395.04/246.26	MAYBE
395.04/246.28	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
395.04/246.28	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
395.04/246.28	
395.04/246.28	
395.04/246.28	Outermost Termination of the given OTRS could not be shown:
395.04/246.28	
395.04/246.28	(0) OTRS
395.04/246.28	(1) Thiemann-SpecialC-Transformation [EQUIVALENT, 0 ms]
395.04/246.28	(2) QTRS
395.04/246.28	    (3) DependencyPairsProof [EQUIVALENT, 3 ms]
395.04/246.28	    (4) QDP
395.04/246.28	    (5) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	    (6) AND
395.04/246.28	        (7) QDP
395.04/246.28	            (8) UsableRulesProof [EQUIVALENT, 0 ms]
395.04/246.28	            (9) QDP
395.04/246.28	            (10) QReductionProof [EQUIVALENT, 0 ms]
395.04/246.28	            (11) QDP
395.04/246.28	            (12) UsableRulesReductionPairsProof [EQUIVALENT, 14 ms]
395.04/246.28	            (13) QDP
395.04/246.28	            (14) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	            (15) TRUE
395.04/246.28	        (16) QDP
395.04/246.28	            (17) UsableRulesProof [EQUIVALENT, 0 ms]
395.04/246.28	            (18) QDP
395.04/246.28	                (19) QReductionProof [EQUIVALENT, 0 ms]
395.04/246.28	                (20) QDP
395.04/246.28	                (21) TransformationProof [EQUIVALENT, 0 ms]
395.04/246.28	                (22) QDP
395.04/246.28	                (23) QDPOrderProof [EQUIVALENT, 0 ms]
395.04/246.28	                (24) QDP
395.04/246.28	            (25) UsableRulesProof [EQUIVALENT, 0 ms]
395.04/246.28	            (26) QDP
395.04/246.28	                (27) QReductionProof [EQUIVALENT, 0 ms]
395.04/246.28	                (28) QDP
395.04/246.28	(29) Trivial-Transformation [SOUND, 0 ms]
395.04/246.28	(30) QTRS
395.04/246.28	    (31) DependencyPairsProof [EQUIVALENT, 0 ms]
395.04/246.28	    (32) QDP
395.04/246.28	    (33) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	    (34) QDP
395.04/246.28	    (35) UsableRulesProof [EQUIVALENT, 0 ms]
395.04/246.28	    (36) QDP
395.04/246.28	    (37) NonTerminationLoopProof [COMPLETE, 0 ms]
395.04/246.28	    (38) NO
395.04/246.28	(39) Raffelsieper-Zantema-Transformation [SOUND, 0 ms]
395.04/246.28	(40) QTRS
395.04/246.28	    (41) DependencyPairsProof [EQUIVALENT, 21 ms]
395.04/246.28	    (42) QDP
395.04/246.28	    (43) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	    (44) QDP
395.04/246.28	    (45) UsableRulesProof [EQUIVALENT, 7 ms]
395.04/246.28	    (46) QDP
395.04/246.28	    (47) TransformationProof [EQUIVALENT, 4 ms]
395.04/246.28	    (48) QDP
395.04/246.28	    (49) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	    (50) QDP
395.04/246.28	    (51) UsableRulesProof [EQUIVALENT, 0 ms]
395.04/246.28	    (52) QDP
395.04/246.28	    (53) TransformationProof [EQUIVALENT, 0 ms]
395.04/246.28	    (54) QDP
395.04/246.28	    (55) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	    (56) QDP
395.04/246.28	    (57) TransformationProof [EQUIVALENT, 0 ms]
395.04/246.28	    (58) QDP
395.04/246.28	    (59) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	    (60) QDP
395.04/246.28	    (61) TransformationProof [EQUIVALENT, 0 ms]
395.04/246.28	    (62) QDP
395.04/246.28	    (63) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	    (64) QDP
395.04/246.28	    (65) TransformationProof [EQUIVALENT, 25 ms]
395.04/246.28	    (66) QDP
395.04/246.28	    (67) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	    (68) QDP
395.04/246.28	    (69) TransformationProof [EQUIVALENT, 0 ms]
395.04/246.28	    (70) QDP
395.04/246.28	    (71) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	    (72) QDP
395.04/246.28	    (73) TransformationProof [EQUIVALENT, 38 ms]
395.04/246.28	    (74) QDP
395.04/246.28	    (75) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	    (76) QDP
395.04/246.28	    (77) TransformationProof [EQUIVALENT, 0 ms]
395.04/246.28	    (78) QDP
395.04/246.28	    (79) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	    (80) QDP
395.04/246.28	    (81) TransformationProof [EQUIVALENT, 28 ms]
395.04/246.28	    (82) QDP
395.04/246.28	    (83) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	    (84) QDP
395.04/246.28	    (85) MRRProof [EQUIVALENT, 166 ms]
395.04/246.28	    (86) QDP
395.04/246.28	    (87) QDPOrderProof [EQUIVALENT, 1170 ms]
395.04/246.28	    (88) QDP
395.04/246.28	    (89) QDPOrderProof [EQUIVALENT, 614 ms]
395.04/246.28	    (90) QDP
395.04/246.28	    (91) QDPOrderProof [EQUIVALENT, 3525 ms]
395.04/246.28	    (92) QDP
395.04/246.28	    (93) QDPOrderProof [EQUIVALENT, 4619 ms]
395.04/246.28	    (94) QDP
395.04/246.28	    (95) SplitQDPProof [EQUIVALENT, 0 ms]
395.04/246.28	    (96) AND
395.04/246.28	        (97) QDP
395.04/246.28	            (98) SemLabProof [SOUND, 0 ms]
