_1)) -> go_up(b(x_1))
354.42/147.62	   redex_a(a(x)) -> result_a(c)
354.42/147.62	   redex_a(u(x)) -> result_a(u(a(x)))
354.42/147.62	   check_a(result_a(x)) -> go_up(x)
354.42/147.62	   check_a(redex_a(x_1)) -> in_a_1(reduce(x_1))
354.42/147.62	   in_a_1(go_up(x_1)) -> go_up(a(x_1))
354.42/147.62	
354.42/147.62	The set Q consists of the following terms:
354.42/147.62	
354.42/147.62	   reduce(a(x0))
354.42/147.62	   reduce(b(x0))
354.42/147.62	   reduce(d(x0))
354.42/147.62	   redex_a(a(x0))
354.42/147.62	   redex_b(u(x0))
354.42/147.62	   redex_d(a(x0))
354.42/147.62	   redex_d(b(x0))
354.42/147.62	   redex_a(u(x0))
354.42/147.62	   check_a(result_a(x0))
354.42/147.62	   check_b(result_b(x0))
354.42/147.62	   check_d(result_d(x0))
354.42/147.62	   check_a(redex_a(x0))
354.42/147.62	   check_b(redex_b(x0))
354.42/147.62	   check_d(redex_d(x0))
354.42/147.62	   reduce(u(x0))
354.42/147.62	   in_a_1(go_up(x0))
354.42/147.62	   in_b_1(go_up(x0))
354.42/147.62	   in_u_1(go_up(x0))
354.42/147.62	   in_d_1(go_up(x0))
354.42/147.62	
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(39) Trivial-Transformation (SOUND)
354.42/147.62	We applied the Trivial transformation  to transform the outermost TRS to a standard TRS.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(40)
354.42/147.62	Obligation:
354.42/147.62	Q restricted rewrite system:
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   a(a(x)) -> c
354.42/147.62	   b(u(x)) -> b(d(x))
354.42/147.62	   d(a(x)) -> a(d(x))
354.42/147.62	   d(b(x)) -> u(a(b(x)))
354.42/147.62	   a(u(x)) -> u(a(x))
354.42/147.62	
354.42/147.62	Q is empty.
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(41) DependencyPairsProof (EQUIVALENT)
354.42/147.62	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(42)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   B(u(x)) -> B(d(x))
354.42/147.62	   B(u(x)) -> D(x)
354.42/147.62	   D(a(x)) -> A(d(x))
354.42/147.62	   D(a(x)) -> D(x)
354.42/147.62	   D(b(x)) -> A(b(x))
354.42/147.62	   A(u(x)) -> A(x)
354.42/147.62	
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   a(a(x)) -> c
354.42/147.62	   b(u(x)) -> b(d(x))
354.42/147.62	   d(a(x)) -> a(d(x))
354.42/147.62	   d(b(x)) -> u(a(b(x)))
354.42/147.62	   a(u(x)) -> u(a(x))
354.42/147.62	
354.42/147.62	Q is empty.
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(43) DependencyGraphProof (EQUIVALENT)
354.42/147.62	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 3 less nodes.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(44)
354.42/147.62	Complex Obligation (AND)
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(45)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   A(u(x)) -> A(x)
354.42/147.62	
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   a(a(x)) -> c
354.42/147.62	   b(u(x)) -> b(d(x))
354.42/147.62	   d(a(x)) -> a(d(x))
354.42/147.62	   d(b(x)) -> u(a(b(x)))
354.42/147.62	   a(u(x)) -> u(a(x))
354.42/147.62	
354.42/147.62	Q is empty.
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(46) UsableRulesProof (EQUIVALENT)
354.42/147.62	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(47)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   A(u(x)) -> A(x)
354.42/147.62	
354.42/147.62	R is empty.
354.42/147.62	Q is empty.
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(48) QDPSizeChangeProof (EQUIVALENT)
354.42/147.62	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. 
354.42/147.62	
354.42/147.62	From the DPs we obtained the following set of size-change graphs:
354.42/147.62	*A(u(x)) -> A(x)
354.42/147.62	The graph contains the following edges 1 > 1
354.42/147.62	
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(49)
354.42/147.62	YES
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(50)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   D(a(x)) -> D(x)
354.42/147.62	
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   a(a(x)) -> c
354.42/147.62	   b(u(x)) -> b(d(x))
354.42/147.62	   d(a(x)) -> a(d(x))
354.42/147.62	   d(b(x)) -> u(a(b(x)))
354.42/147.62	   a(u(x)) -> u(a(x))
354.42/147.62	
354.42/147.62	Q is empty.
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(51) UsableRulesProof (EQUIVALENT)
354.42/147.62	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(52)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   D(a(x)) -> D(x)
354.42/147.62	
354.42/147.62	R is empty.
354.42/147.62	Q is empty.
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(53) QDPSizeChangeProof (EQUIVALENT)
354.42/147.62	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. 
354.42/147.62	
354.42/147.62	From the DPs we obtained the following set of size-change graphs:
354.42/147.62	*D(a(x)) -> D(x)
354.42/147.62	The graph contains the following edges 1 > 1
354.42/147.62	
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(54)
354.42/147.62	YES
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(55)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   B(u(x)) -> B(d(x))
354.42/147.62	
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   a(a(x)) -> c
354.42/147.62	   b(u(x)) -> b(d(x))
354.42/147.62	   d(a(x)) -> a(d(x))
354.42/147.62	   d(b(x)) -> u(a(b(x)))
354.42/147.62	   a(u(x)) -> u(a(x))
354.42/147.62	
354.42/147.62	Q is empty.
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(56) TransformationProof (EQUIVALENT)
354.42/147.62	By narrowing [LPAR04] the rule B(u(x)) -> B(d(x)) at position [0] we obtained the following new rules [LPAR04]:
354.42/147.62	
354.42/147.62	   (B(u(a(x0))) -> B(a(d(x0))),B(u(a(x0))) -> B(a(d(x0))))
354.42/147.62	   (B(u(b(x0))) -> B(u(a(b(x0)))),B(u(b(x0))) -> B(u(a(b(x0)))))
354.42/147.62	
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(57)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   B(u(a(x0))) -> B(a(d(x0)))
354.42/147.62	   B(u(b(x0))) -> B(u(a(b(x0))))
354.42/147.62	
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   a(a(x)) -> c
354.42/147.62	   b(u(x)) -> b(d(x))
354.42/147.62	   d(a(x)) -> a(d(x))
354.42/147.62	   d(b(x)) -> u(a(b(x)))
354.42/147.62	   a(u(x)) -> u(a(x))
354.42/147.62	
354.42/147.62	Q is empty.
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(58) QDPOrderProof (EQUIVALENT)
354.42/147.62	We use the reduction pair processor [LPAR04,JAR06].
354.42/147.62	
354.42/147.62	
354.42/147.62	The following pairs can be oriented strictly and are deleted.
354.42/147.62	
354.42/147.62	   B(u(b(x0))) -> B(u(a(b(x0))))
354.42/147.62	The remaining pairs can at least be oriented weakly.
354.42/147.62	Used ordering:  Polynomial interpretation [POLO]:
354.42/147.62	
354.42/147.62	   POL(B(x_1)) = x_1
354.42/147.62	   POL(a(x_1)) = 0
354.42/147.62	   POL(b(x_1)) = 1 + x_1
354.42/147.62	   POL(c) = 0
354.42/147.62	   POL(d(x_1)) = x_1
354.42/147.62	   POL(u(x_1)) = x_1
354.42/147.62	
354.42/147.62	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
354.42/147.62	
354.42/147.62	   a(a(x)) -> c
354.42/147.62	   a(u(x)) -> u(a(x))
354.42/147.62	
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(59)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   B(u(a(x0))) -> B(a(d(x0)))
354.42/147.62	
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   a(a(x)) -> c
354.42/147.62	   b(u(x)) -> b(d(x))
354.42/147.62	   d(a(x)) -> a(d(x))
354.42/147.62	   d(b(x)) -> u(a(b(x)))
354.42/147.62	   a(u(x)) -> u(a(x))
354.42/147.62	
354.42/147.62	Q is empty.
354.42/147.62	We have to consider all minimal (P,Q,R)-chains.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(60) Raffelsieper-Zantema-Transformation (SOUND)
354.42/147.62	We applied the Raffelsieper-Zantema transformation  to transform the outermost TRS to a standard TRS.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(61)
354.42/147.62	Obligation:
354.42/147.62	Q restricted rewrite system:
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   down(a(a(x))) -> up(c)
354.42/147.62	   down(b(u(x))) -> up(b(d(x)))
354.42/147.62	   down(d(a(x))) -> up(a(d(x)))
354.42/147.62	   down(d(b(x))) -> up(u(a(b(x))))
354.42/147.62	   down(a(u(x))) -> up(u(a(x)))
354.42/147.62	   top(up(x)) -> top(down(x))
354.42/147.62	   down(u(y2)) -> u_flat(down(y2))
354.42/147.62	   down(a(c)) -> a_flat(down(c))
354.42/147.62	   down(a(b(y6))) -> a_flat(down(b(y6)))
354.42/147.62	   down(a(d(y8))) -> a_flat(down(d(y8)))
354.42/147.62	   down(a(fresh_constant)) -> a_flat(down(fresh_constant))
354.42/147.62	   down(b(a(y10))) -> b_flat(down(a(y10)))
354.42/147.62	   down(b(c)) -> b_flat(down(c))
354.42/147.62	   down(b(b(y11))) -> b_flat(down(b(y11)))
354.42/147.62	   down(b(d(y13))) -> b_flat(down(d(y13)))
354.42/147.62	   down(b(fresh_constant)) -> b_flat(down(fresh_constant))
354.42/147.62	   down(d(c)) -> d_flat(down(c))
354.42/147.62	   down(d(u(y17))) -> d_flat(down(u(y17)))
354.42/147.62	   down(d(d(y18))) -> d_flat(down(d(y18)))
354.42/147.62	   down(d(fresh_constant)) -> d_flat(down(fresh_constant))
354.42/147.62	   a_flat(up(x_1)) -> up(a(x_1))
354.42/147.62	   b_flat(up(x_1)) -> up(b(x_1))
354.42/147.62	   u_flat(up(x_1)) -> up(u(x_1))
354.42/147.62	   d_flat(up(x_1)) -> up(d(x_1))
354.42/147.62	
354.42/147.62	Q is empty.
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(62) AAECC Innermost (EQUIVALENT)
354.42/147.62	We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is 
354.42/147.62	   down(u(y2)) -> u_flat(down(y2))
354.42/147.62	   down(a(c)) -> a_flat(down(c))
354.42/147.62	   down(a(b(y6))) -> a_flat(down(b(y6)))
354.42/147.62	   down(a(d(y8))) -> a_flat(down(d(y8)))
354.42/147.62	   down(a(fresh_constant)) -> a_flat(down(fresh_constant))
354.42/147.62	   down(b(a(y10))) -> b_flat(down(a(y10)))
354.42/147.62	   down(b(c)) -> b_flat(down(c))
354.42/147.62	   down(b(b(y11))) -> b_flat(down(b(y11)))
354.42/147.62	   down(b(d(y13))) -> b_flat(down(d(y13)))
354.42/147.62	   down(b(fresh_constant)) -> b_flat(down(fresh_constant))
354.42/147.62	   down(d(c)) -> d_flat(down(c))
354.42/147.62	   down(d(u(y17))) -> d_flat(down(u(y17)))
354.42/147.62	   down(d(d(y18))) -> d_flat(down(d(y18)))
354.42/147.62	   down(d(fresh_constant)) -> d_flat(down(fresh_constant))
354.42/147.62	   a_flat(up(x_1)) -> up(a(x_1))
354.42/147.62	   b_flat(up(x_1)) -> up(b(x_1))
354.42/147.62	   u_flat(up(x_1)) -> up(u(x_1))
354.42/147.62	   d_flat(up(x_1)) -> up(d(x_1))
354.42/147.62	   down(a(a(x))) -> up(c)
354.42/147.62	   down(b(u(x))) -> up(b(d(x)))
354.42/147.62	   down(d(a(x))) -> up(a(d(x)))
354.42/147.62	   down(d(b(x))) -> up(u(a(b(x))))
354.42/147.62	   down(a(u(x))) -> up(u(a(x)))
354.42/147.62	
354.42/147.62	The TRS R 2 is 
354.42/147.62	   top(up(x)) -> top(down(x))
354.42/147.62	
354.42/147.62	The signature Sigma is {top_1}
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(63)
354.42/147.62	Obligation:
354.42/147.62	Q restricted rewrite system:
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   down(a(a(x))) -> up(c)
354.42/147.62	   down(b(u(x))) -> up(b(d(x)))
354.42/147.62	   down(d(a(x))) -> up(a(d(x)))
354.42/147.62	   down(d(b(x))) -> up(u(a(b(x))))
354.42/147.62	   down(a(u(x))) -> up(u(a(x)))
354.42/147.62	   top(up(x)) -> top(down(x))
354.42/147.62	   down(u(y2)) -> u_flat(down(y2))
354.42/147.62	   down(a(c)) -> a_flat(down(c))
354.42/147.62	   down(a(b(y6))) -> a_flat(down(b(y6)))
354.42/147.62	   down(a(d(y8))) -> a_flat(down(d(y8)))
354.42/147.62	   down(a(fresh_constant)) -> a_flat(down(fresh_constant))
354.42/147.62	   down(b(a(y10))) -> b_flat(down(a(y10)))
354.42/147.62	   down(b(c)) -> b_flat(down(c))
354.42/147.62	   down(b(b(y11))) -> b_flat(down(b(y11)))
354.42/147.62	   down(b(d(y13))) -> b_flat(down(d(y13)))
354.42/147.62	   down(b(fresh_constant)) -> b_flat(down(fresh_constant))
354.42/147.62	   down(d(c)) -> d_flat(down(c))
354.42/147.62	   down(d(u(y17))) -> d_flat(down(u(y17)))
354.42/147.62	   down(d(d(y18))) -> d_flat(down(d(y18)))
354.42/147.62	   down(d(fresh_constant)) -> d_flat(down(fresh_constant))
354.42/147.62	   a_flat(up(x_1)) -> up(a(x_1))
354.42/147.62	   b_flat(up(x_1)) -> up(b(x_1))
354.42/147.62	   u_flat(up(x_1)) -> up(u(x_1))
354.42/147.62	   d_flat(up(x_1)) -> up(d(x_1))
354.42/147.62	
354.42/147.62	The set Q consists of the following terms:
354.42/147.62	
354.42/147.62	   down(a(a(x0)))
354.42/147.62	   down(b(u(x0)))
354.42/147.62	   down(d(a(x0)))
354.42/147.62	   down(d(b(x0)))
354.42/147.62	   down(a(u(x0)))
354.42/147.62	   top(up(x0))
354.42/147.62	   down(u(x0))
354.42/147.62	   down(a(c))
354.42/147.62	   down(a(b(x0)))
354.42/147.62	   down(a(d(x0)))
354.42/147.62	   down(a(fresh_constant))
354.42/147.62	   down(b(a(x0)))
354.42/147.62	   down(b(c))
354.42/147.62	   down(b(b(x0)))
354.42/147.62	   down(b(d(x0)))
354.42/147.62	   down(b(fresh_constant))
354.42/147.62	   down(d(c))
354.42/147.62	   down(d(u(x0)))
354.42/147.62	   down(d(d(x0)))
354.42/147.62	   down(d(fresh_constant))
354.42/147.62	   a_flat(up(x0))
354.42/147.62	   b_flat(up(x0))
354.42/147.62	   u_flat(up(x0))
354.42/147.62	   d_flat(up(x0))
354.42/147.62	
354.42/147.62	
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(64) DependencyPairsProof (EQUIVALENT)
354.42/147.62	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
354.42/147.62	----------------------------------------
354.42/147.62	
354.42/147.62	(65)
354.42/147.62	Obligation:
354.42/147.62	Q DP problem:
354.42/147.62	The TRS P consists of the following rules:
354.42/147.62	
354.42/147.62	   TOP(up(x)) -> TOP(down(x))
354.42/147.62	   TOP(up(x)) -> DOWN(x)
354.42/147.62	   DOWN(u(y2)) -> U_FLAT(down(y2))
354.42/147.62	   DOWN(u(y2)) -> DOWN(y2)
354.42/147.62	   DOWN(a(c)) -> A_FLAT(down(c))
354.42/147.62	   DOWN(a(c)) -> DOWN(c)
354.42/147.62	   DOWN(a(b(y6))) -> A_FLAT(down(b(y6)))
354.42/147.62	   DOWN(a(b(y6))) -> DOWN(b(y6))
354.42/147.62	   DOWN(a(d(y8))) -> A_FLAT(down(d(y8)))
354.42/147.62	   DOWN(a(d(y8))) -> DOWN(d(y8))
354.42/147.62	   DOWN(a(fresh_constant)) -> A_FLAT(down(fresh_constant))
354.42/147.62	   DOWN(a(fresh_constant)) -> DOWN(fresh_constant)
354.42/147.62	   DOWN(b(a(y10))) -> B_FLAT(down(a(y10)))
354.42/147.62	   DOWN(b(a(y10))) -> DOWN(a(y10))
354.42/147.62	   DOWN(b(c)) -> B_FLAT(down(c))
354.42/147.62	   DOWN(b(c)) -> DOWN(c)
354.42/147.62	   DOWN(b(b(y11))) -> B_FLAT(down(b(y11)))
354.42/147.62	   DOWN(b(b(y11))) -> DOWN(b(y11))
354.42/147.62	   DOWN(b(d(y13))) -> B_FLAT(down(d(y13)))
354.42/147.62	   DOWN(b(d(y13))) -> DOWN(d(y13))
354.42/147.62	   DOWN(b(fresh_constant)) -> B_FLAT(down(fresh_constant))
354.42/147.62	   DOWN(b(fresh_constant)) -> DOWN(fresh_constant)
354.42/147.62	   DOWN(d(c)) -> D_FLAT(down(c))
354.42/147.62	   DOWN(d(c)) -> DOWN(c)
354.42/147.62	   DOWN(d(u(y17))) -> D_FLAT(down(u(y17)))
354.42/147.62	   DOWN(d(u(y17))) -> DOWN(u(y17))
354.42/147.62	   DOWN(d(d(y18))) -> D_FLAT(down(d(y18)))
354.42/147.62	   DOWN(d(d(y18))) -> DOWN(d(y18))
354.42/147.62	   DOWN(d(fresh_constant)) -> D_FLAT(down(fresh_constant))
354.42/147.62	   DOWN(d(fresh_constant)) -> DOWN(fresh_constant)
354.42/147.62	
354.42/147.62	The TRS R consists of the following rules:
354.42/147.62	
354.42/147.62	   down(a(a(x))) -> up(c)
354.42/147.62	   down(b(u(x))) -> up(b(d(x)))
354.42/147.62	   down(d(a(x))) -> up(a(d(x)))
354.42/147.62	   down(d(b(x))) -> up(u(a(b(x))))
354.42/147.62	   down(a(u(x))) -> up(u(a(x)))
354.42/147.62	   top(up(x)) -> top(down(x))
354.42/147.62	   down(u(y2)) -> u_flat(down(y2))
354.42/147.62	   down(a(c)) -> a_flat(down(c))
354.42/147.62	   down(a(b(y6))) -> a_flat(down(b(y6)))
354.42/147.62	   down(a(d(y8))) -> a_flat(down(d(y8)))
354.42/147.62	   down(a(fresh_constant)) -> a_flat(down(fresh_constant))
354.42/147.62	   down(b(a(y10))) -> b_flat(down(a(y10)))
354.42/147.62	   down(b(c)) -> b_flat(down(c))
354.42/147.62	   down(b(b(y11))) -> b_flat(down(b(y11)))
354.42/147.62	   down(b(d(y13))) -> b_flat(down(d(y13)))
354.42/147.62	   down(b(fresh_constant)) -> b_flat(down(fresh_constant))
354.42/147.62	   down(d(c)) -> d_flat(down(c))
354.42/147.62	   down(d(u(y17))) -> d_flat(down(u(y17)))
354.42/147.62	   down(d(d(y18))) -> d_flat(down(d(y18)))
354.42/147.62	   down(d(fresh_constant)) -> d_flat(down(fresh_constant))
354.42/147.62	   a_flat(up(x_1)) -> up(a(x_1))
354.42/147.62	   b_flat(up(x_1)) -> up(b(x_1))
354.42/147.62	   u_flat(up(x_1)) -> up(u(x_1))
354.42/147.62	   d_flat(up(x_1)) -> up(d(x_1))
354.42/147.62	
354.42/147.62	The set Q consists of the following terms:
354.42/147.62	
354.42/147.62	   down(a(a(x0)))
354.42/147.62	   down(b(u(x0)))
354.42/147.62	   down(d(a(x0)))
354.42/147.62	   down(d(b(x0)))
354.42/147.62	   down(a(u(x0)))
354.42/147.62	   top(up(x0))
354.42/147.62	   down(u(x0))
354.42/147.62	   down(a(c))
354.42/147.62	   down(a(b(x0)))
354.42/147.62	   down(a(d(x0)))
354.42/147.63	   down(a(fresh_constant))
354.42/147.63	   down(b(a(x0)))
354.42/147.63	   down(b(c))
354.42/147.63	   down(b(b(x0)))
354.42/147.63	   down(b(d(x0)))
354.42/147.63	   down(b(fresh_constant))
354.42/147.63	   down(d(c))
354.42/147.63	   down(d(u(x0)))
354.42/147.63	   down(d(d(x0)))
354.42/147.63	   down(d(fresh_constant))
354.42/147.63	   a_flat(up(x0))
354.42/147.63	   b_flat(up(x0))
354.42/147.63	   u_flat(up(x0))
354.42/147.63	   d_flat(up(x0))
354.42/147.63	
354.42/147.63	We have to consider all minimal (P,Q,R)-chains.
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(66) DependencyGraphProof (EQUIVALENT)
354.42/147.63	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 21 less nodes.