395.04/246.28	            (99) QDP
395.04/246.28	            (100) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
395.04/246.28	            (101) QDP
395.04/246.28	            (102) MRRProof [EQUIVALENT, 11 ms]
395.04/246.28	            (103) QDP
395.04/246.28	            (104) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	            (105) QDP
395.04/246.28	            (106) QDPOrderProof [EQUIVALENT, 0 ms]
395.04/246.28	            (107) QDP
395.04/246.28	            (108) QDPOrderProof [EQUIVALENT, 0 ms]
395.04/246.28	            (109) QDP
395.04/246.28	            (110) PisEmptyProof [SOUND, 0 ms]
395.04/246.28	            (111) TRUE
395.04/246.28	        (112) QDP
395.04/246.28	            (113) SplitQDPProof [EQUIVALENT, 0 ms]
395.04/246.28	            (114) AND
395.04/246.28	                (115) QDP
395.04/246.28	                    (116) SemLabProof [SOUND, 0 ms]
395.04/246.28	                    (117) QDP
395.04/246.28	                    (118) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
395.04/246.28	                    (119) QDP
395.04/246.28	                    (120) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	                    (121) QDP
395.04/246.28	                    (122) MRRProof [EQUIVALENT, 2 ms]
395.04/246.28	                    (123) QDP
395.04/246.28	                    (124) QDPOrderProof [EQUIVALENT, 16 ms]
395.04/246.28	                    (125) QDP
395.04/246.28	                    (126) QDPOrderProof [EQUIVALENT, 0 ms]
395.04/246.28	                    (127) QDP
395.04/246.28	                    (128) QDPOrderProof [EQUIVALENT, 0 ms]
395.04/246.28	                    (129) QDP
395.04/246.28	                    (130) QDPOrderProof [EQUIVALENT, 10 ms]
395.04/246.28	                    (131) QDP
395.04/246.28	                    (132) PisEmptyProof [SOUND, 0 ms]
395.04/246.28	                    (133) TRUE
395.04/246.28	                (134) QDP
395.04/246.28	                    (135) SplitQDPProof [EQUIVALENT, 0 ms]
395.04/246.28	                    (136) AND
395.04/246.28	                        (137) QDP
395.04/246.28	                            (138) SemLabProof [SOUND, 0 ms]
395.04/246.28	                            (139) QDP
395.04/246.28	                            (140) MRRProof [EQUIVALENT, 0 ms]
395.04/246.28	                            (141) QDP
395.04/246.28	                            (142) QDPOrderProof [EQUIVALENT, 18 ms]
395.04/246.28	                            (143) QDP
395.04/246.28	                            (144) QDPOrderProof [EQUIVALENT, 18 ms]
395.04/246.28	                            (145) QDP
395.04/246.28	                            (146) PisEmptyProof [SOUND, 0 ms]
395.04/246.28	                            (147) TRUE
395.04/246.28	                        (148) QDP
395.04/246.28	                            (149) TransformationProof [EQUIVALENT, 0 ms]
395.04/246.28	                            (150) QDP
395.04/246.28	                            (151) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	                            (152) QDP
395.04/246.28	                            (153) TransformationProof [EQUIVALENT, 0 ms]
395.04/246.28	                            (154) QDP
395.04/246.28	                            (155) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	                            (156) QDP
395.04/246.28	                            (157) TransformationProof [EQUIVALENT, 11 ms]
395.04/246.28	                            (158) QDP
395.04/246.28	                            (159) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	                            (160) QDP
395.04/246.28	                            (161) SplitQDPProof [EQUIVALENT, 0 ms]
395.04/246.28	                            (162) AND
395.04/246.28	                                (163) QDP
395.04/246.28	                                    (164) SemLabProof [SOUND, 0 ms]
395.04/246.28	                                    (165) QDP
395.04/246.28	                                    (166) MRRProof [EQUIVALENT, 0 ms]
395.04/246.28	                                    (167) QDP
395.04/246.28	                                    (168) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	                                    (169) QDP
395.04/246.28	                                    (170) MRRProof [EQUIVALENT, 0 ms]
395.04/246.28	                                    (171) QDP
395.04/246.28	                                    (172) MRRProof [EQUIVALENT, 6 ms]
395.04/246.28	                                    (173) QDP
395.04/246.28	                                    (174) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	                                    (175) QDP