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(67)
354.42/147.63	Complex Obligation (AND)
354.42/147.63	
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(68)
354.42/147.63	Obligation:
354.42/147.63	Q DP problem:
354.42/147.63	The TRS P consists of the following rules:
354.42/147.63	
354.42/147.63	   DOWN(a(b(y6))) -> DOWN(b(y6))
354.42/147.63	   DOWN(b(a(y10))) -> DOWN(a(y10))
354.42/147.63	   DOWN(a(d(y8))) -> DOWN(d(y8))
354.42/147.63	   DOWN(d(u(y17))) -> DOWN(u(y17))
354.42/147.63	   DOWN(u(y2)) -> DOWN(y2)
354.42/147.63	   DOWN(b(b(y11))) -> DOWN(b(y11))
354.42/147.63	   DOWN(b(d(y13))) -> DOWN(d(y13))
354.42/147.63	   DOWN(d(d(y18))) -> DOWN(d(y18))
354.42/147.63	
354.42/147.63	The TRS R consists of the following rules:
354.42/147.63	
354.42/147.63	   down(a(a(x))) -> up(c)
354.42/147.63	   down(b(u(x))) -> up(b(d(x)))
354.42/147.63	   down(d(a(x))) -> up(a(d(x)))
354.42/147.63	   down(d(b(x))) -> up(u(a(b(x))))
354.42/147.63	   down(a(u(x))) -> up(u(a(x)))
354.42/147.63	   top(up(x)) -> top(down(x))
354.42/147.63	   down(u(y2)) -> u_flat(down(y2))
354.42/147.63	   down(a(c)) -> a_flat(down(c))
354.42/147.63	   down(a(b(y6))) -> a_flat(down(b(y6)))
354.42/147.63	   down(a(d(y8))) -> a_flat(down(d(y8)))
354.42/147.63	   down(a(fresh_constant)) -> a_flat(down(fresh_constant))
354.42/147.63	   down(b(a(y10))) -> b_flat(down(a(y10)))
354.42/147.63	   down(b(c)) -> b_flat(down(c))
354.42/147.63	   down(b(b(y11))) -> b_flat(down(b(y11)))
354.42/147.63	   down(b(d(y13))) -> b_flat(down(d(y13)))
354.42/147.63	   down(b(fresh_constant)) -> b_flat(down(fresh_constant))
354.42/147.63	   down(d(c)) -> d_flat(down(c))
354.42/147.63	   down(d(u(y17))) -> d_flat(down(u(y17)))
354.42/147.63	   down(d(d(y18))) -> d_flat(down(d(y18)))
354.42/147.63	   down(d(fresh_constant)) -> d_flat(down(fresh_constant))
354.42/147.63	   a_flat(up(x_1)) -> up(a(x_1))
354.42/147.63	   b_flat(up(x_1)) -> up(b(x_1))
354.42/147.63	   u_flat(up(x_1)) -> up(u(x_1))
354.42/147.63	   d_flat(up(x_1)) -> up(d(x_1))
354.42/147.63	
354.42/147.63	The set Q consists of the following terms:
354.42/147.63	
354.42/147.63	   down(a(a(x0)))
354.42/147.63	   down(b(u(x0)))
354.42/147.63	   down(d(a(x0)))
354.42/147.63	   down(d(b(x0)))
354.42/147.63	   down(a(u(x0)))
354.42/147.63	   top(up(x0))
354.42/147.63	   down(u(x0))
354.42/147.63	   down(a(c))
354.42/147.63	   down(a(b(x0)))
354.42/147.63	   down(a(d(x0)))
354.42/147.63	   down(a(fresh_constant))
354.42/147.63	   down(b(a(x0)))
354.42/147.63	   down(b(c))
354.42/147.63	   down(b(b(x0)))
354.42/147.63	   down(b(d(x0)))
354.42/147.63	   down(b(fresh_constant))
354.42/147.63	   down(d(c))
354.42/147.63	   down(d(u(x0)))
354.42/147.63	   down(d(d(x0)))
354.42/147.63	   down(d(fresh_constant))
354.42/147.63	   a_flat(up(x0))
354.42/147.63	   b_flat(up(x0))
354.42/147.63	   u_flat(up(x0))
354.42/147.63	   d_flat(up(x0))
354.42/147.63	
354.42/147.63	We have to consider all minimal (P,Q,R)-chains.
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(69) UsableRulesProof (EQUIVALENT)
354.42/147.63	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.
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(70)
354.42/147.63	Obligation:
354.42/147.63	Q DP problem:
354.42/147.63	The TRS P consists of the following rules:
354.42/147.63	
354.42/147.63	   DOWN(a(b(y6))) -> DOWN(b(y6))
354.42/147.63	   DOWN(b(a(y10))) -> DOWN(a(y10))
354.42/147.63	   DOWN(a(d(y8))) -> DOWN(d(y8))
354.42/147.63	   DOWN(d(u(y17))) -> DOWN(u(y17))
354.42/147.63	   DOWN(u(y2)) -> DOWN(y2)
354.42/147.63	   DOWN(b(b(y11))) -> DOWN(b(y11))
354.42/147.63	   DOWN(b(d(y13))) -> DOWN(d(y13))
354.42/147.63	   DOWN(d(d(y18))) -> DOWN(d(y18))
354.42/147.63	
354.42/147.63	R is empty.
354.42/147.63	The set Q consists of the following terms:
354.42/147.63	
354.42/147.63	   down(a(a(x0)))
354.42/147.63	   down(b(u(x0)))
354.42/147.63	   down(d(a(x0)))
354.42/147.63	   down(d(b(x0)))
354.42/147.63	   down(a(u(x0)))
354.42/147.63	   top(up(x0))
354.42/147.63	   down(u(x0))
354.42/147.63	   down(a(c))
354.42/147.63	   down(a(b(x0)))
354.42/147.63	   down(a(d(x0)))
354.42/147.63	   down(a(fresh_constant))
354.42/147.63	   down(b(a(x0)))
354.42/147.63	   down(b(c))
354.42/147.63	   down(b(b(x0)))
354.42/147.63	   down(b(d(x0)))
354.42/147.63	   down(b(fresh_constant))
354.42/147.63	   down(d(c))
354.42/147.63	   down(d(u(x0)))
354.42/147.63	   down(d(d(x0)))
354.42/147.63	   down(d(fresh_constant))
354.42/147.63	   a_flat(up(x0))
354.42/147.63	   b_flat(up(x0))
354.42/147.63	   u_flat(up(x0))
354.42/147.63	   d_flat(up(x0))
354.42/147.63	
354.42/147.63	We have to consider all minimal (P,Q,R)-chains.
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(71) QReductionProof (EQUIVALENT)
354.42/147.63	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
354.42/147.63	
354.42/147.63	   down(a(a(x0)))
354.42/147.63	   down(b(u(x0)))
354.42/147.63	   down(d(a(x0)))
354.42/147.63	   down(d(b(x0)))
354.42/147.63	   down(a(u(x0)))
354.42/147.63	   top(up(x0))
354.42/147.63	   down(u(x0))
354.42/147.63	   down(a(c))
354.42/147.63	   down(a(b(x0)))
354.42/147.63	   down(a(d(x0)))
354.42/147.63	   down(a(fresh_constant))
354.42/147.63	   down(b(a(x0)))
354.42/147.63	   down(b(c))
354.42/147.63	   down(b(b(x0)))
354.42/147.63	   down(b(d(x0)))
354.42/147.63	   down(b(fresh_constant))
354.42/147.63	   down(d(c))
354.42/147.63	   down(d(u(x0)))
354.42/147.63	   down(d(d(x0)))
354.42/147.63	   down(d(fresh_constant))
354.42/147.63	   a_flat(up(x0))
354.42/147.63	   b_flat(up(x0))
354.42/147.63	   u_flat(up(x0))
354.42/147.63	   d_flat(up(x0))
354.42/147.63	
354.42/147.63	
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(72)
354.42/147.63	Obligation:
354.42/147.63	Q DP problem:
354.42/147.63	The TRS P consists of the following rules:
354.42/147.63	
354.42/147.63	   DOWN(a(b(y6))) -> DOWN(b(y6))
354.42/147.63	   DOWN(b(a(y10))) -> DOWN(a(y10))
354.42/147.63	   DOWN(a(d(y8))) -> DOWN(d(y8))
354.42/147.63	   DOWN(d(u(y17))) -> DOWN(u(y17))
354.42/147.63	   DOWN(u(y2)) -> DOWN(y2)
354.42/147.63	   DOWN(b(b(y11))) -> DOWN(b(y11))
354.42/147.63	   DOWN(b(d(y13))) -> DOWN(d(y13))
354.42/147.63	   DOWN(d(d(y18))) -> DOWN(d(y18))
354.42/147.63	
354.42/147.63	R is empty.
354.42/147.63	Q is empty.
354.42/147.63	We have to consider all minimal (P,Q,R)-chains.
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(73) QDPSizeChangeProof (EQUIVALENT)
354.42/147.63	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. 
354.42/147.63	
354.42/147.63	From the DPs we obtained the following set of size-change graphs:
354.42/147.63	*DOWN(b(a(y10))) -> DOWN(a(y10))
354.42/147.63	The graph contains the following edges 1 > 1
354.42/147.63	
354.42/147.63	
354.42/147.63	*DOWN(u(y2)) -> DOWN(y2)
354.42/147.63	The graph contains the following edges 1 > 1
354.42/147.63	
354.42/147.63	
354.42/147.63	*DOWN(b(b(y11))) -> DOWN(b(y11))
354.42/147.63	The graph contains the following edges 1 > 1
354.42/147.63	
354.42/147.63	
354.42/147.63	*DOWN(b(d(y13))) -> DOWN(d(y13))
354.42/147.63	The graph contains the following edges 1 > 1
354.42/147.63	
354.42/147.63	
354.42/147.63	*DOWN(a(b(y6))) -> DOWN(b(y6))
354.42/147.63	The graph contains the following edges 1 > 1
354.42/147.63	
354.42/147.63	
354.42/147.63	*DOWN(a(d(y8))) -> DOWN(d(y8))
354.42/147.63	The graph contains the following edges 1 > 1
354.42/147.63	
354.42/147.63	
354.42/147.63	*DOWN(d(d(y18))) -> DOWN(d(y18))
354.42/147.63	The graph contains the following edges 1 > 1
354.42/147.63	
354.42/147.63	
354.42/147.63	*DOWN(d(u(y17))) -> DOWN(u(y17))
354.42/147.63	The graph contains the following edges 1 > 1
354.42/147.63	
354.42/147.63	
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(74)
354.42/147.63	YES
354.42/147.63	
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(75)
354.42/147.63	Obligation:
354.42/147.63	Q DP problem:
354.42/147.63	The TRS P consists of the following rules:
354.42/147.63	
354.42/147.63	   TOP(up(x)) -> TOP(down(x))
354.42/147.63	
354.42/147.63	The TRS R consists of the following rules:
354.42/147.63	
354.42/147.63	   down(a(a(x))) -> up(c)
354.42/147.63	   down(b(u(x))) -> up(b(d(x)))
354.42/147.63	   down(d(a(x))) -> up(a(d(x)))
354.42/147.63	   down(d(b(x))) -> up(u(a(b(x))))
354.42/147.63	   down(a(u(x))) -> up(u(a(x)))
354.42/147.63	   top(up(x)) -> top(down(x))
354.42/147.63	   down(u(y2)) -> u_flat(down(y2))
354.42/147.63	   down(a(c)) -> a_flat(down(c))
354.42/147.63	   down(a(b(y6))) -> a_flat(down(b(y6)))
354.42/147.63	   down(a(d(y8))) -> a_flat(down(d(y8)))
354.42/147.63	   down(a(fresh_constant)) -> a_flat(down(fresh_constant))
354.42/147.63	   down(b(a(y10))) -> b_flat(down(a(y10)))
354.42/147.63	   down(b(c)) -> b_flat(down(c))
354.42/147.63	   down(b(b(y11))) -> b_flat(down(b(y11)))
354.42/147.63	   down(b(d(y13))) -> b_flat(down(d(y13)))
354.42/147.63	   down(b(fresh_constant)) -> b_flat(down(fresh_constant))
354.42/147.63	   down(d(c)) -> d_flat(down(c))
354.42/147.63	   down(d(u(y17))) -> d_flat(down(u(y17)))
354.42/147.63	   down(d(d(y18))) -> d_flat(down(d(y18)))
354.42/147.63	   down(d(fresh_constant)) -> d_flat(down(fresh_constant))
354.42/147.63	   a_flat(up(x_1)) -> up(a(x_1))
354.42/147.63	   b_flat(up(x_1)) -> up(b(x_1))
354.42/147.63	   u_flat(up(x_1)) -> up(u(x_1))
354.42/147.63	   d_flat(up(x_1)) -> up(d(x_1))
354.42/147.63	
354.42/147.63	The set Q consists of the following terms:
354.42/147.63	
354.42/147.63	   down(a(a(x0)))
354.42/147.63	   down(b(u(x0)))
354.42/147.63	   down(d(a(x0)))
354.42/147.63	   down(d(b(x0)))
354.42/147.63	   down(a(u(x0)))
354.42/147.63	   top(up(x0))
354.42/147.63	   down(u(x0))
354.42/147.63	   down(a(c))
354.42/147.63	   down(a(b(x0)))
354.42/147.63	   down(a(d(x0)))
354.42/147.63	   down(a(fresh_constant))
354.42/147.63	   down(b(a(x0)))
354.42/147.63	   down(b(c))
354.42/147.63	   down(b(b(x0)))
354.42/147.63	   down(b(d(x0)))
354.42/147.63	   down(b(fresh_constant))
354.42/147.63	   down(d(c))
354.42/147.63	   down(d(u(x0)))
354.42/147.63	   down(d(d(x0)))
354.42/147.63	   down(d(fresh_constant))
354.42/147.63	   a_flat(up(x0))
354.42/147.63	   b_flat(up(x0))
354.42/147.63	   u_flat(up(x0))
354.42/147.63	   d_flat(up(x0))
354.42/147.63	
354.42/147.63	We have to consider all minimal (P,Q,R)-chains.
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(76) UsableRulesProof (EQUIVALENT)
354.42/147.63	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.
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(77)
354.42/147.63	Obligation:
354.42/147.63	Q DP problem:
354.42/147.63	The TRS P consists of the following rules:
354.42/147.63	
354.42/147.63	   TOP(up(x)) -> TOP(down(x))
354.42/147.63	
354.42/147.63	The TRS R consists of the following rules:
354.42/147.63	
354.42/147.63	   down(a(a(x))) -> up(c)
354.42/147.63	   down(b(u(x))) -> up(b(d(x)))
354.42/147.63	   down(d(a(x))) -> up(a(d(x)))
354.42/147.63	   down(d(b(x))) -> up(u(a(b(x))))
354.42/147.63	   down(a(u(x))) -> up(u(a(x)))
354.42/147.63	   down(u(y2)) -> u_flat(down(y2))
354.42/147.63	   down(a(c)) -> a_flat(down(c))
354.42/147.63	   down(a(b(y6))) -> a_flat(down(b(y6)))
354.42/147.63	   down(a(d(y8))) -> a_flat(down(d(y8)))
354.42/147.63	   down(a(fresh_constant)) -> a_flat(down(fresh_constant))
354.42/147.63	   down(b(a(y10))) -> b_flat(down(a(y10)))
354.42/147.63	   down(b(c)) -> b_flat(down(c))
354.42/147.63	   down(b(b(y11))) -> b_flat(down(b(y11)))
354.42/147.63	   down(b(d(y13))) -> b_flat(down(d(y13)))
354.42/147.63	   down(b(fresh_constant)) -> b_flat(down(fresh_constant))
354.42/147.63	   down(d(c)) -> d_flat(down(c))
354.42/147.63	   down(d(u(y17))) -> d_flat(down(u(y17)))
354.42/147.63	   down(d(d(y18))) -> d_flat(down(d(y18)))
354.42/147.63	   down(d(fresh_constant)) -> d_flat(down(fresh_constant))
354.42/147.63	   d_flat(up(x_1)) -> up(d(x_1))
354.42/147.63	   b_flat(up(x_1)) -> up(b(x_1))
354.42/147.63	   a_flat(up(x_1)) -> up(a(x_1))
354.42/147.63	   u_flat(up(x_1)) -> up(u(x_1))
354.42/147.63	
354.42/147.63	The set Q consists of the following terms:
354.42/147.63	
354.42/147.63	   down(a(a(x0)))
354.42/147.63	   down(b(u(x0)))
354.42/147.63	   down(d(a(x0)))
354.42/147.63	   down(d(b(x0)))
354.42/147.63	   down(a(u(x0)))
354.42/147.63	   top(up(x0))
354.42/147.63	   down(u(x0))
354.42/147.63	   down(a(c))
354.42/147.63	   down(a(b(x0)))
354.42/147.63	   down(a(d(x0)))
354.42/147.63	   down(a(fresh_constant))
354.42/147.63	   down(b(a(x0)))
354.42/147.63	   down(b(c))
354.42/147.63	   down(b(b(x0)))
354.42/147.63	   down(b(d(x0)))
354.42/147.63	   down(b(fresh_constant))
354.42/147.63	   down(d(c))
354.42/147.63	   down(d(u(x0)))
354.42/147.63	   down(d(d(x0)))
354.42/147.63	   down(d(fresh_constant))
354.42/147.63	   a_flat(up(x0))
354.42/147.63	   b_flat(up(x0))
354.42/147.63	   u_flat(up(x0))
354.42/147.63	   d_flat(up(x0))
354.42/147.63	
354.42/147.63	We have to consider all minimal (P,Q,R)-chains.
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(78) QReductionProof (EQUIVALENT)
354.42/147.63	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
354.42/147.63	
354.42/147.63	   top(up(x0))
354.42/147.63	
354.42/147.63	
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(79)
354.42/147.63	Obligation:
354.42/147.63	Q DP problem:
354.42/147.63	The TRS P consists of the following rules:
354.42/147.63	
354.42/147.63	   TOP(up(x)) -> TOP(down(x))
354.42/147.63	
354.42/147.63	The TRS R consists of the following rules:
354.42/147.63	
354.42/147.63	   down(a(a(x))) -> up(c)
354.42/147.63	   down(b(u(x))) -> up(b(d(x)))
354.42/147.63	   down(d(a(x))) -> up(a(d(x)))
354.42/147.63	   down(d(b(x))) -> up(u(a(b(x))))
354.42/147.63	   down(a(u(x))) -> up(u(a(x)))
354.42/147.63	   down(u(y2)) -> u_flat(down(y2))
354.42/147.63	   down(a(c)) -> a_flat(down(c))
354.42/147.63	   down(a(b(y6))) -> a_flat(down(b(y6)))
354.42/147.63	   down(a(d(y8))) -> a_flat(down(d(y8)))
354.42/147.63	   down(a(fresh_constant)) -> a_flat(down(fresh_constant))
354.42/147.63	   down(b(a(y10))) -> b_flat(down(a(y10)))
354.42/147.63	   down(b(c)) -> b_flat(down(c))
354.42/147.63	   down(b(b(y11))) -> b_flat(down(b(y11)))
354.42/147.63	   down(b(d(y13))) -> b_flat(down(d(y13)))
354.42/147.63	   down(b(fresh_constant)) -> b_flat(down(fresh_constant))
354.42/147.63	   down(d(c)) -> d_flat(down(c))
354.42/147.63	   down(d(u(y17))) -> d_flat(down(u(y17)))
354.42/147.63	   down(d(d(y18))) -> d_flat(down(d(y18)))
354.42/147.63	   down(d(fresh_constant)) -> d_flat(down(fresh_constant))
354.42/147.63	   d_flat(up(x_1)) -> up(d(x_1))
354.42/147.63	   b_flat(up(x_1)) -> up(b(x_1))
354.42/147.63	   a_flat(up(x_1)) -> up(a(x_1))
354.42/147.63	   u_flat(up(x_1)) -> up(u(x_1))
354.42/147.63	
354.42/147.63	The set Q consists of the following terms:
354.42/147.63	
354.42/147.63	   down(a(a(x0)))
354.42/147.63	   down(b(u(x0)))
354.42/147.63	   down(d(a(x0)))
354.42/147.63	   down(d(b(x0)))
354.42/147.63	   down(a(u(x0)))
354.42/147.63	   down(u(x0))
354.42/147.63	   down(a(c))
354.42/147.63	   down(a(b(x0)))
354.42/147.63	   down(a(d(x0)))
354.42/147.63	   down(a(fresh_constant))
354.42/147.63	   down(b(a(x0)))
354.42/147.63	   down(b(c))
354.42/147.63	   down(b(b(x0)))
354.42/147.63	   down(b(d(x0)))
354.42/147.63	   down(b(fresh_constant))
354.42/147.63	   down(d(c))
354.42/147.63	   down(d(u(x0)))
354.42/147.63	   down(d(d(x0)))
354.42/147.63	   down(d(fresh_constant))
354.42/147.63	   a_flat(up(x0))
354.42/147.63	   b_flat(up(x0))
354.42/147.63	   u_flat(up(x0))
354.42/147.63	   d_flat(up(x0))
354.42/147.63	
354.42/147.63	We have to consider all minimal (P,Q,R)-chains.
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(80) TransformationProof (EQUIVALENT)
354.42/147.63	By narrowing [LPAR04] the rule TOP(up(x)) -> TOP(down(x)) at position [0] we obtained the following new rules [LPAR04]:
354.42/147.63	
354.42/147.63	   (TOP(up(a(a(x0)))) -> TOP(up(c)),TOP(up(a(a(x0)))) -> TOP(up(c)))
354.42/147.63	   (TOP(up(b(u(x0)))) -> TOP(up(b(d(x0)))),TOP(up(b(u(x0)))) -> TOP(up(b(d(x0)))))
354.42/147.63	   (TOP(up(d(a(x0)))) -> TOP(up(a(d(x0)))),TOP(up(d(a(x0)))) -> TOP(up(a(d(x0)))))
354.42/147.63	   (TOP(up(d(b(x0)))) -> TOP(up(u(a(b(x0))))),TOP(up(d(b(x0)))) -> TOP(up(u(a(b(x0))))))
354.42/147.63	   (TOP(up(a(u(x0)))) -> TOP(up(u(a(x0)))),TOP(up(a(u(x0)))) -> TOP(up(u(a(x0)))))
354.42/147.63	   (TOP(up(u(x0))) -> TOP(u_flat(down(x0))),TOP(up(u(x0))) -> TOP(u_flat(down(x0))))
354.42/147.63	   (TOP(up(a(c))) -> TOP(a_flat(down(c))),TOP(up(a(c))) -> TOP(a_flat(down(c))))
354.42/147.63	   (TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0)))),TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0)))))
354.42/147.63	   (TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0)))),TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0)))))
354.42/147.63	   (TOP(up(a(fresh_constant))) -> TOP(a_flat(down(fresh_constant))),TOP(up(a(fresh_constant))) -> TOP(a_flat(down(fresh_constant))))
354.42/147.63	   (TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0)))),TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0)))))
354.42/147.63	   (TOP(up(b(c))) -> TOP(b_flat(down(c))),TOP(up(b(c))) -> TOP(b_flat(down(c))))
354.42/147.63	   (TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))),TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0)))))
354.42/147.63	   (TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0)))),TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0)))))
354.42/147.63	   (TOP(up(b(fresh_constant))) -> TOP(b_flat(down(fresh_constant))),TOP(up(b(fresh_constant))) -> TOP(b_flat(down(fresh_constant))))
354.42/147.63	   (TOP(up(d(c))) -> TOP(d_flat(down(c))),TOP(up(d(c))) -> TOP(d_flat(down(c))))
354.42/147.63	   (TOP(up(d(u(x0)))) -> TOP(d_flat(down(u(x0)))),TOP(up(d(u(x0)))) -> TOP(d_flat(down(u(x0)))))
354.42/147.63	   (TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0)))),TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0)))))
354.42/147.63	   (TOP(up(d(fresh_constant))) -> TOP(d_flat(down(fresh_constant))),TOP(up(d(fresh_constant))) -> TOP(d_flat(down(fresh_constant))))
354.42/147.63	
354.42/147.63	
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(81)
354.42/147.63	Obligation:
354.42/147.63	Q DP problem:
354.42/147.63	The TRS P consists of the following rules:
354.42/147.63	
354.42/147.63	   TOP(up(a(a(x0)))) -> TOP(up(c))
354.42/147.63	   TOP(up(b(u(x0)))) -> TOP(up(b(d(x0))))
354.42/147.63	   TOP(up(d(a(x0)))) -> TOP(up(a(d(x0))))
354.42/147.63	   TOP(up(d(b(x0)))) -> TOP(up(u(a(b(x0)))))
354.42/147.63	   TOP(up(a(u(x0)))) -> TOP(up(u(a(x0))))
354.42/147.63	   TOP(up(u(x0))) -> TOP(u_flat(down(x0)))
354.42/147.63	   TOP(up(a(c))) -> TOP(a_flat(down(c)))
354.42/147.63	   TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0))))
354.42/147.63	   TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0))))
354.42/147.63	   TOP(up(a(fresh_constant))) -> TOP(a_flat(down(fresh_constant)))
354.42/147.63	   TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0))))
354.42/147.63	   TOP(up(b(c))) -> TOP(b_flat(down(c)))
354.42/147.63	   TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0))))
354.42/147.63	   TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0))))
354.42/147.63	   TOP(up(b(fresh_constant))) -> TOP(b_flat(down(fresh_constant)))
354.42/147.63	   TOP(up(d(c))) -> TOP(d_flat(down(c)))
354.42/147.63	   TOP(up(d(u(x0)))) -> TOP(d_flat(down(u(x0))))
354.42/147.63	   TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0))))
354.42/147.63	   TOP(up(d(fresh_constant))) -> TOP(d_flat(down(fresh_constant)))
354.42/147.63	
354.42/147.63	The TRS R consists of the following rules:
354.42/147.63	
354.42/147.63	   down(a(a(x))) -> up(c)
354.42/147.63	   down(b(u(x))) -> up(b(d(x)))
354.42/147.63	   down(d(a(x))) -> up(a(d(x)))
354.42/147.63	   down(d(b(x))) -> up(u(a(b(x))))
354.42/147.63	   down(a(u(x))) -> up(u(a(x)))
354.42/147.63	   down(u(y2)) -> u_flat(down(y2))
354.42/147.63	   down(a(c)) -> a_flat(down(c))
354.42/147.63	   down(a(b(y6))) -> a_flat(down(b(y6)))
354.42/147.63	   down(a(d(y8))) -> a_flat(down(d(y8)))
354.42/147.63	   down(a(fresh_constant)) -> a_flat(down(fresh_constant))
354.42/147.63	   down(b(a(y10))) -> b_flat(down(a(y10)))
354.42/147.63	   down(b(c)) -> b_flat(down(c))
354.42/147.63	   down(b(b(y11))) -> b_flat(down(b(y11)))
354.42/147.63	   down(b(d(y13))) -> b_flat(down(d(y13)))
354.42/147.63	   down(b(fresh_constant)) -> b_flat(down(fresh_constant))
354.42/147.63	   down(d(c)) -> d_flat(down(c))
354.42/147.63	   down(d(u(y17))) -> d_flat(down(u(y17)))
354.42/147.63	   down(d(d(y18))) -> d_flat(down(d(y18)))
354.42/147.63	   down(d(fresh_constant)) -> d_flat(down(fresh_constant))
354.42/147.63	   d_flat(up(x_1)) -> up(d(x_1))
354.42/147.63	   b_flat(up(x_1)) -> up(b(x_1))
354.42/147.63	   a_flat(up(x_1)) -> up(a(x_1))
354.42/147.63	   u_flat(up(x_1)) -> up(u(x_1))
354.42/147.63	
354.42/147.63	The set Q consists of the following terms:
354.42/147.63	
354.42/147.63	   down(a(a(x0)))
354.42/147.63	   down(b(u(x0)))
354.42/147.63	   down(d(a(x0)))
354.42/147.63	   down(d(b(x0)))
354.42/147.63	   down(a(u(x0)))
354.42/147.63	   down(u(x0))
354.42/147.63	   down(a(c))
354.42/147.63	   down(a(b(x0)))
354.42/147.63	   down(a(d(x0)))
354.42/147.63	   down(a(fresh_constant))
354.42/147.63	   down(b(a(x0)))
354.42/147.63	   down(b(c))
354.42/147.63	   down(b(b(x0)))
354.42/147.63	   down(b(d(x0)))
354.42/147.63	   down(b(fresh_constant))
354.42/147.63	   down(d(c))
354.42/147.63	   down(d(u(x0)))
354.42/147.63	   down(d(d(x0)))
354.42/147.63	   down(d(fresh_constant))
354.42/147.63	   a_flat(up(x0))
354.42/147.63	   b_flat(up(x0))
354.42/147.63	   u_flat(up(x0))
354.42/147.63	   d_flat(up(x0))
354.42/147.63	
354.42/147.63	We have to consider all minimal (P,Q,R)-chains.
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(82) DependencyGraphProof (EQUIVALENT)
354.42/147.63	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes.
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(83)
354.42/147.63	Obligation:
354.42/147.63	Q DP problem:
354.42/147.63	The TRS P consists of the following rules:
354.42/147.63	
354.42/147.63	   TOP(up(b(u(x0)))) -> TOP(up(b(d(x0))))
354.42/147.63	   TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0))))
354.42/147.63	   TOP(up(d(a(x0)))) -> TOP(up(a(d(x0))))
354.42/147.63	   TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0))))
354.42/147.63	   TOP(up(d(b(x0)))) -> TOP(up(u(a(b(x0)))))
354.42/147.63	   TOP(up(u(x0))) -> TOP(u_flat(down(x0)))
354.42/147.63	   TOP(up(a(u(x0)))) -> TOP(up(u(a(x0))))
354.42/147.63	   TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0))))
354.42/147.63	   TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0))))
354.42/147.63	   TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0))))
354.42/147.63	   TOP(up(d(u(x0)))) -> TOP(d_flat(down(u(x0))))
354.42/147.63	   TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0))))
354.42/147.63	
354.42/147.63	The TRS R consists of the following rules:
354.42/147.63	
354.42/147.63	   down(a(a(x))) -> up(c)
354.42/147.63	   down(b(u(x))) -> up(b(d(x)))
354.42/147.63	   down(d(a(x))) -> up(a(d(x)))
354.42/147.63	   down(d(b(x))) -> up(u(a(b(x))))
354.42/147.63	   down(a(u(x))) -> up(u(a(x)))
354.42/147.63	   down(u(y2)) -> u_flat(down(y2))
354.42/147.63	   down(a(c)) -> a_flat(down(c))
354.42/147.63	   down(a(b(y6))) -> a_flat(down(b(y6)))
354.42/147.63	   down(a(d(y8))) -> a_flat(down(d(y8)))
354.42/147.63	   down(a(fresh_constant)) -> a_flat(down(fresh_constant))
354.42/147.63	   down(b(a(y10))) -> b_flat(down(a(y10)))
354.42/147.63	   down(b(c)) -> b_flat(down(c))
354.42/147.63	   down(b(b(y11))) -> b_flat(down(b(y11)))
354.42/147.63	   down(b(d(y13))) -> b_flat(down(d(y13)))
354.42/147.63	   down(b(fresh_constant)) -> b_flat(down(fresh_constant))
354.42/147.63	   down(d(c)) -> d_flat(down(c))
354.42/147.63	   down(d(u(y17))) -> d_flat(down(u(y17)))
354.42/147.63	   down(d(d(y18))) -> d_flat(down(d(y18)))
354.42/147.63	   down(d(fresh_constant)) -> d_flat(down(fresh_constant))
354.42/147.63	   d_flat(up(x_1)) -> up(d(x_1))
354.42/147.63	   b_flat(up(x_1)) -> up(b(x_1))
354.42/147.63	   a_flat(up(x_1)) -> up(a(x_1))
354.42/147.63	   u_flat(up(x_1)) -> up(u(x_1))
354.42/147.63	
354.42/147.63	The set Q consists of the following terms:
354.42/147.63	
354.42/147.63	   down(a(a(x0)))
354.42/147.63	   down(b(u(x0)))
354.42/147.63	   down(d(a(x0)))
354.42/147.63	   down(d(b(x0)))
354.42/147.63	   down(a(u(x0)))
354.42/147.63	   down(u(x0))
354.42/147.63	   down(a(c))
354.42/147.63	   down(a(b(x0)))
354.42/147.63	   down(a(d(x0)))
354.42/147.63	   down(a(fresh_constant))
354.42/147.63	   down(b(a(x0)))
354.42/147.63	   down(b(c))
354.42/147.63	   down(b(b(x0)))
354.42/147.63	   down(b(d(x0)))
354.42/147.63	   down(b(fresh_constant))
354.42/147.63	   down(d(c))
354.42/147.63	   down(d(u(x0)))
354.42/147.63	   down(d(d(x0)))
354.42/147.63	   down(d(fresh_constant))
354.42/147.63	   a_flat(up(x0))
354.42/147.63	   b_flat(up(x0))
354.42/147.63	   u_flat(up(x0))
354.42/147.63	   d_flat(up(x0))
354.42/147.63	
354.42/147.63	We have to consider all minimal (P,Q,R)-chains.
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(84) TransformationProof (EQUIVALENT)
354.42/147.63	By rewriting [LPAR04] the rule TOP(up(d(u(x0)))) -> TOP(d_flat(down(u(x0)))) at position [0,0] we obtained the following new rules [LPAR04]:
354.42/147.63	
354.42/147.63	   (TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0)))),TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0)))))
354.42/147.63	
354.42/147.63	
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(85)
354.42/147.63	Obligation:
354.42/147.63	Q DP problem:
354.42/147.63	The TRS P consists of the following rules:
354.42/147.63	
354.42/147.63	   TOP(up(b(u(x0)))) -> TOP(up(b(d(x0))))
354.42/147.63	   TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0))))
354.42/147.63	   TOP(up(d(a(x0)))) -> TOP(up(a(d(x0))))
354.42/147.63	   TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0))))
354.42/147.63	   TOP(up(d(b(x0)))) -> TOP(up(u(a(b(x0)))))
354.42/147.63	   TOP(up(u(x0))) -> TOP(u_flat(down(x0)))
354.42/147.63	   TOP(up(a(u(x0)))) -> TOP(up(u(a(x0))))
354.42/147.63	   TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0))))
354.42/147.63	   TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0))))
354.42/147.63	   TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0))))
354.42/147.63	   TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0))))
354.42/147.63	   TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0))))
354.42/147.63	
354.42/147.63	The TRS R consists of the following rules:
354.42/147.63	
354.42/147.63	   down(a(a(x))) -> up(c)
354.42/147.63	   down(b(u(x))) -> up(b(d(x)))
354.42/147.63	   down(d(a(x))) -> up(a(d(x)))
354.42/147.63	   down(d(b(x))) -> up(u(a(b(x))))
354.42/147.63	   down(a(u(x))) -> up(u(a(x)))
354.42/147.63	   down(u(y2)) -> u_flat(down(y2))
354.42/147.63	   down(a(c)) -> a_flat(down(c))
354.42/147.63	   down(a(b(y6))) -> a_flat(down(b(y6)))
354.42/147.63	   down(a(d(y8))) -> a_flat(down(d(y8)))
354.42/147.63	   down(a(fresh_constant)) -> a_flat(down(fresh_constant))
354.42/147.63	   down(b(a(y10))) -> b_flat(down(a(y10)))
354.42/147.63	   down(b(c)) -> b_flat(down(c))
354.42/147.63	   down(b(b(y11))) -> b_flat(down(b(y11)))
354.42/147.63	   down(b(d(y13))) -> b_flat(down(d(y13)))
354.42/147.63	   down(b(fresh_constant)) -> b_flat(down(fresh_constant))
354.42/147.63	   down(d(c)) -> d_flat(down(c))
354.42/147.63	   down(d(u(y17))) -> d_flat(down(u(y17)))
354.42/147.63	   down(d(d(y18))) -> d_flat(down(d(y18)))
354.42/147.63	   down(d(fresh_constant)) -> d_flat(down(fresh_constant))
354.42/147.63	   d_flat(up(x_1)) -> up(d(x_1))
354.42/147.63	   b_flat(up(x_1)) -> up(b(x_1))
354.42/147.63	   a_flat(up(x_1)) -> up(a(x_1))
354.42/147.63	   u_flat(up(x_1)) -> up(u(x_1))
354.42/147.63	
354.42/147.63	The set Q consists of the following terms:
354.42/147.63	
354.42/147.63	   down(a(a(x0)))
354.42/147.63	   down(b(u(x0)))
354.42/147.63	   down(d(a(x0)))
354.42/147.63	   down(d(b(x0)))
354.42/147.63	   down(a(u(x0)))
354.42/147.63	   down(u(x0))
354.42/147.63	   down(a(c))
354.42/147.63	   down(a(b(x0)))
354.42/147.63	   down(a(d(x0)))
354.42/147.63	   down(a(fresh_constant))
354.42/147.63	   down(b(a(x0)))
354.42/147.63	   down(b(c))
354.42/147.63	   down(b(b(x0)))
354.42/147.63	   down(b(d(x0)))
354.42/147.63	   down(b(fresh_constant))
354.42/147.63	   down(d(c))
354.42/147.63	   down(d(u(x0)))
354.42/147.63	   down(d(d(x0)))
354.42/147.63	   down(d(fresh_constant))
354.42/147.63	   a_flat(up(x0))
354.42/147.63	   b_flat(up(x0))
354.42/147.63	   u_flat(up(x0))
354.42/147.63	   d_flat(up(x0))
354.42/147.63	
354.42/147.63	We have to consider all minimal (P,Q,R)-chains.
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(86) QDPOrderProof (EQUIVALENT)
354.42/147.63	We use the reduction pair processor [LPAR04,JAR06].
354.42/147.63	
354.42/147.63	
354.42/147.63	The following pairs can be oriented strictly and are deleted.
354.42/147.63	
354.42/147.63	   TOP(up(d(b(x0)))) -> TOP(up(u(a(b(x0)))))
354.42/147.63	   TOP(up(a(u(x0)))) -> TOP(up(u(a(x0))))
354.42/147.63	The remaining pairs can at least be oriented weakly.
354.42/147.63	Used ordering:  Polynomial interpretation [POLO]:
354.42/147.63	
354.42/147.63	   POL(TOP(x_1)) = x_1
354.42/147.63	   POL(a(x_1)) = 1
354.42/147.63	   POL(a_flat(x_1)) = 1
354.42/147.63	   POL(b(x_1)) = 0
354.42/147.63	   POL(b_flat(x_1)) = 0
354.42/147.63	   POL(c) = 0
354.42/147.63	   POL(d(x_1)) = 1
354.42/147.63	   POL(d_flat(x_1)) = 1
354.42/147.63	   POL(down(x_1)) = 0
354.42/147.63	   POL(fresh_constant) = 0
354.42/147.63	   POL(u(x_1)) = 0
354.42/147.63	   POL(u_flat(x_1)) = 0
354.42/147.63	   POL(up(x_1)) = x_1
354.42/147.63	
354.42/147.63	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
354.42/147.63	
354.42/147.63	   b_flat(up(x_1)) -> up(b(x_1))
354.42/147.63	   a_flat(up(x_1)) -> up(a(x_1))
354.42/147.63	   u_flat(up(x_1)) -> up(u(x_1))
354.42/147.63	   d_flat(up(x_1)) -> up(d(x_1))
354.42/147.63	
354.42/147.63	
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(87)
354.42/147.63	Obligation:
354.42/147.63	Q DP problem:
354.42/147.63	The TRS P consists of the following rules:
354.42/147.63	
354.42/147.63	   TOP(up(b(u(x0)))) -> TOP(up(b(d(x0))))
354.42/147.63	   TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0))))
354.42/147.63	   TOP(up(d(a(x0)))) -> TOP(up(a(d(x0))))
354.42/147.63	   TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0))))
354.42/147.63	   TOP(up(u(x0))) -> TOP(u_flat(down(x0)))
354.42/147.63	   TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0))))
354.42/147.63	   TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0))))
354.42/147.63	   TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0))))
354.42/147.63	   TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0))))
354.42/147.63	   TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0))))
354.42/147.63	
354.42/147.63	The TRS R consists of the following rules:
354.42/147.63	
354.42/147.63	   down(a(a(x))) -> up(c)
354.42/147.63	   down(b(u(x))) -> up(b(d(x)))
354.42/147.63	   down(d(a(x))) -> up(a(d(x)))
354.42/147.63	   down(d(b(x))) -> up(u(a(b(x))))
354.42/147.63	   down(a(u(x))) -> up(u(a(x)))
354.42/147.63	   down(u(y2)) -> u_flat(down(y2))
354.42/147.63	   down(a(c)) -> a_flat(down(c))
354.42/147.63	   down(a(b(y6))) -> a_flat(down(b(y6)))
354.42/147.63	   down(a(d(y8))) -> a_flat(down(d(y8)))
354.42/147.63	   down(a(fresh_constant)) -> a_flat(down(fresh_constant))
354.42/147.63	   down(b(a(y10))) -> b_flat(down(a(y10)))
354.42/147.63	   down(b(c)) -> b_flat(down(c))
354.42/147.63	   down(b(b(y11))) -> b_flat(down(b(y11)))
354.42/147.63	   down(b(d(y13))) -> b_flat(down(d(y13)))
354.42/147.63	   down(b(fresh_constant)) -> b_flat(down(fresh_constant))
354.42/147.63	   down(d(c)) -> d_flat(down(c))
354.42/147.63	   down(d(u(y17))) -> d_flat(down(u(y17)))
354.42/147.63	   down(d(d(y18))) -> d_flat(down(d(y18)))
354.42/147.63	   down(d(fresh_constant)) -> d_flat(down(fresh_constant))
354.42/147.63	   d_flat(up(x_1)) -> up(d(x_1))
354.42/147.63	   b_flat(up(x_1)) -> up(b(x_1))
354.42/147.63	   a_flat(up(x_1)) -> up(a(x_1))
354.42/147.63	   u_flat(up(x_1)) -> up(u(x_1))
354.42/147.63	
354.42/147.63	The set Q consists of the following terms:
354.42/147.63	
354.42/147.63	   down(a(a(x0)))
354.42/147.63	   down(b(u(x0)))
354.42/147.63	   down(d(a(x0)))
354.42/147.63	   down(d(b(x0)))
354.42/147.63	   down(a(u(x0)))
354.42/147.63	   down(u(x0))
354.42/147.63	   down(a(c))
354.42/147.63	   down(a(b(x0)))
354.42/147.63	   down(a(d(x0)))
354.42/147.63	   down(a(fresh_constant))
354.42/147.63	   down(b(a(x0)))
354.42/147.63	   down(b(c))
354.42/147.63	   down(b(b(x0)))
354.42/147.63	   down(b(d(x0)))
354.42/147.63	   down(b(fresh_constant))
354.42/147.63	   down(d(c))
354.42/147.63	   down(d(u(x0)))
354.42/147.63	   down(d(d(x0)))
354.42/147.63	   down(d(fresh_constant))
354.42/147.63	   a_flat(up(x0))
354.42/147.63	   b_flat(up(x0))
354.42/147.63	   u_flat(up(x0))
354.42/147.63	   d_flat(up(x0))
354.42/147.63	
354.42/147.63	We have to consider all minimal (P,Q,R)-chains.
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(88) QDPOrderProof (EQUIVALENT)
354.42/147.63	We use the reduction pair processor [LPAR04,JAR06].
354.42/147.63	
354.42/147.63	
354.42/147.63	The following pairs can be oriented strictly and are deleted.
354.42/147.63	
354.42/147.63	   TOP(up(d(a(x0)))) -> TOP(up(a(d(x0))))
354.42/147.63	The remaining pairs can at least be oriented weakly.
354.42/147.63	Used ordering:  Polynomial interpretation [POLO]:
354.42/147.63	
354.42/147.63	   POL(TOP(x_1)) = x_1
354.42/147.63	   POL(a(x_1)) = 0
354.42/147.63	   POL(a_flat(x_1)) = 0
354.42/147.63	   POL(b(x_1)) = 0
354.42/147.63	   POL(b_flat(x_1)) = 0
354.42/147.63	   POL(c) = 0
354.42/147.63	   POL(d(x_1)) = 1
354.42/147.63	   POL(d_flat(x_1)) = 1
354.42/147.63	   POL(down(x_1)) = 0
354.42/147.63	   POL(fresh_constant) = 0
354.42/147.63	   POL(u(x_1)) = 0
354.42/147.63	   POL(u_flat(x_1)) = 0
354.42/147.63	   POL(up(x_1)) = x_1
354.42/147.63	
354.42/147.63	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
354.42/147.63	
354.42/147.63	   b_flat(up(x_1)) -> up(b(x_1))
354.42/147.63	   a_flat(up(x_1)) -> up(a(x_1))
354.42/147.63	   u_flat(up(x_1)) -> up(u(x_1))
354.42/147.63	   d_flat(up(x_1)) -> up(d(x_1))
354.42/147.63	
354.42/147.63	
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(89)
354.42/147.63	Obligation:
354.42/147.63	Q DP problem:
354.42/147.63	The TRS P consists of the following rules:
354.42/147.63	
354.42/147.63	   TOP(up(b(u(x0)))) -> TOP(up(b(d(x0))))
354.42/147.63	   TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0))))
354.42/147.63	   TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0))))
354.42/147.63	   TOP(up(u(x0))) -> TOP(u_flat(down(x0)))
354.42/147.63	   TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0))))
354.42/147.63	   TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0))))
354.42/147.63	   TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0))))
354.42/147.63	   TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0))))
354.42/147.63	   TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0))))
354.42/147.63	
354.42/147.63	The TRS R consists of the following rules:
354.42/147.63	
354.42/147.63	   down(a(a(x))) -> up(c)
354.42/147.63	   down(b(u(x))) -> up(b(d(x)))
354.42/147.63	   down(d(a(x))) -> up(a(d(x)))
354.42/147.63	   down(d(b(x))) -> up(u(a(b(x))))
354.42/147.63	   down(a(u(x))) -> up(u(a(x)))
354.42/147.63	   down(u(y2)) -> u_flat(down(y2))
354.42/147.63	   down(a(c)) -> a_flat(down(c))
354.42/147.63	   down(a(b(y6))) -> a_flat(down(b(y6)))
354.42/147.63	   down(a(d(y8))) -> a_flat(down(d(y8)))
354.42/147.63	   down(a(fresh_constant)) -> a_flat(down(fresh_constant))
354.42/147.63	   down(b(a(y10))) -> b_flat(down(a(y10)))
354.42/147.63	   down(b(c)) -> b_flat(down(c))
354.42/147.63	   down(b(b(y11))) -> b_flat(down(b(y11)))
354.42/147.63	   down(b(d(y13))) -> b_flat(down(d(y13)))
354.42/147.63	   down(b(fresh_constant)) -> b_flat(down(fresh_constant))
354.42/147.63	   down(d(c)) -> d_flat(down(c))
354.42/147.63	   down(d(u(y17))) -> d_flat(down(u(y17)))
354.42/147.63	   down(d(d(y18))) -> d_flat(down(d(y18)))
354.42/147.63	   down(d(fresh_constant)) -> d_flat(down(fresh_constant))
354.42/147.63	   d_flat(up(x_1)) -> up(d(x_1))
354.42/147.63	   b_flat(up(x_1)) -> up(b(x_1))
354.42/147.63	   a_flat(up(x_1)) -> up(a(x_1))
354.42/147.63	   u_flat(up(x_1)) -> up(u(x_1))
354.42/147.63	
354.42/147.63	The set Q consists of the following terms:
354.42/147.63	
354.42/147.63	   down(a(a(x0)))
354.42/147.63	   down(b(u(x0)))
354.42/147.63	   down(d(a(x0)))
354.42/147.63	   down(d(b(x0)))
354.42/147.63	   down(a(u(x0)))
354.42/147.63	   down(u(x0))
354.42/147.63	   down(a(c))
354.42/147.63	   down(a(b(x0)))
354.42/147.63	   down(a(d(x0)))
354.42/147.63	   down(a(fresh_constant))
354.42/147.63	   down(b(a(x0)))
354.42/147.63	   down(b(c))
354.42/147.63	   down(b(b(x0)))
354.42/147.63	   down(b(d(x0)))
354.42/147.63	   down(b(fresh_constant))
354.42/147.63	   down(d(c))
354.42/147.63	   down(d(u(x0)))
354.42/147.63	   down(d(d(x0)))
354.42/147.63	   down(d(fresh_constant))
354.42/147.63	   a_flat(up(x0))
354.42/147.63	   b_flat(up(x0))
354.42/147.63	   u_flat(up(x0))
354.42/147.63	   d_flat(up(x0))
354.42/147.63	
354.42/147.63	We have to consider all minimal (P,Q,R)-chains.