395.04/246.28	                                    (176) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
395.04/246.28	                                    (177) QDP
395.04/246.28	                                    (178) QDPOrderProof [EQUIVALENT, 14 ms]
395.04/246.28	                                    (179) QDP
395.04/246.28	                                    (180) QDPOrderProof [EQUIVALENT, 13 ms]
395.04/246.28	                                    (181) QDP
395.04/246.28	                                    (182) PisEmptyProof [SOUND, 0 ms]
395.04/246.28	                                    (183) TRUE
395.04/246.28	                                (184) QDP
395.04/246.28	                                    (185) SplitQDPProof [EQUIVALENT, 0 ms]
395.04/246.28	                                    (186) AND
395.04/246.28	                                        (187) QDP
395.04/246.28	                                            (188) SemLabProof [SOUND, 0 ms]
395.04/246.28	                                            (189) QDP
395.04/246.28	                                            (190) MRRProof [EQUIVALENT, 0 ms]
395.04/246.28	                                            (191) QDP
395.04/246.28	                                            (192) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	                                            (193) QDP
395.04/246.28	                                            (194) MRRProof [EQUIVALENT, 6 ms]
395.04/246.28	                                            (195) QDP
395.04/246.28	                                            (196) MRRProof [EQUIVALENT, 0 ms]
395.04/246.28	                                            (197) QDP
395.04/246.28	                                            (198) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	                                            (199) QDP
395.04/246.28	                                            (200) UsableRulesReductionPairsProof [EQUIVALENT, 8 ms]
395.04/246.28	                                            (201) QDP
395.04/246.28	                                            (202) QDPOrderProof [EQUIVALENT, 0 ms]
395.04/246.28	                                            (203) QDP
395.04/246.28	                                            (204) UsableRulesReductionPairsProof [EQUIVALENT, 6 ms]
395.04/246.28	                                            (205) QDP
395.04/246.28	                                            (206) PisEmptyProof [SOUND, 0 ms]
395.04/246.28	                                            (207) TRUE
395.04/246.28	                                        (208) QDP
395.04/246.28	                                            (209) SplitQDPProof [EQUIVALENT, 0 ms]
395.04/246.28	                                            (210) AND
395.04/246.28	                                                (211) QDP
395.04/246.28	                                                    (212) SemLabProof [SOUND, 0 ms]
395.04/246.28	                                                    (213) QDP
395.04/246.28	                                                    (214) MRRProof [EQUIVALENT, 1 ms]
395.04/246.28	                                                    (215) QDP
395.04/246.28	                                                    (216) DependencyGraphProof [EQUIVALENT, 0 ms]
395.04/246.28	                                                    (217) QDP
395.04/246.28	                                                    (218) MRRProof [EQUIVALENT, 0 ms]
395.04/246.28	                                                    (219) QDP
395.04/246.28	                                                    (220) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
395.04/246.28	                                                    (221) QDP
395.04/246.28	                                                    (222) PisEmptyProof [SOUND, 0 ms]
395.04/246.28	                                                    (223) TRUE
395.04/246.28	                                                (224) QDP
395.04/246.28	
395.04/246.28	
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(0)
395.04/246.28	Obligation:
395.04/246.28	Term rewrite system R:
395.04/246.28	The TRS R consists of the following rules:
395.04/246.28	
395.04/246.28	   a -> f(a)
395.04/246.28	   a -> g(a)
395.04/246.28	   f(f(f(x))) -> b
395.04/246.28	   g(g(g(x))) -> b
395.04/246.28	   g(f(f(g(x)))) -> b
395.04/246.28	   f(g(g(f(x)))) -> b
395.04/246.28	   g(f(g(f(x)))) -> b
395.04/246.28	
395.04/246.28	
395.04/246.28	
395.04/246.28	Outermost Strategy.
395.04/246.28	
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(1) Thiemann-SpecialC-Transformation (EQUIVALENT)
395.04/246.28	We applied the Thiemann-SpecialC transformation  to transform the outermost TRS to an innermost TRS.