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(90) MNOCProof (EQUIVALENT)
354.42/147.63	We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.
354.42/147.63	----------------------------------------
354.42/147.63	
354.42/147.63	(91)
354.42/147.63	Obligation:
354.42/147.63	Q DP problem:
354.42/147.63	The TRS P consists of the following rules:
354.42/147.63	
354.42/147.63	   TOP(up(b(u(x0)))) -> TOP(up(b(d(x0))))
354.42/147.63	   TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0))))
354.42/147.63	   TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0))))
354.42/147.63	   TOP(up(u(x0))) -> TOP(u_flat(down(x0)))
354.42/147.63	   TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0))))
354.42/147.63	   TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0))))
354.42/147.63	   TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0))))
354.42/147.63	   TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0))))
354.42/147.63	   TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0))))
354.42/147.63	
354.42/147.63	The TRS R consists of the following rules:
354.42/147.63	
354.42/147.63	   down(a(a(x))) -> up(c)
354.42/147.63	   down(b(u(x))) -> up(b(d(x)))
354.42/147.63	   down(d(a(x))) -> up(a(d(x)))
354.42/147.63	   down(d(b(x))) -> up(u(a(b(x))))
354.42/147.63	   down(a(u(x))) -> up(u(a(x)))
354.42/147.63	   down(u(y2)) -> u_flat(down(y2))
354.42/147.63	   down(a(c)) -> a_flat(down(c))
354.42/147.63	   down(a(b(y6))) -> a_flat(down(b(y6)))
354.42/147.63	   down(a(d(y8))) -> a_flat(down(d(y8)))
354.42/147.63	   down(a(fresh_constant)) -> a_flat(down(fresh_constant))
354.42/147.63	   down(b(a(y10))) -> b_flat(down(a(y10)))
354.42/147.63	   down(b(c)) -> b_flat(down(c))
354.42/147.63	   down(b(b(y11))) -> b_flat(down(b(y11)))
354.42/147.63	   down(b(d(y13))) -> b_flat(down(d(y13)))
354.42/147.63	   down(b(fresh_constant)) -> b_flat(down(fresh_constant))
354.42/147.63	   down(d(c)) -> d_flat(down(c))
354.42/147.63	   down(d(u(y17))) -> d_flat(down(u(y17)))
354.42/147.64	   down(d(d(y18))) -> d_flat(down(d(y18)))
354.42/147.64	   down(d(fresh_constant)) -> d_flat(down(fresh_constant))
354.42/147.64	   d_flat(up(x_1)) -> up(d(x_1))
354.42/147.64	   b_flat(up(x_1)) -> up(b(x_1))
354.42/147.64	   a_flat(up(x_1)) -> up(a(x_1))
354.42/147.64	   u_flat(up(x_1)) -> up(u(x_1))
354.42/147.64	
354.42/147.64	Q is empty.
354.42/147.64	We have to consider all (P,Q,R)-chains.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(92) SplitQDPProof (EQUIVALENT)
354.42/147.64	We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem
354.42/147.64	
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(93)
354.42/147.64	Complex Obligation (AND)
354.42/147.64	
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(94)
354.42/147.64	Obligation:
354.42/147.64	Q DP problem:
354.42/147.64	The TRS P consists of the following rules:
354.42/147.64	
354.42/147.64	   TOP(up(b(u(x0)))) -> TOP(up(b(d(x0))))
354.42/147.64	   TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0))))
354.42/147.64	   TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0))))
354.42/147.64	   TOP(up(u(x0))) -> TOP(u_flat(down(x0)))
354.42/147.64	   TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0))))
354.42/147.64	   TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0))))
354.42/147.64	   TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0))))
354.42/147.64	   TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0))))
354.42/147.64	   TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0))))
354.42/147.64	
354.42/147.64	The TRS R consists of the following rules:
354.42/147.64	
354.42/147.64	   down(a(a(x))) -> up(c)
354.42/147.64	   down(b(u(x))) -> up(b(d(x)))
354.42/147.64	   down(d(a(x))) -> up(a(d(x)))
354.42/147.64	   down(d(b(x))) -> up(u(a(b(x))))
354.42/147.64	   down(a(u(x))) -> up(u(a(x)))
354.42/147.64	   down(u(y2)) -> u_flat(down(y2))
354.42/147.64	   down(a(c)) -> a_flat(down(c))
354.42/147.64	   down(a(b(y6))) -> a_flat(down(b(y6)))
354.42/147.64	   down(a(d(y8))) -> a_flat(down(d(y8)))
354.42/147.64	   down(a(fresh_constant)) -> a_flat(down(fresh_constant))
354.42/147.64	   down(b(a(y10))) -> b_flat(down(a(y10)))
354.42/147.64	   down(b(c)) -> b_flat(down(c))
354.42/147.64	   down(b(b(y11))) -> b_flat(down(b(y11)))
354.42/147.64	   down(b(d(y13))) -> b_flat(down(d(y13)))
354.42/147.64	   down(b(fresh_constant)) -> b_flat(down(fresh_constant))
354.42/147.64	   down(d(c)) -> d_flat(down(c))
354.42/147.64	   down(d(u(y17))) -> d_flat(down(u(y17)))
354.42/147.64	   down(d(d(y18))) -> d_flat(down(d(y18)))
354.42/147.64	   down(d(fresh_constant)) -> d_flat(down(fresh_constant))
354.42/147.64	   d_flat(up(x_1)) -> up(d(x_1))
354.42/147.64	   b_flat(up(x_1)) -> up(b(x_1))
354.42/147.64	   a_flat(up(x_1)) -> up(a(x_1))
354.42/147.64	   u_flat(up(x_1)) -> up(u(x_1))
354.42/147.64	
354.42/147.64	The set Q consists of the following terms:
354.42/147.64	
354.42/147.64	   down(a(a(x0)))
354.42/147.64	   down(b(u(x0)))
354.42/147.64	   down(d(a(x0)))
354.42/147.64	   down(d(b(x0)))
354.42/147.64	   down(a(u(x0)))
354.42/147.64	   down(u(x0))
354.42/147.64	   down(a(c))
354.42/147.64	   down(a(b(x0)))
354.42/147.64	   down(a(d(x0)))
354.42/147.64	   down(a(fresh_constant))
354.42/147.64	   down(b(a(x0)))
354.42/147.64	   down(b(c))
354.42/147.64	   down(b(b(x0)))
354.42/147.64	   down(b(d(x0)))
354.42/147.64	   down(b(fresh_constant))
354.42/147.64	   down(d(c))
354.42/147.64	   down(d(u(x0)))
354.42/147.64	   down(d(d(x0)))
354.42/147.64	   down(d(fresh_constant))
354.42/147.64	   a_flat(up(x0))
354.42/147.64	   b_flat(up(x0))
354.42/147.64	   u_flat(up(x0))
354.42/147.64	   d_flat(up(x0))
354.42/147.64	
354.42/147.64	We have to consider all minimal (P,Q,R)-chains.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(95) SemLabProof (SOUND)
354.42/147.64	We found the following model for the rules of the TRSs R and P.
354.42/147.64	Interpretation over the domain with elements from 0 to 1.
354.42/147.64	a: 0
354.42/147.64	c: 0
354.42/147.64	TOP: 0
354.42/147.64	u: 0
354.42/147.64	b: 0
354.42/147.64	d: 0
354.42/147.64	down: 0
354.42/147.64	d_flat: 0
354.42/147.64	fresh_constant: 1
354.42/147.64	up: 0
354.42/147.64	u_flat: 0
354.42/147.64	b_flat: 0
354.42/147.64	a_flat: 0
354.42/147.64	By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(96)
354.42/147.64	Obligation:
354.42/147.64	Q DP problem:
354.42/147.64	The TRS P consists of the following rules:
354.42/147.64	
354.42/147.64	   TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(d.1(x0)))) -> TOP.0(b_flat.0(down.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0)))
354.42/147.64	   TOP.0(up.0(u.1(x0))) -> TOP.0(u_flat.0(down.1(x0)))
354.42/147.64	   TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0))))
354.42/147.64	   TOP.0(up.0(a.0(b.1(x0)))) -> TOP.0(a_flat.0(down.0(b.1(x0))))
354.42/147.64	   TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(a.0(d.1(x0)))) -> TOP.0(a_flat.0(down.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(a.1(x0)))) -> TOP.0(b_flat.0(down.0(a.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0))))
354.42/147.64	   TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(d.0(d.1(x0)))) -> TOP.0(d_flat.0(down.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0))))
354.42/147.64	   TOP.0(up.0(d.0(u.1(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.1(x0))))
354.42/147.64	
354.42/147.64	The TRS R consists of the following rules:
354.42/147.64	
354.42/147.64	   down.0(a.0(a.0(x))) -> up.0(c.)
354.42/147.64	   down.0(a.0(a.1(x))) -> up.0(c.)
354.42/147.64	   down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x)))
354.42/147.64	   down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x)))
354.42/147.64	   down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x)))
354.42/147.64	   down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x)))
354.42/147.64	   down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x))))
354.42/147.64	   down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x))))
354.42/147.64	   down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x)))
354.42/147.64	   down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x)))
354.42/147.64	   down.0(u.0(y2)) -> u_flat.0(down.0(y2))
354.42/147.64	   down.0(u.1(y2)) -> u_flat.0(down.1(y2))
354.42/147.64	   down.0(a.0(c.)) -> a_flat.0(down.0(c.))
354.42/147.64	   down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6)))
354.42/147.64	   down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6)))
354.42/147.64	   down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8)))
354.42/147.64	   down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8)))
354.42/147.64	   down.0(a.1(fresh_constant.)) -> a_flat.0(down.1(fresh_constant.))
354.42/147.64	   down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10)))
354.42/147.64	   down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10)))
354.42/147.64	   down.0(b.0(c.)) -> b_flat.0(down.0(c.))
354.42/147.64	   down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11)))
354.42/147.64	   down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11)))
354.42/147.64	   down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13)))
354.42/147.64	   down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13)))
354.42/147.64	   down.0(b.1(fresh_constant.)) -> b_flat.0(down.1(fresh_constant.))
354.42/147.64	   down.0(d.0(c.)) -> d_flat.0(down.0(c.))
354.42/147.64	   down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17)))
354.42/147.64	   down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17)))
354.42/147.64	   down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18)))
354.42/147.64	   down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18)))
354.42/147.64	   down.0(d.1(fresh_constant.)) -> d_flat.0(down.1(fresh_constant.))
354.42/147.64	   d_flat.0(up.0(x_1)) -> up.0(d.0(x_1))
354.42/147.64	   d_flat.0(up.1(x_1)) -> up.0(d.1(x_1))
354.42/147.64	   b_flat.0(up.0(x_1)) -> up.0(b.0(x_1))
354.42/147.64	   b_flat.0(up.1(x_1)) -> up.0(b.1(x_1))
354.42/147.64	   a_flat.0(up.0(x_1)) -> up.0(a.0(x_1))
354.42/147.64	   a_flat.0(up.1(x_1)) -> up.0(a.1(x_1))
354.42/147.64	   u_flat.0(up.0(x_1)) -> up.0(u.0(x_1))
354.42/147.64	   u_flat.0(up.1(x_1)) -> up.0(u.1(x_1))
354.42/147.64	
354.42/147.64	The set Q consists of the following terms:
354.42/147.64	
354.42/147.64	   down.0(a.0(a.0(x0)))
354.42/147.64	   down.0(a.0(a.1(x0)))
354.42/147.64	   down.0(b.0(u.0(x0)))
354.42/147.64	   down.0(b.0(u.1(x0)))
354.42/147.64	   down.0(d.0(a.0(x0)))
354.42/147.64	   down.0(d.0(a.1(x0)))
354.42/147.64	   down.0(d.0(b.0(x0)))
354.42/147.64	   down.0(d.0(b.1(x0)))
354.42/147.64	   down.0(a.0(u.0(x0)))
354.42/147.64	   down.0(a.0(u.1(x0)))
354.42/147.64	   down.0(u.0(x0))
354.42/147.64	   down.0(u.1(x0))
354.42/147.64	   down.0(a.0(c.))
354.42/147.64	   down.0(a.0(b.0(x0)))
354.42/147.64	   down.0(a.0(b.1(x0)))
354.42/147.64	   down.0(a.0(d.0(x0)))
354.42/147.64	   down.0(a.0(d.1(x0)))
354.42/147.64	   down.0(a.1(fresh_constant.))
354.42/147.64	   down.0(b.0(a.0(x0)))
354.42/147.64	   down.0(b.0(a.1(x0)))
354.42/147.64	   down.0(b.0(c.))
354.42/147.64	   down.0(b.0(b.0(x0)))
354.42/147.64	   down.0(b.0(b.1(x0)))
354.42/147.64	   down.0(b.0(d.0(x0)))
354.42/147.64	   down.0(b.0(d.1(x0)))
354.42/147.64	   down.0(b.1(fresh_constant.))
354.42/147.64	   down.0(d.0(c.))
354.42/147.64	   down.0(d.0(u.0(x0)))
354.42/147.64	   down.0(d.0(u.1(x0)))
354.42/147.64	   down.0(d.0(d.0(x0)))
354.42/147.64	   down.0(d.0(d.1(x0)))
354.42/147.64	   down.0(d.1(fresh_constant.))
354.42/147.64	   a_flat.0(up.0(x0))
354.42/147.64	   a_flat.0(up.1(x0))
354.42/147.64	   b_flat.0(up.0(x0))
354.42/147.64	   b_flat.0(up.1(x0))
354.42/147.64	   u_flat.0(up.0(x0))
354.42/147.64	   u_flat.0(up.1(x0))
354.42/147.64	   d_flat.0(up.0(x0))
354.42/147.64	   d_flat.0(up.1(x0))
354.42/147.64	
354.42/147.64	We have to consider all minimal (P,Q,R)-chains.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(97) DependencyGraphProof (EQUIVALENT)
354.42/147.64	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(98)
354.42/147.64	Obligation:
354.42/147.64	Q DP problem:
354.42/147.64	The TRS P consists of the following rules:
354.42/147.64	
354.42/147.64	   TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(d.1(x0)))) -> TOP.0(b_flat.0(down.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0))))
354.42/147.64	   TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0)))
354.42/147.64	   TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0))))
354.42/147.64	   TOP.0(up.0(a.0(b.1(x0)))) -> TOP.0(a_flat.0(down.0(b.1(x0))))
354.42/147.64	   TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(a.0(d.1(x0)))) -> TOP.0(a_flat.0(down.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(a.1(x0)))) -> TOP.0(b_flat.0(down.0(a.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0))))
354.42/147.64	   TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(d.0(d.1(x0)))) -> TOP.0(d_flat.0(down.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0))))
354.42/147.64	
354.42/147.64	The TRS R consists of the following rules:
354.42/147.64	
354.42/147.64	   down.0(a.0(a.0(x))) -> up.0(c.)
354.42/147.64	   down.0(a.0(a.1(x))) -> up.0(c.)
354.42/147.64	   down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x)))
354.42/147.64	   down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x)))
354.42/147.64	   down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x)))
354.42/147.64	   down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x)))
354.42/147.64	   down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x))))
354.42/147.64	   down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x))))
354.42/147.64	   down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x)))
354.42/147.64	   down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x)))
354.42/147.64	   down.0(u.0(y2)) -> u_flat.0(down.0(y2))
354.42/147.64	   down.0(u.1(y2)) -> u_flat.0(down.1(y2))
354.42/147.64	   down.0(a.0(c.)) -> a_flat.0(down.0(c.))
354.42/147.64	   down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6)))
354.42/147.64	   down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6)))
354.42/147.64	   down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8)))
354.42/147.64	   down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8)))
354.42/147.64	   down.0(a.1(fresh_constant.)) -> a_flat.0(down.1(fresh_constant.))
354.42/147.64	   down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10)))
354.42/147.64	   down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10)))
354.42/147.64	   down.0(b.0(c.)) -> b_flat.0(down.0(c.))
354.42/147.64	   down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11)))
354.42/147.64	   down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11)))
354.42/147.64	   down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13)))
354.42/147.64	   down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13)))
354.42/147.64	   down.0(b.1(fresh_constant.)) -> b_flat.0(down.1(fresh_constant.))
354.42/147.64	   down.0(d.0(c.)) -> d_flat.0(down.0(c.))
354.42/147.64	   down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17)))
354.42/147.64	   down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17)))
354.42/147.64	   down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18)))
354.42/147.64	   down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18)))
354.42/147.64	   down.0(d.1(fresh_constant.)) -> d_flat.0(down.1(fresh_constant.))
354.42/147.64	   d_flat.0(up.0(x_1)) -> up.0(d.0(x_1))
354.42/147.64	   d_flat.0(up.1(x_1)) -> up.0(d.1(x_1))
354.42/147.64	   b_flat.0(up.0(x_1)) -> up.0(b.0(x_1))
354.42/147.64	   b_flat.0(up.1(x_1)) -> up.0(b.1(x_1))
354.42/147.64	   a_flat.0(up.0(x_1)) -> up.0(a.0(x_1))
354.42/147.64	   a_flat.0(up.1(x_1)) -> up.0(a.1(x_1))
354.42/147.64	   u_flat.0(up.0(x_1)) -> up.0(u.0(x_1))
354.42/147.64	   u_flat.0(up.1(x_1)) -> up.0(u.1(x_1))
354.42/147.64	
354.42/147.64	The set Q consists of the following terms:
354.42/147.64	
354.42/147.64	   down.0(a.0(a.0(x0)))
354.42/147.64	   down.0(a.0(a.1(x0)))
354.42/147.64	   down.0(b.0(u.0(x0)))
354.42/147.64	   down.0(b.0(u.1(x0)))
354.42/147.64	   down.0(d.0(a.0(x0)))
354.42/147.64	   down.0(d.0(a.1(x0)))
354.42/147.64	   down.0(d.0(b.0(x0)))
354.42/147.64	   down.0(d.0(b.1(x0)))
354.42/147.64	   down.0(a.0(u.0(x0)))
354.42/147.64	   down.0(a.0(u.1(x0)))
354.42/147.64	   down.0(u.0(x0))
354.42/147.64	   down.0(u.1(x0))
354.42/147.64	   down.0(a.0(c.))
354.42/147.64	   down.0(a.0(b.0(x0)))
354.42/147.64	   down.0(a.0(b.1(x0)))
354.42/147.64	   down.0(a.0(d.0(x0)))
354.42/147.64	   down.0(a.0(d.1(x0)))
354.42/147.64	   down.0(a.1(fresh_constant.))
354.42/147.64	   down.0(b.0(a.0(x0)))
354.42/147.64	   down.0(b.0(a.1(x0)))
354.42/147.64	   down.0(b.0(c.))
354.42/147.64	   down.0(b.0(b.0(x0)))
354.42/147.64	   down.0(b.0(b.1(x0)))
354.42/147.64	   down.0(b.0(d.0(x0)))
354.42/147.64	   down.0(b.0(d.1(x0)))
354.42/147.64	   down.0(b.1(fresh_constant.))
354.42/147.64	   down.0(d.0(c.))
354.42/147.64	   down.0(d.0(u.0(x0)))
354.42/147.64	   down.0(d.0(u.1(x0)))
354.42/147.64	   down.0(d.0(d.0(x0)))
354.42/147.64	   down.0(d.0(d.1(x0)))
354.42/147.64	   down.0(d.1(fresh_constant.))
354.42/147.64	   a_flat.0(up.0(x0))
354.42/147.64	   a_flat.0(up.1(x0))
354.42/147.64	   b_flat.0(up.0(x0))
354.42/147.64	   b_flat.0(up.1(x0))
354.42/147.64	   u_flat.0(up.0(x0))
354.42/147.64	   u_flat.0(up.1(x0))
354.42/147.64	   d_flat.0(up.0(x0))
354.42/147.64	   d_flat.0(up.1(x0))
354.42/147.64	
354.42/147.64	We have to consider all minimal (P,Q,R)-chains.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(99) UsableRulesReductionPairsProof (EQUIVALENT)
354.42/147.64	By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
354.42/147.64	
354.42/147.64	No dependency pairs are removed.
354.42/147.64	
354.42/147.64	The following rules are removed from R:
354.42/147.64	
354.42/147.64	   d_flat.0(up.1(x_1)) -> up.0(d.1(x_1))
354.42/147.64	   b_flat.0(up.1(x_1)) -> up.0(b.1(x_1))
354.42/147.64	   a_flat.0(up.1(x_1)) -> up.0(a.1(x_1))
354.42/147.64	   u_flat.0(up.1(x_1)) -> up.0(u.1(x_1))
354.42/147.64	Used ordering: POLO with Polynomial interpretation [POLO]:
354.42/147.64	
354.42/147.64	   POL(TOP.0(x_1)) = x_1
354.42/147.64	   POL(a.0(x_1)) = x_1
354.42/147.64	   POL(a.1(x_1)) = x_1
354.42/147.64	   POL(a_flat.0(x_1)) = x_1
354.42/147.64	   POL(b.0(x_1)) = 1 + x_1
354.42/147.64	   POL(b.1(x_1)) = 1 + x_1
354.42/147.64	   POL(b_flat.0(x_1)) = 1 + x_1
354.42/147.64	   POL(c.) = 0
354.42/147.64	   POL(d.0(x_1)) = 1 + x_1
354.42/147.64	   POL(d.1(x_1)) = 1 + x_1
354.42/147.64	   POL(d_flat.0(x_1)) = 1 + x_1
354.42/147.64	   POL(down.0(x_1)) = x_1
354.42/147.64	   POL(down.1(x_1)) = x_1
354.42/147.64	   POL(fresh_constant.) = 0
354.42/147.64	   POL(u.0(x_1)) = 1 + x_1
354.42/147.64	   POL(u.1(x_1)) = 1 + x_1
354.42/147.64	   POL(u_flat.0(x_1)) = 1 + x_1
354.42/147.64	   POL(up.0(x_1)) = x_1
354.42/147.64	   POL(up.1(x_1)) = 1 + x_1
354.42/147.64	
354.42/147.64	
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(100)
354.42/147.64	Obligation:
354.42/147.64	Q DP problem:
354.42/147.64	The TRS P consists of the following rules:
354.42/147.64	
354.42/147.64	   TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(d.1(x0)))) -> TOP.0(b_flat.0(down.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0))))
354.42/147.64	   TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0)))
354.42/147.64	   TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0))))
354.42/147.64	   TOP.0(up.0(a.0(b.1(x0)))) -> TOP.0(a_flat.0(down.0(b.1(x0))))
354.42/147.64	   TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(a.0(d.1(x0)))) -> TOP.0(a_flat.0(down.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(a.1(x0)))) -> TOP.0(b_flat.0(down.0(a.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0))))
354.42/147.64	   TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(d.0(d.1(x0)))) -> TOP.0(d_flat.0(down.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0))))
354.42/147.64	
354.42/147.64	The TRS R consists of the following rules:
354.42/147.64	
354.42/147.64	   down.0(a.0(a.0(x))) -> up.0(c.)
354.42/147.64	   down.0(a.0(a.1(x))) -> up.0(c.)
354.42/147.64	   down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x)))
354.42/147.64	   down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x)))
354.42/147.64	   down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x)))
354.42/147.64	   down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x)))
354.42/147.64	   down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x))))
354.42/147.64	   down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x))))
354.42/147.64	   down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x)))
354.42/147.64	   down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x)))
354.42/147.64	   down.0(u.0(y2)) -> u_flat.0(down.0(y2))
354.42/147.64	   down.0(u.1(y2)) -> u_flat.0(down.1(y2))
354.42/147.64	   down.0(a.0(c.)) -> a_flat.0(down.0(c.))
354.42/147.64	   down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6)))
354.42/147.64	   down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6)))
354.42/147.64	   down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8)))
354.42/147.64	   down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8)))
354.42/147.64	   down.0(a.1(fresh_constant.)) -> a_flat.0(down.1(fresh_constant.))
354.42/147.64	   down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10)))
354.42/147.64	   down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10)))
354.42/147.64	   down.0(b.0(c.)) -> b_flat.0(down.0(c.))
354.42/147.64	   down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11)))
354.42/147.64	   down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11)))
354.42/147.64	   down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13)))
354.42/147.64	   down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13)))
354.42/147.64	   down.0(b.1(fresh_constant.)) -> b_flat.0(down.1(fresh_constant.))
354.42/147.64	   down.0(d.0(c.)) -> d_flat.0(down.0(c.))
354.42/147.64	   down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17)))
354.42/147.64	   down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17)))
354.42/147.64	   down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18)))
354.42/147.64	   down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18)))
354.42/147.64	   down.0(d.1(fresh_constant.)) -> d_flat.0(down.1(fresh_constant.))
354.42/147.64	   u_flat.0(up.0(x_1)) -> up.0(u.0(x_1))
354.42/147.64	   d_flat.0(up.0(x_1)) -> up.0(d.0(x_1))
354.42/147.64	   b_flat.0(up.0(x_1)) -> up.0(b.0(x_1))
354.42/147.64	   a_flat.0(up.0(x_1)) -> up.0(a.0(x_1))
354.42/147.64	
354.42/147.64	The set Q consists of the following terms:
354.42/147.64	
354.42/147.64	   down.0(a.0(a.0(x0)))
354.42/147.64	   down.0(a.0(a.1(x0)))
354.42/147.64	   down.0(b.0(u.0(x0)))
354.42/147.64	   down.0(b.0(u.1(x0)))
354.42/147.64	   down.0(d.0(a.0(x0)))
354.42/147.64	   down.0(d.0(a.1(x0)))
354.42/147.64	   down.0(d.0(b.0(x0)))
354.42/147.64	   down.0(d.0(b.1(x0)))
354.42/147.64	   down.0(a.0(u.0(x0)))
354.42/147.64	   down.0(a.0(u.1(x0)))
354.42/147.64	   down.0(u.0(x0))
354.42/147.64	   down.0(u.1(x0))
354.42/147.64	   down.0(a.0(c.))
354.42/147.64	   down.0(a.0(b.0(x0)))
354.42/147.64	   down.0(a.0(b.1(x0)))
354.42/147.64	   down.0(a.0(d.0(x0)))
354.42/147.64	   down.0(a.0(d.1(x0)))
354.42/147.64	   down.0(a.1(fresh_constant.))
354.42/147.64	   down.0(b.0(a.0(x0)))
354.42/147.64	   down.0(b.0(a.1(x0)))
354.42/147.64	   down.0(b.0(c.))
354.42/147.64	   down.0(b.0(b.0(x0)))
354.42/147.64	   down.0(b.0(b.1(x0)))
354.42/147.64	   down.0(b.0(d.0(x0)))
354.42/147.64	   down.0(b.0(d.1(x0)))
354.42/147.64	   down.0(b.1(fresh_constant.))
354.42/147.64	   down.0(d.0(c.))
354.42/147.64	   down.0(d.0(u.0(x0)))
354.42/147.64	   down.0(d.0(u.1(x0)))
354.42/147.64	   down.0(d.0(d.0(x0)))
354.42/147.64	   down.0(d.0(d.1(x0)))
354.42/147.64	   down.0(d.1(fresh_constant.))
354.42/147.64	   a_flat.0(up.0(x0))
354.42/147.64	   a_flat.0(up.1(x0))
354.42/147.64	   b_flat.0(up.0(x0))
354.42/147.64	   b_flat.0(up.1(x0))
354.42/147.64	   u_flat.0(up.0(x0))
354.42/147.64	   u_flat.0(up.1(x0))
354.42/147.64	   d_flat.0(up.0(x0))
354.42/147.64	   d_flat.0(up.1(x0))
354.42/147.64	
354.42/147.64	We have to consider all minimal (P,Q,R)-chains.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(101) MRRProof (EQUIVALENT)
354.42/147.64	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
354.42/147.64	
354.42/147.64	
354.42/147.64	Strictly oriented rules of the TRS R:
354.42/147.64	
354.42/147.64	   down.0(u.1(y2)) -> u_flat.0(down.1(y2))
354.42/147.64	   down.0(a.1(fresh_constant.)) -> a_flat.0(down.1(fresh_constant.))
354.42/147.64	   down.0(b.1(fresh_constant.)) -> b_flat.0(down.1(fresh_constant.))
354.42/147.64	   down.0(d.1(fresh_constant.)) -> d_flat.0(down.1(fresh_constant.))
354.42/147.64	
354.42/147.64	Used ordering: Polynomial interpretation [POLO]:
354.42/147.64	
354.42/147.64	   POL(TOP.0(x_1)) = x_1
354.42/147.64	   POL(a.0(x_1)) = x_1
354.42/147.64	   POL(a.1(x_1)) = x_1
354.42/147.64	   POL(a_flat.0(x_1)) = x_1
354.42/147.64	   POL(b.0(x_1)) = 1 + x_1
354.42/147.64	   POL(b.1(x_1)) = 1 + x_1
354.42/147.64	   POL(b_flat.0(x_1)) = 1 + x_1
354.42/147.64	   POL(c.) = 0
354.42/147.64	   POL(d.0(x_1)) = x_1
354.42/147.64	   POL(d.1(x_1)) = x_1
354.42/147.64	   POL(d_flat.0(x_1)) = x_1
354.42/147.64	   POL(down.0(x_1)) = 1 + x_1
354.42/147.64	   POL(down.1(x_1)) = x_1
354.42/147.64	   POL(fresh_constant.) = 0
354.42/147.64	   POL(u.0(x_1)) = x_1
354.42/147.64	   POL(u.1(x_1)) = x_1
354.42/147.64	   POL(u_flat.0(x_1)) = x_1
354.42/147.64	   POL(up.0(x_1)) = 1 + x_1
354.42/147.64	
354.42/147.64	
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(102)
354.42/147.64	Obligation:
354.42/147.64	Q DP problem:
354.42/147.64	The TRS P consists of the following rules:
354.42/147.64	
354.42/147.64	   TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(d.1(x0)))) -> TOP.0(b_flat.0(down.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0))))
354.42/147.64	   TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0)))
354.42/147.64	   TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0))))
354.42/147.64	   TOP.0(up.0(a.0(b.1(x0)))) -> TOP.0(a_flat.0(down.0(b.1(x0))))
354.42/147.64	   TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(a.0(d.1(x0)))) -> TOP.0(a_flat.0(down.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(a.1(x0)))) -> TOP.0(b_flat.0(down.0(a.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0))))
354.42/147.64	   TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(d.0(d.1(x0)))) -> TOP.0(d_flat.0(down.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0))))
354.42/147.64	
354.42/147.64	The TRS R consists of the following rules:
354.42/147.64	
354.42/147.64	   down.0(a.0(a.0(x))) -> up.0(c.)
354.42/147.64	   down.0(a.0(a.1(x))) -> up.0(c.)
354.42/147.64	   down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x)))
354.42/147.64	   down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x)))
354.42/147.64	   down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x)))
354.42/147.64	   down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x)))
354.42/147.64	   down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x))))
354.42/147.64	   down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x))))
354.42/147.64	   down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x)))
354.42/147.64	   down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x)))
354.42/147.64	   down.0(u.0(y2)) -> u_flat.0(down.0(y2))
354.42/147.64	   down.0(a.0(c.)) -> a_flat.0(down.0(c.))
354.42/147.64	   down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6)))
354.42/147.64	   down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6)))
354.42/147.64	   down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8)))
354.42/147.64	   down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8)))
354.42/147.64	   down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10)))
354.42/147.64	   down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10)))
354.42/147.64	   down.0(b.0(c.)) -> b_flat.0(down.0(c.))
354.42/147.64	   down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11)))
354.42/147.64	   down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11)))
354.42/147.64	   down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13)))
354.42/147.64	   down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13)))
354.42/147.64	   down.0(d.0(c.)) -> d_flat.0(down.0(c.))
354.42/147.64	   down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17)))
354.42/147.64	   down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17)))
354.42/147.64	   down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18)))
354.42/147.64	   down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18)))
354.42/147.64	   u_flat.0(up.0(x_1)) -> up.0(u.0(x_1))
354.42/147.64	   d_flat.0(up.0(x_1)) -> up.0(d.0(x_1))
354.42/147.64	   b_flat.0(up.0(x_1)) -> up.0(b.0(x_1))
354.42/147.64	   a_flat.0(up.0(x_1)) -> up.0(a.0(x_1))
354.42/147.64	
354.42/147.64	The set Q consists of the following terms:
354.42/147.64	
354.42/147.64	   down.0(a.0(a.0(x0)))
354.42/147.64	   down.0(a.0(a.1(x0)))
354.42/147.64	   down.0(b.0(u.0(x0)))
354.42/147.64	   down.0(b.0(u.1(x0)))
354.42/147.64	   down.0(d.0(a.0(x0)))
354.42/147.64	   down.0(d.0(a.1(x0)))
354.42/147.64	   down.0(d.0(b.0(x0)))
354.42/147.64	   down.0(d.0(b.1(x0)))
354.42/147.64	   down.0(a.0(u.0(x0)))
354.42/147.64	   down.0(a.0(u.1(x0)))
354.42/147.64	   down.0(u.0(x0))
354.42/147.64	   down.0(u.1(x0))
354.42/147.64	   down.0(a.0(c.))
354.42/147.64	   down.0(a.0(b.0(x0)))
354.42/147.64	   down.0(a.0(b.1(x0)))
354.42/147.64	   down.0(a.0(d.0(x0)))
354.42/147.64	   down.0(a.0(d.1(x0)))
354.42/147.64	   down.0(a.1(fresh_constant.))
354.42/147.64	   down.0(b.0(a.0(x0)))
354.42/147.64	   down.0(b.0(a.1(x0)))
354.42/147.64	   down.0(b.0(c.))
354.42/147.64	   down.0(b.0(b.0(x0)))
354.42/147.64	   down.0(b.0(b.1(x0)))
354.42/147.64	   down.0(b.0(d.0(x0)))
354.42/147.64	   down.0(b.0(d.1(x0)))
354.42/147.64	   down.0(b.1(fresh_constant.))
354.42/147.64	   down.0(d.0(c.))
354.42/147.64	   down.0(d.0(u.0(x0)))
354.42/147.64	   down.0(d.0(u.1(x0)))
354.42/147.64	   down.0(d.0(d.0(x0)))
354.42/147.64	   down.0(d.0(d.1(x0)))
354.42/147.64	   down.0(d.1(fresh_constant.))
354.42/147.64	   a_flat.0(up.0(x0))
354.42/147.64	   a_flat.0(up.1(x0))
354.42/147.64	   b_flat.0(up.0(x0))
354.42/147.64	   b_flat.0(up.1(x0))
354.42/147.64	   u_flat.0(up.0(x0))
354.42/147.64	   u_flat.0(up.1(x0))
354.42/147.64	   d_flat.0(up.0(x0))
354.42/147.64	   d_flat.0(up.1(x0))
354.42/147.64	
354.42/147.64	We have to consider all minimal (P,Q,R)-chains.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(103) DependencyGraphProof (EQUIVALENT)
354.42/147.64	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(104)
354.42/147.64	Obligation:
354.42/147.64	Q DP problem:
354.42/147.64	The TRS P consists of the following rules:
354.42/147.64	
354.42/147.64	   TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0)))
354.42/147.64	   TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0))))
354.42/147.64	   TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0))))
354.42/147.64	   TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0))))
354.42/147.64	
354.42/147.64	The TRS R consists of the following rules:
354.42/147.64	
354.42/147.64	   down.0(a.0(a.0(x))) -> up.0(c.)
354.42/147.64	   down.0(a.0(a.1(x))) -> up.0(c.)
354.42/147.64	   down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x)))
354.42/147.64	   down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x)))
354.42/147.64	   down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x)))
354.42/147.64	   down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x)))
354.42/147.64	   down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x))))
354.42/147.64	   down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x))))
354.42/147.64	   down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x)))
354.42/147.64	   down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x)))
354.42/147.64	   down.0(u.0(y2)) -> u_flat.0(down.0(y2))
354.42/147.64	   down.0(a.0(c.)) -> a_flat.0(down.0(c.))
354.42/147.64	   down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6)))
354.42/147.64	   down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6)))
354.42/147.64	   down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8)))
354.42/147.64	   down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8)))
354.42/147.64	   down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10)))
354.42/147.64	   down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10)))
354.42/147.64	   down.0(b.0(c.)) -> b_flat.0(down.0(c.))
354.42/147.64	   down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11)))
354.42/147.64	   down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11)))
354.42/147.64	   down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13)))
354.42/147.64	   down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13)))
354.42/147.64	   down.0(d.0(c.)) -> d_flat.0(down.0(c.))
354.42/147.64	   down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17)))
354.42/147.64	   down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17)))
354.42/147.64	   down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18)))
354.42/147.64	   down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18)))
354.42/147.64	   u_flat.0(up.0(x_1)) -> up.0(u.0(x_1))
354.42/147.64	   d_flat.0(up.0(x_1)) -> up.0(d.0(x_1))
354.42/147.64	   b_flat.0(up.0(x_1)) -> up.0(b.0(x_1))
354.42/147.64	   a_flat.0(up.0(x_1)) -> up.0(a.0(x_1))
354.42/147.64	
354.42/147.64	The set Q consists of the following terms:
354.42/147.64	
354.42/147.64	   down.0(a.0(a.0(x0)))
354.42/147.64	   down.0(a.0(a.1(x0)))
354.42/147.64	   down.0(b.0(u.0(x0)))
354.42/147.64	   down.0(b.0(u.1(x0)))
354.42/147.64	   down.0(d.0(a.0(x0)))
354.42/147.64	   down.0(d.0(a.1(x0)))
354.42/147.64	   down.0(d.0(b.0(x0)))
354.42/147.64	   down.0(d.0(b.1(x0)))
354.42/147.64	   down.0(a.0(u.0(x0)))
354.42/147.64	   down.0(a.0(u.1(x0)))
354.42/147.64	   down.0(u.0(x0))
354.42/147.64	   down.0(u.1(x0))
354.42/147.64	   down.0(a.0(c.))
354.42/147.64	   down.0(a.0(b.0(x0)))
354.42/147.64	   down.0(a.0(b.1(x0)))
354.42/147.64	   down.0(a.0(d.0(x0)))
354.42/147.64	   down.0(a.0(d.1(x0)))
354.42/147.64	   down.0(a.1(fresh_constant.))
354.42/147.64	   down.0(b.0(a.0(x0)))
354.42/147.64	   down.0(b.0(a.1(x0)))
354.42/147.64	   down.0(b.0(c.))
354.42/147.64	   down.0(b.0(b.0(x0)))
354.42/147.64	   down.0(b.0(b.1(x0)))
354.42/147.64	   down.0(b.0(d.0(x0)))
354.42/147.64	   down.0(b.0(d.1(x0)))
354.42/147.64	   down.0(b.1(fresh_constant.))
354.42/147.64	   down.0(d.0(c.))
354.42/147.64	   down.0(d.0(u.0(x0)))
354.42/147.64	   down.0(d.0(u.1(x0)))
354.42/147.64	   down.0(d.0(d.0(x0)))
354.42/147.64	   down.0(d.0(d.1(x0)))
354.42/147.64	   down.0(d.1(fresh_constant.))
354.42/147.64	   a_flat.0(up.0(x0))
354.42/147.64	   a_flat.0(up.1(x0))
354.42/147.64	   b_flat.0(up.0(x0))
354.42/147.64	   b_flat.0(up.1(x0))
354.42/147.64	   u_flat.0(up.0(x0))
354.42/147.64	   u_flat.0(up.1(x0))
354.42/147.64	   d_flat.0(up.0(x0))
354.42/147.64	   d_flat.0(up.1(x0))
354.42/147.64	
354.42/147.64	We have to consider all minimal (P,Q,R)-chains.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(105) MRRProof (EQUIVALENT)
354.42/147.64	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
354.42/147.64	
354.42/147.64	
354.42/147.64	Strictly oriented rules of the TRS R:
354.42/147.64	
354.42/147.64	   down.0(a.0(a.1(x))) -> up.0(c.)
354.42/147.64	   down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x)))
354.42/147.64	   down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x)))
354.42/147.64	
354.42/147.64	Used ordering: Polynomial interpretation [POLO]:
354.42/147.64	
354.42/147.64	   POL(TOP.0(x_1)) = x_1
354.42/147.64	   POL(a.0(x_1)) = x_1
354.42/147.64	   POL(a.1(x_1)) = 1 + x_1
354.42/147.64	   POL(a_flat.0(x_1)) = x_1
354.42/147.64	   POL(b.0(x_1)) = x_1
354.42/147.64	   POL(b.1(x_1)) = x_1
354.42/147.64	   POL(b_flat.0(x_1)) = x_1
354.42/147.64	   POL(c.) = 0
354.42/147.64	   POL(d.0(x_1)) = x_1
354.42/147.64	   POL(d.1(x_1)) = x_1
354.42/147.64	   POL(d_flat.0(x_1)) = x_1
354.42/147.64	   POL(down.0(x_1)) = x_1
354.42/147.64	   POL(u.0(x_1)) = x_1
354.42/147.64	   POL(u.1(x_1)) = 1 + x_1
354.42/147.64	   POL(u_flat.0(x_1)) = x_1
354.42/147.64	   POL(up.0(x_1)) = x_1
354.42/147.64	
354.42/147.64	
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(106)
354.42/147.64	Obligation:
354.42/147.64	Q DP problem:
354.42/147.64	The TRS P consists of the following rules:
354.42/147.64	
354.42/147.64	   TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0)))
354.42/147.64	   TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0))))
354.42/147.64	   TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0))))
354.42/147.64	   TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0))))
354.42/147.64	
354.42/147.64	The TRS R consists of the following rules:
354.42/147.64	
354.42/147.64	   down.0(a.0(a.0(x))) -> up.0(c.)
354.42/147.64	   down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x)))
354.42/147.64	   down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x)))
354.42/147.64	   down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x))))
354.42/147.64	   down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x))))
354.42/147.64	   down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x)))
354.42/147.64	   down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x)))
354.42/147.64	   down.0(u.0(y2)) -> u_flat.0(down.0(y2))
354.42/147.64	   down.0(a.0(c.)) -> a_flat.0(down.0(c.))
354.42/147.64	   down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6)))
354.42/147.64	   down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6)))
354.42/147.64	   down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8)))
354.42/147.64	   down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8)))
354.42/147.64	   down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10)))
354.42/147.64	   down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10)))
354.42/147.64	   down.0(b.0(c.)) -> b_flat.0(down.0(c.))
354.42/147.64	   down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11)))
354.42/147.64	   down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11)))
354.42/147.64	   down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13)))
354.42/147.64	   down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13)))
354.42/147.64	   down.0(d.0(c.)) -> d_flat.0(down.0(c.))
354.42/147.64	   down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17)))
354.42/147.64	   down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17)))
354.42/147.64	   down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18)))
354.42/147.64	   down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18)))
354.42/147.64	   u_flat.0(up.0(x_1)) -> up.0(u.0(x_1))
354.42/147.64	   d_flat.0(up.0(x_1)) -> up.0(d.0(x_1))
354.42/147.64	   b_flat.0(up.0(x_1)) -> up.0(b.0(x_1))
354.42/147.64	   a_flat.0(up.0(x_1)) -> up.0(a.0(x_1))
354.42/147.64	
354.42/147.64	The set Q consists of the following terms:
354.42/147.64	
354.42/147.64	   down.0(a.0(a.0(x0)))
354.42/147.64	   down.0(a.0(a.1(x0)))
354.42/147.64	   down.0(b.0(u.0(x0)))
354.42/147.64	   down.0(b.0(u.1(x0)))
354.42/147.64	   down.0(d.0(a.0(x0)))
354.42/147.64	   down.0(d.0(a.1(x0)))
354.42/147.64	   down.0(d.0(b.0(x0)))
354.42/147.64	   down.0(d.0(b.1(x0)))
354.42/147.64	   down.0(a.0(u.0(x0)))
354.42/147.64	   down.0(a.0(u.1(x0)))
354.42/147.64	   down.0(u.0(x0))
354.42/147.64	   down.0(u.1(x0))
354.42/147.64	   down.0(a.0(c.))
354.42/147.64	   down.0(a.0(b.0(x0)))
354.42/147.64	   down.0(a.0(b.1(x0)))
354.42/147.64	   down.0(a.0(d.0(x0)))
354.42/147.64	   down.0(a.0(d.1(x0)))
354.42/147.64	   down.0(a.1(fresh_constant.))
354.42/147.64	   down.0(b.0(a.0(x0)))
354.42/147.64	   down.0(b.0(a.1(x0)))
354.42/147.64	   down.0(b.0(c.))
354.42/147.64	   down.0(b.0(b.0(x0)))
354.42/147.64	   down.0(b.0(b.1(x0)))
354.42/147.64	   down.0(b.0(d.0(x0)))
354.42/147.64	   down.0(b.0(d.1(x0)))
354.42/147.64	   down.0(b.1(fresh_constant.))
354.42/147.64	   down.0(d.0(c.))
354.42/147.64	   down.0(d.0(u.0(x0)))
354.42/147.64	   down.0(d.0(u.1(x0)))
354.42/147.64	   down.0(d.0(d.0(x0)))
354.42/147.64	   down.0(d.0(d.1(x0)))
354.42/147.64	   down.0(d.1(fresh_constant.))
354.42/147.64	   a_flat.0(up.0(x0))
354.42/147.64	   a_flat.0(up.1(x0))
354.42/147.64	   b_flat.0(up.0(x0))
354.42/147.64	   b_flat.0(up.1(x0))
354.42/147.64	   u_flat.0(up.0(x0))
354.42/147.64	   u_flat.0(up.1(x0))
354.42/147.64	   d_flat.0(up.0(x0))
354.42/147.64	   d_flat.0(up.1(x0))
354.42/147.64	
354.42/147.64	We have to consider all minimal (P,Q,R)-chains.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(107) MRRProof (EQUIVALENT)
354.42/147.64	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
354.42/147.64	
354.42/147.64	
354.42/147.64	Strictly oriented rules of the TRS R:
354.42/147.64	
354.42/147.64	   down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x)))
354.42/147.64	
354.42/147.64	Used ordering: Polynomial interpretation [POLO]:
354.42/147.64	
354.42/147.64	   POL(TOP.0(x_1)) = x_1
354.42/147.64	   POL(a.0(x_1)) = x_1
354.42/147.64	   POL(a.1(x_1)) = x_1
354.42/147.64	   POL(a_flat.0(x_1)) = x_1
354.42/147.64	   POL(b.0(x_1)) = x_1
354.42/147.64	   POL(b.1(x_1)) = x_1
354.42/147.64	   POL(b_flat.0(x_1)) = x_1
354.42/147.64	   POL(c.) = 0
354.42/147.64	   POL(d.0(x_1)) = x_1
354.42/147.64	   POL(d.1(x_1)) = x_1
354.42/147.64	   POL(d_flat.0(x_1)) = x_1
354.42/147.64	   POL(down.0(x_1)) = x_1
354.42/147.64	   POL(u.0(x_1)) = x_1
354.42/147.64	   POL(u.1(x_1)) = 1 + x_1
354.42/147.64	   POL(u_flat.0(x_1)) = x_1
354.42/147.64	   POL(up.0(x_1)) = x_1
354.42/147.64	
354.42/147.64	
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(108)
354.42/147.64	Obligation:
354.42/147.64	Q DP problem:
354.42/147.64	The TRS P consists of the following rules:
354.42/147.64	
354.42/147.64	   TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0)))
354.42/147.64	   TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0))))
354.42/147.64	   TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0))))
354.42/147.64	   TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0))))
354.42/147.64	
354.42/147.64	The TRS R consists of the following rules:
354.42/147.64	
354.42/147.64	   down.0(a.0(a.0(x))) -> up.0(c.)
354.42/147.64	   down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x)))
354.42/147.64	   down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x)))
354.42/147.64	   down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x))))
354.42/147.64	   down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x))))
354.42/147.64	   down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x)))
354.42/147.64	   down.0(u.0(y2)) -> u_flat.0(down.0(y2))
354.42/147.64	   down.0(a.0(c.)) -> a_flat.0(down.0(c.))
354.42/147.64	   down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6)))
354.42/147.64	   down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6)))
354.42/147.64	   down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8)))
354.42/147.64	   down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8)))
354.42/147.64	   down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10)))
354.42/147.64	   down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10)))
354.42/147.64	   down.0(b.0(c.)) -> b_flat.0(down.0(c.))
354.42/147.64	   down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11)))
354.42/147.64	   down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11)))
354.42/147.64	   down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13)))
354.42/147.64	   down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13)))
354.42/147.64	   down.0(d.0(c.)) -> d_flat.0(down.0(c.))
354.42/147.64	   down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17)))
354.42/147.64	   down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17)))
354.42/147.64	   down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18)))
354.42/147.64	   down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18)))
354.42/147.64	   u_flat.0(up.0(x_1)) -> up.0(u.0(x_1))
354.42/147.64	   d_flat.0(up.0(x_1)) -> up.0(d.0(x_1))
354.42/147.64	   b_flat.0(up.0(x_1)) -> up.0(b.0(x_1))
354.42/147.64	   a_flat.0(up.0(x_1)) -> up.0(a.0(x_1))
354.42/147.64	
354.42/147.64	The set Q consists of the following terms:
354.42/147.64	
354.42/147.64	   down.0(a.0(a.0(x0)))
354.42/147.64	   down.0(a.0(a.1(x0)))
354.42/147.64	   down.0(b.0(u.0(x0)))
354.42/147.64	   down.0(b.0(u.1(x0)))
354.42/147.64	   down.0(d.0(a.0(x0)))
354.42/147.64	   down.0(d.0(a.1(x0)))
354.42/147.64	   down.0(d.0(b.0(x0)))
354.42/147.64	   down.0(d.0(b.1(x0)))
354.42/147.64	   down.0(a.0(u.0(x0)))
354.42/147.64	   down.0(a.0(u.1(x0)))
354.42/147.64	   down.0(u.0(x0))
354.42/147.64	   down.0(u.1(x0))
354.42/147.64	   down.0(a.0(c.))
354.42/147.64	   down.0(a.0(b.0(x0)))
354.42/147.64	   down.0(a.0(b.1(x0)))
354.42/147.64	   down.0(a.0(d.0(x0)))
354.42/147.64	   down.0(a.0(d.1(x0)))
354.42/147.64	   down.0(a.1(fresh_constant.))
354.42/147.64	   down.0(b.0(a.0(x0)))
354.42/147.64	   down.0(b.0(a.1(x0)))
354.42/147.64	   down.0(b.0(c.))
354.42/147.64	   down.0(b.0(b.0(x0)))
354.42/147.64	   down.0(b.0(b.1(x0)))
354.42/147.64	   down.0(b.0(d.0(x0)))
354.42/147.64	   down.0(b.0(d.1(x0)))
354.42/147.64	   down.0(b.1(fresh_constant.))
354.42/147.64	   down.0(d.0(c.))
354.42/147.64	   down.0(d.0(u.0(x0)))
354.42/147.64	   down.0(d.0(u.1(x0)))
354.42/147.64	   down.0(d.0(d.0(x0)))
354.42/147.64	   down.0(d.0(d.1(x0)))
354.42/147.64	   down.0(d.1(fresh_constant.))
354.42/147.64	   a_flat.0(up.0(x0))
354.42/147.64	   a_flat.0(up.1(x0))
354.42/147.64	   b_flat.0(up.0(x0))
354.42/147.64	   b_flat.0(up.1(x0))
354.42/147.64	   u_flat.0(up.0(x0))
354.42/147.64	   u_flat.0(up.1(x0))
354.42/147.64	   d_flat.0(up.0(x0))
354.42/147.64	   d_flat.0(up.1(x0))
354.42/147.64	
354.42/147.64	We have to consider all minimal (P,Q,R)-chains.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(109) PisEmptyProof (SOUND)
354.42/147.64	The TRS P is empty. Hence, there is no (P,Q,R) chain.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(110)
354.42/147.64	TRUE
354.42/147.64	
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(111)
354.42/147.64	Obligation:
354.42/147.64	Q DP problem:
354.42/147.64	The TRS P consists of the following rules:
354.42/147.64	
354.42/147.64	   TOP(up(b(u(x0)))) -> TOP(up(b(d(x0))))
354.42/147.64	   TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0))))
354.42/147.64	   TOP(up(u(x0))) -> TOP(u_flat(down(x0)))
354.42/147.64	   TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0))))
354.42/147.64	   TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0))))
354.42/147.64	   TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0))))
354.42/147.64	   TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0))))
354.42/147.64	   TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0))))
354.42/147.64	   TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0))))
354.42/147.64	
354.42/147.64	The TRS R consists of the following rules:
354.42/147.64	
354.42/147.64	   down(a(a(x))) -> up(c)
354.42/147.64	   down(b(u(x))) -> up(b(d(x)))
354.42/147.64	   down(d(a(x))) -> up(a(d(x)))
354.42/147.64	   down(d(b(x))) -> up(u(a(b(x))))
354.42/147.64	   down(a(u(x))) -> up(u(a(x)))
354.42/147.64	   down(u(y2)) -> u_flat(down(y2))
354.42/147.64	   down(a(c)) -> a_flat(down(c))
354.42/147.64	   down(a(b(y6))) -> a_flat(down(b(y6)))
354.42/147.64	   down(a(d(y8))) -> a_flat(down(d(y8)))
354.42/147.64	   down(b(a(y10))) -> b_flat(down(a(y10)))
354.42/147.64	   down(b(c)) -> b_flat(down(c))
354.42/147.64	   down(b(b(y11))) -> b_flat(down(b(y11)))
354.42/147.64	   down(b(d(y13))) -> b_flat(down(d(y13)))
354.42/147.64	   down(d(c)) -> d_flat(down(c))
354.42/147.64	   down(d(u(y17))) -> d_flat(down(u(y17)))
354.42/147.64	   down(d(d(y18))) -> d_flat(down(d(y18)))
354.42/147.64	   u_flat(up(x_1)) -> up(u(x_1))
354.42/147.64	   d_flat(up(x_1)) -> up(d(x_1))
354.42/147.64	   b_flat(up(x_1)) -> up(b(x_1))
354.42/147.64	   a_flat(up(x_1)) -> up(a(x_1))
354.42/147.64	
354.42/147.64	The set Q consists of the following terms:
354.42/147.64	
354.42/147.64	   down(a(a(x0)))
354.42/147.64	   down(b(u(x0)))
354.42/147.64	   down(d(a(x0)))
354.42/147.64	   down(d(b(x0)))
354.42/147.64	   down(a(u(x0)))
354.42/147.64	   down(u(x0))
354.42/147.64	   down(a(c))
354.42/147.64	   down(a(b(x0)))
354.42/147.64	   down(a(d(x0)))
354.42/147.64	   down(a(fresh_constant))
354.42/147.64	   down(b(a(x0)))
354.42/147.64	   down(b(c))
354.42/147.64	   down(b(b(x0)))
354.42/147.64	   down(b(d(x0)))
354.42/147.64	   down(b(fresh_constant))
354.42/147.64	   down(d(c))
354.42/147.64	   down(d(u(x0)))
354.42/147.64	   down(d(d(x0)))
354.42/147.64	   down(d(fresh_constant))
354.42/147.64	   a_flat(up(x0))
354.42/147.64	   b_flat(up(x0))
354.42/147.64	   u_flat(up(x0))
354.42/147.64	   d_flat(up(x0))
354.42/147.64	
354.42/147.64	We have to consider all minimal (P,Q,R)-chains.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(112) QReductionProof (EQUIVALENT)
354.42/147.64	We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN].
354.42/147.64	
354.42/147.64	   down(a(fresh_constant))
354.42/147.64	   down(b(fresh_constant))
354.42/147.64	   down(d(fresh_constant))
354.42/147.64	
354.42/147.64	
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(113)
354.42/147.64	Obligation:
354.42/147.64	Q DP problem:
354.42/147.64	The TRS P consists of the following rules:
354.42/147.64	
354.42/147.64	   TOP(up(b(u(x0)))) -> TOP(up(b(d(x0))))
354.42/147.64	   TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0))))
354.42/147.64	   TOP(up(u(x0))) -> TOP(u_flat(down(x0)))
354.42/147.64	   TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0))))
354.42/147.64	   TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0))))
354.42/147.64	   TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0))))
354.42/147.64	   TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0))))
354.42/147.64	   TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0))))
354.42/147.64	   TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0))))
354.42/147.64	
354.42/147.64	The TRS R consists of the following rules:
354.42/147.64	
354.42/147.64	   down(a(a(x))) -> up(c)
354.42/147.64	   down(b(u(x))) -> up(b(d(x)))
354.42/147.64	   down(d(a(x))) -> up(a(d(x)))
354.42/147.64	   down(d(b(x))) -> up(u(a(b(x))))
354.42/147.64	   down(a(u(x))) -> up(u(a(x)))
354.42/147.64	   down(u(y2)) -> u_flat(down(y2))
354.42/147.64	   down(a(c)) -> a_flat(down(c))
354.42/147.64	   down(a(b(y6))) -> a_flat(down(b(y6)))
354.42/147.64	   down(a(d(y8))) -> a_flat(down(d(y8)))
354.42/147.64	   down(b(a(y10))) -> b_flat(down(a(y10)))
354.42/147.64	   down(b(c)) -> b_flat(down(c))
354.42/147.64	   down(b(b(y11))) -> b_flat(down(b(y11)))
354.42/147.64	   down(b(d(y13))) -> b_flat(down(d(y13)))
354.42/147.64	   down(d(c)) -> d_flat(down(c))
354.42/147.64	   down(d(u(y17))) -> d_flat(down(u(y17)))
354.42/147.64	   down(d(d(y18))) -> d_flat(down(d(y18)))
354.42/147.64	   u_flat(up(x_1)) -> up(u(x_1))
354.42/147.64	   d_flat(up(x_1)) -> up(d(x_1))
354.42/147.64	   b_flat(up(x_1)) -> up(b(x_1))
354.42/147.64	   a_flat(up(x_1)) -> up(a(x_1))
354.42/147.64	
354.42/147.64	The set Q consists of the following terms:
354.42/147.64	
354.42/147.64	   down(a(a(x0)))
354.42/147.64	   down(b(u(x0)))
354.42/147.64	   down(d(a(x0)))
354.42/147.64	   down(d(b(x0)))
354.42/147.64	   down(a(u(x0)))
354.42/147.64	   down(u(x0))
354.42/147.64	   down(a(c))
354.42/147.64	   down(a(b(x0)))
354.42/147.64	   down(a(d(x0)))
354.42/147.64	   down(b(a(x0)))
354.42/147.64	   down(b(c))
354.42/147.64	   down(b(b(x0)))
354.42/147.64	   down(b(d(x0)))
354.42/147.64	   down(d(c))
354.42/147.64	   down(d(u(x0)))
354.42/147.64	   down(d(d(x0)))
354.42/147.64	   a_flat(up(x0))
354.42/147.64	   b_flat(up(x0))
354.42/147.64	   u_flat(up(x0))
354.42/147.64	   d_flat(up(x0))
354.42/147.64	
354.42/147.64	We have to consider all (P,Q,R)-chains.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(114) MNOCProof (EQUIVALENT)
354.42/147.64	We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(115)
354.42/147.64	Obligation:
354.42/147.64	Q DP problem:
354.42/147.64	The TRS P consists of the following rules:
354.42/147.64	
354.42/147.64	   TOP(up(b(u(x0)))) -> TOP(up(b(d(x0))))
354.42/147.64	   TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0))))
354.42/147.64	   TOP(up(u(x0))) -> TOP(u_flat(down(x0)))
354.42/147.64	   TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0))))
354.42/147.64	   TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0))))
354.42/147.64	   TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0))))
354.42/147.64	   TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0))))
354.42/147.64	   TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0))))
354.42/147.64	   TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0))))
354.42/147.64	
354.42/147.64	The TRS R consists of the following rules:
354.42/147.64	
354.42/147.64	   down(a(a(x))) -> up(c)
354.42/147.64	   down(b(u(x))) -> up(b(d(x)))
354.42/147.64	   down(d(a(x))) -> up(a(d(x)))
354.42/147.64	   down(d(b(x))) -> up(u(a(b(x))))
354.42/147.64	   down(a(u(x))) -> up(u(a(x)))
354.42/147.64	   down(u(y2)) -> u_flat(down(y2))
354.42/147.64	   down(a(c)) -> a_flat(down(c))
354.42/147.64	   down(a(b(y6))) -> a_flat(down(b(y6)))
354.42/147.64	   down(a(d(y8))) -> a_flat(down(d(y8)))
354.42/147.64	   down(b(a(y10))) -> b_flat(down(a(y10)))
354.42/147.64	   down(b(c)) -> b_flat(down(c))
354.42/147.64	   down(b(b(y11))) -> b_flat(down(b(y11)))
354.42/147.64	   down(b(d(y13))) -> b_flat(down(d(y13)))
354.42/147.64	   down(d(c)) -> d_flat(down(c))
354.42/147.64	   down(d(u(y17))) -> d_flat(down(u(y17)))
354.42/147.64	   down(d(d(y18))) -> d_flat(down(d(y18)))
354.42/147.64	   u_flat(up(x_1)) -> up(u(x_1))
354.42/147.64	   d_flat(up(x_1)) -> up(d(x_1))
354.42/147.64	   b_flat(up(x_1)) -> up(b(x_1))
354.42/147.64	   a_flat(up(x_1)) -> up(a(x_1))
354.42/147.64	
354.42/147.64	Q is empty.
354.42/147.64	We have to consider all (P,Q,R)-chains.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(116) SplitQDPProof (EQUIVALENT)
354.42/147.64	We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem
354.42/147.64	
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(117)
354.42/147.64	Complex Obligation (AND)
354.42/147.64	
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(118)
354.42/147.64	Obligation:
354.42/147.64	Q DP problem:
354.42/147.64	The TRS P consists of the following rules:
354.42/147.64	
354.42/147.64	   TOP(up(b(u(x0)))) -> TOP(up(b(d(x0))))
354.42/147.64	   TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0))))
354.42/147.64	   TOP(up(u(x0))) -> TOP(u_flat(down(x0)))
354.42/147.64	   TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0))))
354.42/147.64	   TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0))))
354.42/147.64	   TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0))))
354.42/147.64	   TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0))))
354.42/147.64	   TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0))))
354.42/147.64	   TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0))))
354.42/147.64	
354.42/147.64	The TRS R consists of the following rules:
354.42/147.64	
354.42/147.64	   down(a(a(x))) -> up(c)
354.42/147.64	   down(b(u(x))) -> up(b(d(x)))
354.42/147.64	   down(d(a(x))) -> up(a(d(x)))
354.42/147.64	   down(d(b(x))) -> up(u(a(b(x))))
354.42/147.64	   down(a(u(x))) -> up(u(a(x)))
354.42/147.64	   down(u(y2)) -> u_flat(down(y2))
354.42/147.64	   down(a(c)) -> a_flat(down(c))
354.42/147.64	   down(a(b(y6))) -> a_flat(down(b(y6)))
354.42/147.64	   down(a(d(y8))) -> a_flat(down(d(y8)))
354.42/147.64	   down(b(a(y10))) -> b_flat(down(a(y10)))
354.42/147.64	   down(b(c)) -> b_flat(down(c))
354.42/147.64	   down(b(b(y11))) -> b_flat(down(b(y11)))
354.42/147.64	   down(b(d(y13))) -> b_flat(down(d(y13)))
354.42/147.64	   down(d(c)) -> d_flat(down(c))
354.42/147.64	   down(d(u(y17))) -> d_flat(down(u(y17)))
354.42/147.64	   down(d(d(y18))) -> d_flat(down(d(y18)))
354.42/147.64	   u_flat(up(x_1)) -> up(u(x_1))
354.42/147.64	   d_flat(up(x_1)) -> up(d(x_1))
354.42/147.64	   b_flat(up(x_1)) -> up(b(x_1))
354.42/147.64	   a_flat(up(x_1)) -> up(a(x_1))
354.42/147.64	
354.42/147.64	The set Q consists of the following terms:
354.42/147.64	
354.42/147.64	   down(a(a(x0)))
354.42/147.64	   down(b(u(x0)))
354.42/147.64	   down(d(a(x0)))
354.42/147.64	   down(d(b(x0)))
354.42/147.64	   down(a(u(x0)))
354.42/147.64	   down(u(x0))
354.42/147.64	   down(a(c))
354.42/147.64	   down(a(b(x0)))
354.42/147.64	   down(a(d(x0)))
354.42/147.64	   down(a(fresh_constant))
354.42/147.64	   down(b(a(x0)))
354.42/147.64	   down(b(c))
354.42/147.64	   down(b(b(x0)))
354.42/147.64	   down(b(d(x0)))
354.42/147.64	   down(b(fresh_constant))
354.42/147.64	   down(d(c))
354.42/147.64	   down(d(u(x0)))
354.42/147.64	   down(d(d(x0)))
354.42/147.64	   down(d(fresh_constant))
354.42/147.64	   a_flat(up(x0))
354.42/147.64	   b_flat(up(x0))
354.42/147.64	   u_flat(up(x0))
354.42/147.64	   d_flat(up(x0))
354.42/147.64	
354.42/147.64	We have to consider all minimal (P,Q,R)-chains.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(119) SemLabProof (SOUND)
354.42/147.64	We found the following model for the rules of the TRSs R and P.
354.42/147.64	Interpretation over the domain with elements from 0 to 1.
354.42/147.64	a: 0
354.42/147.64	c: 1
354.42/147.64	TOP: 0
354.42/147.64	u: 0
354.42/147.64	b: 0
354.42/147.64	d: 0
354.42/147.64	down: 0
354.42/147.64	fresh_constant: 0
354.42/147.64	d_flat: 0
354.42/147.64	up: 0
354.42/147.64	u_flat: 0
354.42/147.64	b_flat: 0
354.42/147.64	a_flat: 0
354.42/147.64	By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(120)
354.42/147.64	Obligation:
354.42/147.64	Q DP problem:
354.42/147.64	The TRS P consists of the following rules:
354.42/147.64	
354.42/147.64	   TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(d.1(x0)))) -> TOP.0(b_flat.0(down.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0)))
354.42/147.64	   TOP.0(up.0(u.1(x0))) -> TOP.0(u_flat.0(down.1(x0)))
354.42/147.64	   TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0))))
354.42/147.64	   TOP.0(up.0(a.0(b.1(x0)))) -> TOP.0(a_flat.0(down.0(b.1(x0))))
354.42/147.64	   TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(a.0(d.1(x0)))) -> TOP.0(a_flat.0(down.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(a.1(x0)))) -> TOP.0(b_flat.0(down.0(a.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0))))
354.42/147.64	   TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(d.0(d.1(x0)))) -> TOP.0(d_flat.0(down.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0))))
354.42/147.64	   TOP.0(up.0(d.0(u.1(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.1(x0))))
354.42/147.64	
354.42/147.64	The TRS R consists of the following rules:
354.42/147.64	
354.42/147.64	   down.0(a.0(a.0(x))) -> up.1(c.)
354.42/147.64	   down.0(a.0(a.1(x))) -> up.1(c.)
354.42/147.64	   down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x)))
354.42/147.64	   down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x)))
354.42/147.64	   down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x)))
354.42/147.64	   down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x)))
354.42/147.64	   down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x))))
354.42/147.64	   down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x))))
354.42/147.64	   down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x)))
354.42/147.64	   down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x)))
354.42/147.64	   down.0(u.0(y2)) -> u_flat.0(down.0(y2))
354.42/147.64	   down.0(u.1(y2)) -> u_flat.0(down.1(y2))
354.42/147.64	   down.0(a.1(c.)) -> a_flat.0(down.1(c.))
354.42/147.64	   down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6)))
354.42/147.64	   down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6)))
354.42/147.64	   down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8)))
354.42/147.64	   down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8)))
354.42/147.64	   down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10)))
354.42/147.64	   down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10)))
354.42/147.64	   down.0(b.1(c.)) -> b_flat.0(down.1(c.))
354.42/147.64	   down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11)))
354.42/147.64	   down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11)))
354.42/147.64	   down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13)))
354.42/147.64	   down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13)))
354.42/147.64	   down.0(d.1(c.)) -> d_flat.0(down.1(c.))
354.42/147.64	   down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17)))
354.42/147.64	   down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17)))
354.42/147.64	   down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18)))
354.42/147.64	   down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18)))
354.42/147.64	   u_flat.0(up.0(x_1)) -> up.0(u.0(x_1))
354.42/147.64	   u_flat.0(up.1(x_1)) -> up.0(u.1(x_1))
354.42/147.64	   d_flat.0(up.0(x_1)) -> up.0(d.0(x_1))
354.42/147.64	   d_flat.0(up.1(x_1)) -> up.0(d.1(x_1))
354.42/147.64	   b_flat.0(up.0(x_1)) -> up.0(b.0(x_1))
354.42/147.64	   b_flat.0(up.1(x_1)) -> up.0(b.1(x_1))
354.42/147.64	   a_flat.0(up.0(x_1)) -> up.0(a.0(x_1))
354.42/147.64	   a_flat.0(up.1(x_1)) -> up.0(a.1(x_1))
354.42/147.64	
354.42/147.64	The set Q consists of the following terms:
354.42/147.64	
354.42/147.64	   down.0(a.0(a.0(x0)))
354.42/147.64	   down.0(a.0(a.1(x0)))
354.42/147.64	   down.0(b.0(u.0(x0)))
354.42/147.64	   down.0(b.0(u.1(x0)))
354.42/147.64	   down.0(d.0(a.0(x0)))
354.42/147.64	   down.0(d.0(a.1(x0)))
354.42/147.64	   down.0(d.0(b.0(x0)))
354.42/147.64	   down.0(d.0(b.1(x0)))
354.42/147.64	   down.0(a.0(u.0(x0)))
354.42/147.64	   down.0(a.0(u.1(x0)))
354.42/147.64	   down.0(u.0(x0))
354.42/147.64	   down.0(u.1(x0))
354.42/147.64	   down.0(a.1(c.))
354.42/147.64	   down.0(a.0(b.0(x0)))
354.42/147.64	   down.0(a.0(b.1(x0)))
354.42/147.64	   down.0(a.0(d.0(x0)))
354.42/147.64	   down.0(a.0(d.1(x0)))
354.42/147.64	   down.0(a.0(fresh_constant.))
354.42/147.64	   down.0(b.0(a.0(x0)))
354.42/147.64	   down.0(b.0(a.1(x0)))
354.42/147.64	   down.0(b.1(c.))
354.42/147.64	   down.0(b.0(b.0(x0)))
354.42/147.64	   down.0(b.0(b.1(x0)))
354.42/147.64	   down.0(b.0(d.0(x0)))
354.42/147.64	   down.0(b.0(d.1(x0)))
354.42/147.64	   down.0(b.0(fresh_constant.))
354.42/147.64	   down.0(d.1(c.))
354.42/147.64	   down.0(d.0(u.0(x0)))
354.42/147.64	   down.0(d.0(u.1(x0)))
354.42/147.64	   down.0(d.0(d.0(x0)))
354.42/147.64	   down.0(d.0(d.1(x0)))
354.42/147.64	   down.0(d.0(fresh_constant.))
354.42/147.64	   a_flat.0(up.0(x0))
354.42/147.64	   a_flat.0(up.1(x0))
354.42/147.64	   b_flat.0(up.0(x0))
354.42/147.64	   b_flat.0(up.1(x0))
354.42/147.64	   u_flat.0(up.0(x0))
354.42/147.64	   u_flat.0(up.1(x0))
354.42/147.64	   d_flat.0(up.0(x0))
354.42/147.64	   d_flat.0(up.1(x0))
354.42/147.64	
354.42/147.64	We have to consider all minimal (P,Q,R)-chains.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(121) DependencyGraphProof (EQUIVALENT)
354.42/147.64	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(122)
354.42/147.64	Obligation:
354.42/147.64	Q DP problem:
354.42/147.64	The TRS P consists of the following rules:
354.42/147.64	
354.42/147.64	   TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(d.1(x0)))) -> TOP.0(b_flat.0(down.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0))))
354.42/147.64	   TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0)))
354.42/147.64	   TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0))))
354.42/147.64	   TOP.0(up.0(a.0(b.1(x0)))) -> TOP.0(a_flat.0(down.0(b.1(x0))))
354.42/147.64	   TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(a.0(d.1(x0)))) -> TOP.0(a_flat.0(down.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(a.1(x0)))) -> TOP.0(b_flat.0(down.0(a.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0))))
354.42/147.64	   TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(d.0(d.1(x0)))) -> TOP.0(d_flat.0(down.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0))))
354.42/147.64	
354.42/147.64	The TRS R consists of the following rules:
354.42/147.64	
354.42/147.64	   down.0(a.0(a.0(x))) -> up.1(c.)
354.42/147.64	   down.0(a.0(a.1(x))) -> up.1(c.)
354.42/147.64	   down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x)))
354.42/147.64	   down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x)))
354.42/147.64	   down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x)))
354.42/147.64	   down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x)))
354.42/147.64	   down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x))))
354.42/147.64	   down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x))))
354.42/147.64	   down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x)))
354.42/147.64	   down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x)))
354.42/147.64	   down.0(u.0(y2)) -> u_flat.0(down.0(y2))
354.42/147.64	   down.0(u.1(y2)) -> u_flat.0(down.1(y2))
354.42/147.64	   down.0(a.1(c.)) -> a_flat.0(down.1(c.))
354.42/147.64	   down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6)))
354.42/147.64	   down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6)))
354.42/147.64	   down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8)))
354.42/147.64	   down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8)))
354.42/147.64	   down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10)))
354.42/147.64	   down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10)))
354.42/147.64	   down.0(b.1(c.)) -> b_flat.0(down.1(c.))
354.42/147.64	   down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11)))
354.42/147.64	   down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11)))
354.42/147.64	   down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13)))
354.42/147.64	   down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13)))
354.42/147.64	   down.0(d.1(c.)) -> d_flat.0(down.1(c.))
354.42/147.64	   down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17)))
354.42/147.64	   down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17)))
354.42/147.64	   down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18)))
354.42/147.64	   down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18)))
354.42/147.64	   u_flat.0(up.0(x_1)) -> up.0(u.0(x_1))
354.42/147.64	   u_flat.0(up.1(x_1)) -> up.0(u.1(x_1))
354.42/147.64	   d_flat.0(up.0(x_1)) -> up.0(d.0(x_1))
354.42/147.64	   d_flat.0(up.1(x_1)) -> up.0(d.1(x_1))
354.42/147.64	   b_flat.0(up.0(x_1)) -> up.0(b.0(x_1))
354.42/147.64	   b_flat.0(up.1(x_1)) -> up.0(b.1(x_1))
354.42/147.64	   a_flat.0(up.0(x_1)) -> up.0(a.0(x_1))
354.42/147.64	   a_flat.0(up.1(x_1)) -> up.0(a.1(x_1))
354.42/147.64	
354.42/147.64	The set Q consists of the following terms:
354.42/147.64	
354.42/147.64	   down.0(a.0(a.0(x0)))
354.42/147.64	   down.0(a.0(a.1(x0)))
354.42/147.64	   down.0(b.0(u.0(x0)))
354.42/147.64	   down.0(b.0(u.1(x0)))
354.42/147.64	   down.0(d.0(a.0(x0)))
354.42/147.64	   down.0(d.0(a.1(x0)))
354.42/147.64	   down.0(d.0(b.0(x0)))
354.42/147.64	   down.0(d.0(b.1(x0)))
354.42/147.64	   down.0(a.0(u.0(x0)))
354.42/147.64	   down.0(a.0(u.1(x0)))
354.42/147.64	   down.0(u.0(x0))
354.42/147.64	   down.0(u.1(x0))
354.42/147.64	   down.0(a.1(c.))
354.42/147.64	   down.0(a.0(b.0(x0)))
354.42/147.64	   down.0(a.0(b.1(x0)))
354.42/147.64	   down.0(a.0(d.0(x0)))
354.42/147.64	   down.0(a.0(d.1(x0)))
354.42/147.64	   down.0(a.0(fresh_constant.))
354.42/147.64	   down.0(b.0(a.0(x0)))
354.42/147.64	   down.0(b.0(a.1(x0)))
354.42/147.64	   down.0(b.1(c.))
354.42/147.64	   down.0(b.0(b.0(x0)))
354.42/147.64	   down.0(b.0(b.1(x0)))
354.42/147.64	   down.0(b.0(d.0(x0)))
354.42/147.64	   down.0(b.0(d.1(x0)))
354.42/147.64	   down.0(b.0(fresh_constant.))
354.42/147.64	   down.0(d.1(c.))
354.42/147.64	   down.0(d.0(u.0(x0)))
354.42/147.64	   down.0(d.0(u.1(x0)))
354.42/147.64	   down.0(d.0(d.0(x0)))
354.42/147.64	   down.0(d.0(d.1(x0)))
354.42/147.64	   down.0(d.0(fresh_constant.))
354.42/147.64	   a_flat.0(up.0(x0))
354.42/147.64	   a_flat.0(up.1(x0))
354.42/147.64	   b_flat.0(up.0(x0))
354.42/147.64	   b_flat.0(up.1(x0))
354.42/147.64	   u_flat.0(up.0(x0))
354.42/147.64	   u_flat.0(up.1(x0))
354.42/147.64	   d_flat.0(up.0(x0))
354.42/147.64	   d_flat.0(up.1(x0))
354.42/147.64	
354.42/147.64	We have to consider all minimal (P,Q,R)-chains.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(123) MRRProof (EQUIVALENT)
354.42/147.64	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
354.42/147.64	
354.42/147.64	
354.42/147.64	Strictly oriented rules of the TRS R:
354.42/147.64	
354.42/147.64	   down.0(u.1(y2)) -> u_flat.0(down.1(y2))
354.42/147.64	   down.0(a.1(c.)) -> a_flat.0(down.1(c.))
354.42/147.64	   down.0(b.1(c.)) -> b_flat.0(down.1(c.))
354.42/147.64	   down.0(d.1(c.)) -> d_flat.0(down.1(c.))
354.42/147.64	
354.42/147.64	Used ordering: Polynomial interpretation [POLO]:
354.42/147.64	
354.42/147.64	   POL(TOP.0(x_1)) = x_1
354.42/147.64	   POL(a.0(x_1)) = x_1
354.42/147.64	   POL(a.1(x_1)) = x_1
354.42/147.64	   POL(a_flat.0(x_1)) = x_1
354.42/147.64	   POL(b.0(x_1)) = x_1
354.42/147.64	   POL(b.1(x_1)) = x_1
354.42/147.64	   POL(b_flat.0(x_1)) = x_1
354.42/147.64	   POL(c.) = 0
354.42/147.64	   POL(d.0(x_1)) = x_1
354.42/147.64	   POL(d.1(x_1)) = x_1
354.42/147.64	   POL(d_flat.0(x_1)) = x_1
354.42/147.64	   POL(down.0(x_1)) = 1 + x_1
354.42/147.64	   POL(down.1(x_1)) = x_1
354.42/147.64	   POL(u.0(x_1)) = x_1
354.42/147.64	   POL(u.1(x_1)) = x_1
354.42/147.64	   POL(u_flat.0(x_1)) = x_1
354.42/147.64	   POL(up.0(x_1)) = 1 + x_1
354.42/147.64	   POL(up.1(x_1)) = 1 + x_1
354.42/147.64	
354.42/147.64	
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(124)
354.42/147.64	Obligation:
354.42/147.64	Q DP problem:
354.42/147.64	The TRS P consists of the following rules:
354.42/147.64	
354.42/147.64	   TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(u.1(x0)))) -> TOP.0(up.0(b.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(d.1(x0)))) -> TOP.0(b_flat.0(down.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0))))
354.42/147.64	   TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0)))
354.42/147.64	   TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0))))
354.42/147.64	   TOP.0(up.0(a.0(b.1(x0)))) -> TOP.0(a_flat.0(down.0(b.1(x0))))
354.42/147.64	   TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(a.0(d.1(x0)))) -> TOP.0(a_flat.0(down.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(a.1(x0)))) -> TOP.0(b_flat.0(down.0(a.1(x0))))
354.42/147.64	   TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(b.1(x0)))) -> TOP.0(b_flat.0(down.0(b.1(x0))))
354.42/147.64	   TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(d.0(d.1(x0)))) -> TOP.0(d_flat.0(down.0(d.1(x0))))
354.42/147.64	   TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0))))
354.42/147.64	
354.42/147.64	The TRS R consists of the following rules:
354.42/147.64	
354.42/147.64	   down.0(a.0(a.0(x))) -> up.1(c.)
354.42/147.64	   down.0(a.0(a.1(x))) -> up.1(c.)
354.42/147.64	   down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x)))
354.42/147.64	   down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x)))
354.42/147.64	   down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x)))
354.42/147.64	   down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x)))
354.42/147.64	   down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x))))
354.42/147.64	   down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x))))
354.42/147.64	   down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x)))
354.42/147.64	   down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x)))
354.42/147.64	   down.0(u.0(y2)) -> u_flat.0(down.0(y2))
354.42/147.64	   down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6)))
354.42/147.64	   down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6)))
354.42/147.64	   down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8)))
354.42/147.64	   down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8)))
354.42/147.64	   down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10)))
354.42/147.64	   down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10)))
354.42/147.64	   down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11)))
354.42/147.64	   down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11)))
354.42/147.64	   down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13)))
354.42/147.64	   down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13)))
354.42/147.64	   down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17)))
354.42/147.64	   down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17)))
354.42/147.64	   down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18)))
354.42/147.64	   down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18)))
354.42/147.64	   u_flat.0(up.0(x_1)) -> up.0(u.0(x_1))
354.42/147.64	   u_flat.0(up.1(x_1)) -> up.0(u.1(x_1))
354.42/147.64	   d_flat.0(up.0(x_1)) -> up.0(d.0(x_1))
354.42/147.64	   d_flat.0(up.1(x_1)) -> up.0(d.1(x_1))
354.42/147.64	   b_flat.0(up.0(x_1)) -> up.0(b.0(x_1))
354.42/147.64	   b_flat.0(up.1(x_1)) -> up.0(b.1(x_1))
354.42/147.64	   a_flat.0(up.0(x_1)) -> up.0(a.0(x_1))
354.42/147.64	   a_flat.0(up.1(x_1)) -> up.0(a.1(x_1))
354.42/147.64	
354.42/147.64	The set Q consists of the following terms:
354.42/147.64	
354.42/147.64	   down.0(a.0(a.0(x0)))
354.42/147.64	   down.0(a.0(a.1(x0)))
354.42/147.64	   down.0(b.0(u.0(x0)))
354.42/147.64	   down.0(b.0(u.1(x0)))
354.42/147.64	   down.0(d.0(a.0(x0)))
354.42/147.64	   down.0(d.0(a.1(x0)))
354.42/147.64	   down.0(d.0(b.0(x0)))
354.42/147.64	   down.0(d.0(b.1(x0)))
354.42/147.64	   down.0(a.0(u.0(x0)))
354.42/147.64	   down.0(a.0(u.1(x0)))
354.42/147.64	   down.0(u.0(x0))
354.42/147.64	   down.0(u.1(x0))
354.42/147.64	   down.0(a.1(c.))
354.42/147.64	   down.0(a.0(b.0(x0)))
354.42/147.64	   down.0(a.0(b.1(x0)))
354.42/147.64	   down.0(a.0(d.0(x0)))
354.42/147.64	   down.0(a.0(d.1(x0)))
354.42/147.64	   down.0(a.0(fresh_constant.))
354.42/147.64	   down.0(b.0(a.0(x0)))
354.42/147.64	   down.0(b.0(a.1(x0)))
354.42/147.64	   down.0(b.1(c.))
354.42/147.64	   down.0(b.0(b.0(x0)))
354.42/147.64	   down.0(b.0(b.1(x0)))
354.42/147.64	   down.0(b.0(d.0(x0)))
354.42/147.64	   down.0(b.0(d.1(x0)))
354.42/147.64	   down.0(b.0(fresh_constant.))
354.42/147.64	   down.0(d.1(c.))
354.42/147.64	   down.0(d.0(u.0(x0)))
354.42/147.64	   down.0(d.0(u.1(x0)))
354.42/147.64	   down.0(d.0(d.0(x0)))
354.42/147.64	   down.0(d.0(d.1(x0)))
354.42/147.64	   down.0(d.0(fresh_constant.))
354.42/147.64	   a_flat.0(up.0(x0))
354.42/147.64	   a_flat.0(up.1(x0))
354.42/147.64	   b_flat.0(up.0(x0))
354.42/147.64	   b_flat.0(up.1(x0))
354.42/147.64	   u_flat.0(up.0(x0))
354.42/147.64	   u_flat.0(up.1(x0))
354.42/147.64	   d_flat.0(up.0(x0))
354.42/147.64	   d_flat.0(up.1(x0))
354.42/147.64	
354.42/147.64	We have to consider all minimal (P,Q,R)-chains.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(125) DependencyGraphProof (EQUIVALENT)
354.42/147.64	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes.
354.42/147.64	----------------------------------------
354.42/147.64	
354.42/147.64	(126)
354.42/147.64	Obligation:
354.42/147.64	Q DP problem:
354.42/147.64	The TRS P consists of the following rules:
354.42/147.64	
354.42/147.64	   TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0)))
354.42/147.64	   TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0))))
354.42/147.64	   TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0))))
354.42/147.64	   TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0))))
354.42/147.64	   TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0))))
354.42/147.64	   TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0))))
354.42/147.64	
354.42/147.64	The TRS R consists of the following rules:
354.42/147.64	
354.42/147.64	   down.0(a.0(a.0(x))) -> up.1(c.)
354.42/147.64	   down.0(a.0(a.1(x))) -> up.1(c.)
354.42/147.64	   down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x)))
354.42/147.64	   down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x)))
354.42/147.64	   down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x)))
354.42/147.64	   down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x)))
354.42/147.64	   down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x))))
354.53/147.65	   down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x))))
354.53/147.65	   down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x)))
354.53/147.65	   down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x)))
354.53/147.65	   down.0(u.0(y2)) -> u_flat.0(down.0(y2))
354.53/147.65	   down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6)))
354.53/147.65	   down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6)))
354.53/147.65	   down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8)))
354.53/147.65	   down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8)))
354.53/147.65	   down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10)))
354.53/147.65	   down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10)))
354.53/147.65	   down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11)))
354.53/147.65	   down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11)))
354.53/147.65	   down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13)))
354.53/147.65	   down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13)))
354.53/147.65	   down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17)))
354.53/147.65	   down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17)))
354.53/147.65	   down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18)))
354.53/147.65	   down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18)))
354.53/147.65	   u_flat.0(up.0(x_1)) -> up.0(u.0(x_1))
354.53/147.65	   u_flat.0(up.1(x_1)) -> up.0(u.1(x_1))
354.53/147.65	   d_flat.0(up.0(x_1)) -> up.0(d.0(x_1))
354.53/147.65	   d_flat.0(up.1(x_1)) -> up.0(d.1(x_1))
354.53/147.65	   b_flat.0(up.0(x_1)) -> up.0(b.0(x_1))
354.53/147.65	   b_flat.0(up.1(x_1)) -> up.0(b.1(x_1))
354.53/147.65	   a_flat.0(up.0(x_1)) -> up.0(a.0(x_1))
354.53/147.65	   a_flat.0(up.1(x_1)) -> up.0(a.1(x_1))
354.53/147.65	
354.53/147.65	The set Q consists of the following terms:
354.53/147.65	
354.53/147.65	   down.0(a.0(a.0(x0)))
354.53/147.65	   down.0(a.0(a.1(x0)))
354.53/147.65	   down.0(b.0(u.0(x0)))
354.53/147.65	   down.0(b.0(u.1(x0)))
354.53/147.65	   down.0(d.0(a.0(x0)))
354.53/147.65	   down.0(d.0(a.1(x0)))
354.53/147.65	   down.0(d.0(b.0(x0)))
354.53/147.65	   down.0(d.0(b.1(x0)))
354.53/147.65	   down.0(a.0(u.0(x0)))
354.53/147.65	   down.0(a.0(u.1(x0)))
354.53/147.65	   down.0(u.0(x0))
354.53/147.65	   down.0(u.1(x0))
354.53/147.65	   down.0(a.1(c.))
354.53/147.65	   down.0(a.0(b.0(x0)))
354.53/147.65	   down.0(a.0(b.1(x0)))
354.53/147.65	   down.0(a.0(d.0(x0)))
354.53/147.65	   down.0(a.0(d.1(x0)))
354.53/147.65	   down.0(a.0(fresh_constant.))
354.53/147.65	   down.0(b.0(a.0(x0)))
354.53/147.65	   down.0(b.0(a.1(x0)))
354.53/147.65	   down.0(b.1(c.))
354.53/147.65	   down.0(b.0(b.0(x0)))
354.53/147.65	   down.0(b.0(b.1(x0)))
354.53/147.65	   down.0(b.0(d.0(x0)))
354.53/147.65	   down.0(b.0(d.1(x0)))
354.53/147.65	   down.0(b.0(fresh_constant.))
354.53/147.65	   down.0(d.1(c.))
354.53/147.65	   down.0(d.0(u.0(x0)))
354.53/147.65	   down.0(d.0(u.1(x0)))
354.53/147.65	   down.0(d.0(d.0(x0)))
354.53/147.65	   down.0(d.0(d.1(x0)))
354.53/147.65	   down.0(d.0(fresh_constant.))
354.53/147.65	   a_flat.0(up.0(x0))
354.53/147.65	   a_flat.0(up.1(x0))
354.53/147.65	   b_flat.0(up.0(x0))
354.53/147.65	   b_flat.0(up.1(x0))
354.53/147.65	   u_flat.0(up.0(x0))
354.53/147.65	   u_flat.0(up.1(x0))
354.53/147.65	   d_flat.0(up.0(x0))
354.53/147.65	   d_flat.0(up.1(x0))
354.53/147.65	
354.53/147.65	We have to consider all minimal (P,Q,R)-chains.
354.53/147.65	----------------------------------------
354.53/147.65	
354.53/147.65	(127) MRRProof (EQUIVALENT)
354.53/147.65	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
354.53/147.65	
354.53/147.65	
354.53/147.65	Strictly oriented rules of the TRS R:
354.53/147.65	
354.53/147.65	   b_flat.0(up.1(x_1)) -> up.0(b.1(x_1))
354.53/147.65	
354.53/147.65	Used ordering: Polynomial interpretation [POLO]:
354.53/147.65	
354.53/147.65	   POL(TOP.0(x_1)) = x_1
354.53/147.65	   POL(a.0(x_1)) = x_1
354.53/147.65	   POL(a.1(x_1)) = x_1
354.53/147.65	   POL(a_flat.0(x_1)) = x_1
354.53/147.65	   POL(b.0(x_1)) = 1 + x_1
354.53/147.65	   POL(b.1(x_1)) = x_1
354.53/147.65	   POL(b_flat.0(x_1)) = 1 + x_1
354.53/147.65	   POL(c.) = 0
354.53/147.65	   POL(d.0(x_1)) = x_1
354.53/147.65	   POL(d.1(x_1)) = x_1
354.53/147.65	   POL(d_flat.0(x_1)) = x_1
354.53/147.65	   POL(down.0(x_1)) = x_1
354.53/147.65	   POL(u.0(x_1)) = x_1
354.53/147.65	   POL(u.1(x_1)) = x_1
354.53/147.65	   POL(u_flat.0(x_1)) = x_1
354.53/147.65	   POL(up.0(x_1)) = x_1
354.53/147.65	   POL(up.1(x_1)) = x_1
354.53/147.65	
354.53/147.65	
354.53/147.65	----------------------------------------
354.53/147.65	
354.53/147.65	(128)
354.53/147.65	Obligation:
354.53/147.65	Q DP problem:
354.53/147.65	The TRS P consists of the following rules:
354.53/147.65	
354.53/147.65	   TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0))))
354.53/147.65	   TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0))))
354.53/147.65	   TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0)))
354.53/147.65	   TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0))))
354.53/147.65	   TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0))))
354.53/147.65	   TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0))))
354.53/147.65	   TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0))))
354.53/147.65	   TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0))))
354.53/147.65	   TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0))))
354.53/147.65	
354.53/147.65	The TRS R consists of the following rules:
354.53/147.65	
354.53/147.65	   down.0(a.0(a.0(x))) -> up.1(c.)
354.53/147.65	   down.0(a.0(a.1(x))) -> up.1(c.)
354.53/147.65	   down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x)))
354.53/147.65	   down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x)))
354.53/147.65	   down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x)))
354.53/147.65	   down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x)))
354.53/147.65	   down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x))))
354.53/147.65	   down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x))))
354.53/147.65	   down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x)))
354.53/147.65	   down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x)))
354.53/147.65	   down.0(u.0(y2)) -> u_flat.0(down.0(y2))
354.53/147.65	   down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6)))
354.53/147.65	   down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6)))
354.53/147.65	   down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8)))
354.53/147.65	   down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8)))
354.53/147.65	   down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10)))
354.53/147.65	   down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10)))
354.53/147.65	   down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11)))
354.53/147.65	   down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11)))
354.53/147.65	   down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13)))
354.53/147.65	   down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13)))
354.53/147.65	   down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17)))
354.53/147.65	   down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17)))
354.53/147.65	   down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18)))
354.53/147.65	   down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18)))
354.53/147.65	   u_flat.0(up.0(x_1)) -> up.0(u.0(x_1))
354.53/147.65	   u_flat.0(up.1(x_1)) -> up.0(u.1(x_1))
354.53/147.65	   d_flat.0(up.0(x_1)) -> up.0(d.0(x_1))
354.53/147.65	   d_flat.0(up.1(x_1)) -> up.0(d.1(x_1))
354.53/147.65	   b_flat.0(up.0(x_1)) -> up.0(b.0(x_1))
354.53/147.65	   a_flat.0(up.0(x_1)) -> up.0(a.0(x_1))
354.53/147.65	   a_flat.0(up.1(x_1)) -> up.0(a.1(x_1))
354.53/147.65	
354.53/147.65	The set Q consists of the following terms:
354.53/147.65	
354.53/147.65	   down.0(a.0(a.0(x0)))
354.53/147.65	   down.0(a.0(a.1(x0)))
354.53/147.65	   down.0(b.0(u.0(x0)))
354.53/147.65	   down.0(b.0(u.1(x0)))
354.53/147.65	   down.0(d.0(a.0(x0)))
354.53/147.65	   down.0(d.0(a.1(x0)))
354.53/147.65	   down.0(d.0(b.0(x0)))
354.53/147.65	   down.0(d.0(b.1(x0)))
354.53/147.65	   down.0(a.0(u.0(x0)))
354.53/147.65	   down.0(a.0(u.1(x0)))
354.53/147.65	   down.0(u.0(x0))
354.53/147.65	   down.0(u.1(x0))
354.53/147.65	   down.0(a.1(c.))
354.53/147.65	   down.0(a.0(b.0(x0)))
354.53/147.65	   down.0(a.0(b.1(x0)))
354.53/147.65	   down.0(a.0(d.0(x0)))
354.53/147.65	   down.0(a.0(d.1(x0)))
354.53/147.65	   down.0(a.0(fresh_constant.))
354.53/147.65	   down.0(b.0(a.0(x0)))
354.53/147.65	   down.0(b.0(a.1(x0)))
354.53/147.65	   down.0(b.1(c.))
354.53/147.65	   down.0(b.0(b.0(x0)))
354.53/147.65	   down.0(b.0(b.1(x0)))
354.53/147.65	   down.0(b.0(d.0(x0)))
354.53/147.65	   down.0(b.0(d.1(x0)))
354.53/147.65	   down.0(b.0(fresh_constant.))
354.53/147.65	   down.0(d.1(c.))
354.53/147.65	   down.0(d.0(u.0(x0)))
354.53/147.65	   down.0(d.0(u.1(x0)))
354.53/147.65	   down.0(d.0(d.0(x0)))
354.53/147.65	   down.0(d.0(d.1(x0)))
354.53/147.65	   down.0(d.0(fresh_constant.))
354.53/147.65	   a_flat.0(up.0(x0))
354.53/147.65	   a_flat.0(up.1(x0))
354.53/147.65	   b_flat.0(up.0(x0))
354.53/147.65	   b_flat.0(up.1(x0))
354.53/147.65	   u_flat.0(up.0(x0))
354.53/147.65	   u_flat.0(up.1(x0))
354.53/147.65	   d_flat.0(up.0(x0))
354.53/147.65	   d_flat.0(up.1(x0))
354.53/147.65	
354.53/147.65	We have to consider all minimal (P,Q,R)-chains.
354.53/147.65	----------------------------------------
354.53/147.65	
354.53/147.65	(129) MRRProof (EQUIVALENT)
354.53/147.65	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
354.53/147.65	
354.53/147.65	
354.53/147.65	Strictly oriented rules of the TRS R:
354.53/147.65	
354.53/147.65	   down.0(b.0(u.1(x))) -> up.0(b.0(d.1(x)))
354.53/147.65	   down.0(d.0(a.1(x))) -> up.0(a.0(d.1(x)))
354.53/147.65	   d_flat.0(up.1(x_1)) -> up.0(d.1(x_1))
354.53/147.65	
354.53/147.65	Used ordering: Polynomial interpretation [POLO]:
354.53/147.65	
354.53/147.65	   POL(TOP.0(x_1)) = x_1
354.53/147.65	   POL(a.0(x_1)) = x_1
354.53/147.65	   POL(a.1(x_1)) = x_1
354.53/147.65	   POL(a_flat.0(x_1)) = x_1
354.53/147.65	   POL(b.0(x_1)) = x_1
354.53/147.65	   POL(b.1(x_1)) = x_1
354.53/147.65	   POL(b_flat.0(x_1)) = x_1
354.53/147.65	   POL(c.) = 0
354.53/147.65	   POL(d.0(x_1)) = 1 + x_1
354.53/147.65	   POL(d.1(x_1)) = x_1
354.53/147.65	   POL(d_flat.0(x_1)) = 1 + x_1
354.53/147.65	   POL(down.0(x_1)) = x_1
354.53/147.65	   POL(u.0(x_1)) = 1 + x_1
354.53/147.65	   POL(u.1(x_1)) = 1 + x_1
354.53/147.65	   POL(u_flat.0(x_1)) = 1 + x_1
354.53/147.65	   POL(up.0(x_1)) = x_1
354.53/147.65	   POL(up.1(x_1)) = x_1
354.53/147.65	
354.53/147.65	
354.53/147.65	----------------------------------------
354.53/147.65	
354.53/147.65	(130)
354.53/147.65	Obligation:
354.53/147.65	Q DP problem:
354.53/147.65	The TRS P consists of the following rules:
354.53/147.65	
354.53/147.65	   TOP.0(up.0(b.0(u.0(x0)))) -> TOP.0(up.0(b.0(d.0(x0))))
354.53/147.65	   TOP.0(up.0(b.0(d.0(x0)))) -> TOP.0(b_flat.0(down.0(d.0(x0))))
354.53/147.65	   TOP.0(up.0(u.0(x0))) -> TOP.0(u_flat.0(down.0(x0)))
354.53/147.65	   TOP.0(up.0(a.0(b.0(x0)))) -> TOP.0(a_flat.0(down.0(b.0(x0))))
354.53/147.65	   TOP.0(up.0(a.0(d.0(x0)))) -> TOP.0(a_flat.0(down.0(d.0(x0))))
354.53/147.65	   TOP.0(up.0(b.0(a.0(x0)))) -> TOP.0(b_flat.0(down.0(a.0(x0))))
354.53/147.65	   TOP.0(up.0(b.0(b.0(x0)))) -> TOP.0(b_flat.0(down.0(b.0(x0))))
354.53/147.65	   TOP.0(up.0(d.0(d.0(x0)))) -> TOP.0(d_flat.0(down.0(d.0(x0))))
354.53/147.65	   TOP.0(up.0(d.0(u.0(x0)))) -> TOP.0(d_flat.0(u_flat.0(down.0(x0))))
354.53/147.65	
354.53/147.65	The TRS R consists of the following rules:
354.53/147.65	
354.53/147.65	   down.0(a.0(a.0(x))) -> up.1(c.)
354.53/147.65	   down.0(a.0(a.1(x))) -> up.1(c.)
354.53/147.65	   down.0(b.0(u.0(x))) -> up.0(b.0(d.0(x)))
354.53/147.65	   down.0(d.0(a.0(x))) -> up.0(a.0(d.0(x)))
354.53/147.65	   down.0(d.0(b.0(x))) -> up.0(u.0(a.0(b.0(x))))
354.53/147.65	   down.0(d.0(b.1(x))) -> up.0(u.0(a.0(b.1(x))))
354.53/147.65	   down.0(a.0(u.0(x))) -> up.0(u.0(a.0(x)))
354.53/147.65	   down.0(a.0(u.1(x))) -> up.0(u.0(a.1(x)))
354.53/147.65	   down.0(u.0(y2)) -> u_flat.0(down.0(y2))
354.53/147.65	   down.0(a.0(b.0(y6))) -> a_flat.0(down.0(b.0(y6)))
354.53/147.65	   down.0(a.0(b.1(y6))) -> a_flat.0(down.0(b.1(y6)))
354.53/147.65	   down.0(a.0(d.0(y8))) -> a_flat.0(down.0(d.0(y8)))
354.53/147.65	   down.0(a.0(d.1(y8))) -> a_flat.0(down.0(d.1(y8)))
354.53/147.65	   down.0(b.0(a.0(y10))) -> b_flat.0(down.0(a.0(y10)))
354.53/147.65	   down.0(b.0(a.1(y10))) -> b_flat.0(down.0(a.1(y10)))
354.53/147.65	   down.0(b.0(b.0(y11))) -> b_flat.0(down.0(b.0(y11)))
354.53/147.65	   down.0(b.0(b.1(y11))) -> b_flat.0(down.0(b.1(y11)))
354.53/147.65	   down.0(b.0(d.0(y13))) -> b_flat.0(down.0(d.0(y13)))
354.53/147.65	   down.0(b.0(d.1(y13))) -> b_flat.0(down.0(d.1(y13)))
354.53/147.65	   down.0(d.0(u.0(y17))) -> d_flat.0(down.0(u.0(y17)))
354.53/147.65	   down.0(d.0(u.1(y17))) -> d_flat.0(down.0(u.1(y17)))
354.53/147.65	   down.0(d.0(d.0(y18))) -> d_flat.0(down.0(d.0(y18)))
354.53/147.65	   down.0(d.0(d.1(y18))) -> d_flat.0(down.0(d.1(y18)))
354.53/147.65	   u_flat.0(up.0(x_1)) -> up.0(u.0(x_1))
354.53/147.65	   u_flat.0(up.1(x_1)) -> up.0(u.1(x_1))
354.53/147.65	   d_flat.0(up.0(x_1)) -> up.0(d.0(x_1))
354.53/147.65	   b_flat.0(up.0(x_1)) -> up.0(b.0(x_1))
354.53/147.65	   a_flat.0(up.0(x_1)) -> up.0(a.0(x_1))
354.53/147.65	   a_flat.0(up.1(x_1)) -> up.0(a.1(x_1))
354.53/147.65	
354.53/147.65	The set Q consists of the following terms:
354.53/147.65	
354.53/147.65	   down.0(a.0(a.0(x0)))
354.53/147.65	   down.0(a.0(a.1(x0)))
354.53/147.65	   down.0(b.0(u.0(x0)))
354.53/147.65	   down.0(b.0(u.1(x0)))
354.53/147.65	   down.0(d.0(a.0(x0)))
354.53/147.65	   down.0(d.0(a.1(x0)))
354.53/147.65	   down.0(d.0(b.0(x0)))
354.53/147.65	   down.0(d.0(b.1(x0)))
354.53/147.65	   down.0(a.0(u.0(x0)))
354.53/147.65	   down.0(a.0(u.1(x0)))
354.53/147.65	   down.0(u.0(x0))
354.53/147.65	   down.0(u.1(x0))
354.53/147.65	   down.0(a.1(c.))
354.53/147.65	   down.0(a.0(b.0(x0)))
354.53/147.65	   down.0(a.0(b.1(x0)))
354.53/147.65	   down.0(a.0(d.0(x0)))
354.53/147.65	   down.0(a.0(d.1(x0)))
354.53/147.65	   down.0(a.0(fresh_constant.))
354.53/147.65	   down.0(b.0(a.0(x0)))
354.53/147.65	   down.0(b.0(a.1(x0)))
354.53/147.65	   down.0(b.1(c.))
354.53/147.65	   down.0(b.0(b.0(x0)))
354.53/147.65	   down.0(b.0(b.1(x0)))
354.53/147.65	   down.0(b.0(d.0(x0)))
354.53/147.65	   down.0(b.0(d.1(x0)))
354.53/147.65	   down.0(b.0(fresh_constant.))
354.53/147.65	   down.0(d.1(c.))
354.53/147.65	   down.0(d.0(u.0(x0)))
354.53/147.65	   down.0(d.0(u.1(x0)))
354.53/147.65	   down.0(d.0(d.0(x0)))
354.53/147.65	   down.0(d.0(d.1(x0)))
354.53/147.65	   down.0(d.0(fresh_constant.))
354.53/147.65	   a_flat.0(up.0(x0))
354.53/147.65	   a_flat.0(up.1(x0))
354.53/147.65	   b_flat.0(up.0(x0))
354.53/147.65	   b_flat.0(up.1(x0))
354.53/147.65	   u_flat.0(up.0(x0))
354.53/147.65	   u_flat.0(up.1(x0))
354.53/147.65	   d_flat.0(up.0(x0))
354.53/147.65	   d_flat.0(up.1(x0))
354.53/147.65	
354.53/147.65	We have to consider all minimal (P,Q,R)-chains.
354.53/147.65	----------------------------------------
354.53/147.65	
354.53/147.65	(131) PisEmptyProof (SOUND)
354.53/147.65	The TRS P is empty. Hence, there is no (P,Q,R) chain.
354.53/147.65	----------------------------------------
354.53/147.65	
354.53/147.65	(132)
354.53/147.65	TRUE
354.53/147.65	
354.53/147.65	----------------------------------------
354.53/147.65	
354.53/147.65	(133)
354.53/147.65	Obligation:
354.53/147.65	Q DP problem:
354.53/147.65	The TRS P consists of the following rules:
354.53/147.65	
354.53/147.65	   TOP(up(b(u(x0)))) -> TOP(up(b(d(x0))))
354.53/147.65	   TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0))))
354.53/147.65	   TOP(up(u(x0))) -> TOP(u_flat(down(x0)))
354.53/147.65	   TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0))))
354.53/147.65	   TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0))))
354.53/147.65	   TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0))))
354.53/147.65	   TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0))))
354.53/147.65	   TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0))))
354.53/147.65	   TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0))))
354.53/147.65	
354.53/147.65	The TRS R consists of the following rules:
354.53/147.65	
354.53/147.65	   down(a(a(x))) -> up(c)
354.53/147.65	   down(b(u(x))) -> up(b(d(x)))
354.53/147.65	   down(d(a(x))) -> up(a(d(x)))
354.53/147.65	   down(d(b(x))) -> up(u(a(b(x))))
354.53/147.65	   down(a(u(x))) -> up(u(a(x)))
354.53/147.65	   down(u(y2)) -> u_flat(down(y2))
354.53/147.65	   down(a(b(y6))) -> a_flat(down(b(y6)))
354.53/147.65	   down(a(d(y8))) -> a_flat(down(d(y8)))
354.53/147.65	   down(b(a(y10))) -> b_flat(down(a(y10)))
354.53/147.65	   down(b(b(y11))) -> b_flat(down(b(y11)))
354.53/147.65	   down(b(d(y13))) -> b_flat(down(d(y13)))
354.53/147.65	   down(d(u(y17))) -> d_flat(down(u(y17)))
354.53/147.65	   down(d(d(y18))) -> d_flat(down(d(y18)))
354.53/147.65	   u_flat(up(x_1)) -> up(u(x_1))
354.53/147.65	   d_flat(up(x_1)) -> up(d(x_1))
354.53/147.65	   b_flat(up(x_1)) -> up(b(x_1))
354.53/147.65	   a_flat(up(x_1)) -> up(a(x_1))
354.53/147.65	
354.53/147.65	The set Q consists of the following terms:
354.53/147.65	
354.53/147.65	   down(a(a(x0)))
354.53/147.65	   down(b(u(x0)))
354.53/147.65	   down(d(a(x0)))
354.53/147.65	   down(d(b(x0)))
354.53/147.65	   down(a(u(x0)))
354.53/147.65	   down(u(x0))
354.53/147.65	   down(a(c))
354.53/147.65	   down(a(b(x0)))
354.53/147.65	   down(a(d(x0)))
354.53/147.65	   down(a(fresh_constant))
354.53/147.65	   down(b(a(x0)))
354.53/147.65	   down(b(c))
354.53/147.65	   down(b(b(x0)))
354.53/147.65	   down(b(d(x0)))
354.53/147.65	   down(b(fresh_constant))
354.53/147.65	   down(d(c))
354.53/147.65	   down(d(u(x0)))
354.53/147.65	   down(d(d(x0)))
354.53/147.65	   down(d(fresh_constant))
354.53/147.65	   a_flat(up(x0))
354.53/147.65	   b_flat(up(x0))
354.53/147.65	   u_flat(up(x0))
354.53/147.65	   d_flat(up(x0))
354.53/147.65	
354.53/147.65	We have to consider all minimal (P,Q,R)-chains.
354.53/147.65	----------------------------------------
354.53/147.65	
354.53/147.65	(134) QReductionProof (EQUIVALENT)
354.53/147.65	We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN].
354.53/147.65	
354.53/147.65	   down(a(fresh_constant))
354.53/147.65	   down(b(fresh_constant))
354.53/147.65	   down(d(fresh_constant))
354.53/147.65	
354.53/147.65	
354.53/147.65	----------------------------------------
354.53/147.65	
354.53/147.65	(135)
354.53/147.65	Obligation:
354.53/147.65	Q DP problem:
354.53/147.65	The TRS P consists of the following rules:
354.53/147.65	
354.53/147.65	   TOP(up(b(u(x0)))) -> TOP(up(b(d(x0))))
354.53/147.65	   TOP(up(b(d(x0)))) -> TOP(b_flat(down(d(x0))))
354.53/147.65	   TOP(up(u(x0))) -> TOP(u_flat(down(x0)))
354.53/147.65	   TOP(up(a(b(x0)))) -> TOP(a_flat(down(b(x0))))
354.53/147.65	   TOP(up(a(d(x0)))) -> TOP(a_flat(down(d(x0))))
354.53/147.65	   TOP(up(b(a(x0)))) -> TOP(b_flat(down(a(x0))))
354.53/147.65	   TOP(up(b(b(x0)))) -> TOP(b_flat(down(b(x0))))
354.53/147.65	   TOP(up(d(d(x0)))) -> TOP(d_flat(down(d(x0))))
354.53/147.65	   TOP(up(d(u(x0)))) -> TOP(d_flat(u_flat(down(x0))))
354.53/147.65	
354.53/147.65	The TRS R consists of the following rules:
354.53/147.65	
354.53/147.65	   down(a(a(x))) -> up(c)
354.53/147.65	   down(b(u(x))) -> up(b(d(x)))
354.53/147.65	   down(d(a(x))) -> up(a(d(x)))
354.53/147.65	   down(d(b(x))) -> up(u(a(b(x))))
354.53/147.65	   down(a(u(x))) -> up(u(a(x)))
354.53/147.65	   down(u(y2)) -> u_flat(down(y2))
354.53/147.65	   down(a(b(y6))) -> a_flat(down(b(y6)))
354.53/147.65	   down(a(d(y8))) -> a_flat(down(d(y8)))
354.53/147.65	   down(b(a(y10))) -> b_flat(down(a(y10)))
354.53/147.65	   down(b(b(y11))) -> b_flat(down(b(y11)))
354.53/147.65	   down(b(d(y13))) -> b_flat(down(d(y13)))
354.53/147.65	   down(d(u(y17))) -> d_flat(down(u(y17)))
354.53/147.65	   down(d(d(y18))) -> d_flat(down(d(y18)))
354.53/147.65	   u_flat(up(x_1)) -> up(u(x_1))
354.53/147.65	   d_flat(up(x_1)) -> up(d(x_1))
354.53/147.65	   b_flat(up(x_1)) -> up(b(x_1))
354.53/147.65	   a_flat(up(x_1)) -> up(a(x_1))
354.53/147.65	
354.53/147.65	The set Q consists of the following terms:
354.53/147.65	
354.53/147.65	   down(a(a(x0)))
354.53/147.65	   down(b(u(x0)))
354.53/147.65	   down(d(a(x0)))
354.53/147.65	   down(d(b(x0)))
354.53/147.65	   down(a(u(x0)))
354.53/147.65	   down(u(x0))
354.53/147.65	   down(a(c))
354.53/147.65	   down(a(b(x0)))
354.53/147.65	   down(a(d(x0)))
354.53/147.65	   down(b(a(x0)))
354.53/147.65	   down(b(c))
354.53/147.65	   down(b(b(x0)))
354.53/147.65	   down(b(d(x0)))
354.53/147.65	   down(d(c))
354.53/147.65	   down(d(u(x0)))
354.53/147.65	   down(d(d(x0)))
354.53/147.65	   a_flat(up(x0))
354.53/147.65	   b_flat(up(x0))
354.53/147.65	   u_flat(up(x0))
354.53/147.65	   d_flat(up(x0))
354.53/147.65	
354.53/147.65	We have to consider all (P,Q,R)-chains.
354.64/147.72	EOF