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(2)
395.04/246.28	Obligation:
395.04/246.28	Q restricted rewrite system:
395.04/246.28	The TRS R consists of the following rules:
395.04/246.28	
395.04/246.28	   top(go_up(x)) -> top(reduce(x))
395.04/246.28	   reduce(f(x_1)) -> check_f(redex_f(x_1))
395.04/246.28	   reduce(g(x_1)) -> check_g(redex_g(x_1))
395.04/246.28	   reduce(a) -> go_up(f(a))
395.04/246.28	   reduce(a) -> go_up(g(a))
395.04/246.28	   redex_f(f(f(x))) -> result_f(b)
395.04/246.28	   redex_g(g(g(x))) -> result_g(b)
395.04/246.28	   redex_g(f(f(g(x)))) -> result_g(b)
395.04/246.28	   redex_f(g(g(f(x)))) -> result_f(b)
395.04/246.28	   redex_g(f(g(f(x)))) -> result_g(b)
395.04/246.28	   check_f(result_f(x)) -> go_up(x)
395.04/246.28	   check_g(result_g(x)) -> go_up(x)
395.04/246.28	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
395.04/246.28	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
395.04/246.28	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
395.04/246.28	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
395.04/246.28	
395.04/246.28	The set Q consists of the following terms:
395.04/246.28	
395.04/246.28	   top(go_up(x0))
395.04/246.28	   reduce(f(x0))
395.04/246.28	   reduce(g(x0))
395.04/246.28	   reduce(a)
395.04/246.28	   redex_f(f(f(x0)))
395.04/246.28	   redex_g(g(g(x0)))
395.04/246.28	   redex_g(f(f(g(x0))))
395.04/246.28	   redex_f(g(g(f(x0))))
395.04/246.28	   redex_g(f(g(f(x0))))
395.04/246.28	   check_f(result_f(x0))
395.04/246.28	   check_g(result_g(x0))
395.04/246.28	   check_f(redex_f(x0))
395.04/246.28	   check_g(redex_g(x0))
395.04/246.28	   in_f_1(go_up(x0))
395.04/246.28	   in_g_1(go_up(x0))
395.04/246.28	
395.04/246.28	
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(3) DependencyPairsProof (EQUIVALENT)
395.04/246.28	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(4)
395.04/246.28	Obligation:
395.04/246.28	Q DP problem:
395.04/246.28	The TRS P consists of the following rules:
395.04/246.28	
395.04/246.28	   TOP(go_up(x)) -> TOP(reduce(x))
395.04/246.28	   TOP(go_up(x)) -> REDUCE(x)
395.04/246.28	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
395.04/246.28	   REDUCE(f(x_1)) -> REDEX_F(x_1)
395.04/246.28	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
395.04/246.28	   REDUCE(g(x_1)) -> REDEX_G(x_1)
395.04/246.28	   CHECK_F(redex_f(x_1)) -> IN_F_1(reduce(x_1))
395.04/246.28	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
395.04/246.28	   CHECK_G(redex_g(x_1)) -> IN_G_1(reduce(x_1))
395.04/246.28	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
395.04/246.28	
395.04/246.28	The TRS R consists of the following rules:
395.04/246.28	
395.04/246.28	   top(go_up(x)) -> top(reduce(x))
395.04/246.28	   reduce(f(x_1)) -> check_f(redex_f(x_1))
395.04/246.28	   reduce(g(x_1)) -> check_g(redex_g(x_1))
395.04/246.28	   reduce(a) -> go_up(f(a))
395.04/246.28	   reduce(a) -> go_up(g(a))
395.04/246.28	   redex_f(f(f(x))) -> result_f(b)
395.04/246.28	   redex_g(g(g(x))) -> result_g(b)
395.04/246.28	   redex_g(f(f(g(x)))) -> result_g(b)
395.04/246.28	   redex_f(g(g(f(x)))) -> result_f(b)
395.04/246.28	   redex_g(f(g(f(x)))) -> result_g(b)
395.04/246.28	   check_f(result_f(x)) -> go_up(x)
395.04/246.28	   check_g(result_g(x)) -> go_up(x)
395.04/246.28	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
395.04/246.28	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
395.04/246.28	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
395.04/246.28	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
395.04/246.28	
395.04/246.28	The set Q consists of the following terms:
395.04/246.28	
395.04/246.28	   top(go_up(x0))
395.04/246.28	   reduce(f(x0))
395.04/246.28	   reduce(g(x0))
395.04/246.28	   reduce(a)
395.04/246.28	   redex_f(f(f(x0)))
395.04/246.28	   redex_g(g(g(x0)))
395.04/246.28	   redex_g(f(f(g(x0))))
395.04/246.28	   redex_f(g(g(f(x0))))
395.04/246.28	   redex_g(f(g(f(x0))))
395.04/246.28	   check_f(result_f(x0))
395.04/246.28	   check_g(result_g(x0))
395.04/246.28	   check_f(redex_f(x0))
395.04/246.28	   check_g(redex_g(x0))
395.04/246.28	   in_f_1(go_up(x0))
395.04/246.28	   in_g_1(go_up(x0))
395.04/246.28	
395.04/246.28	We have to consider all minimal (P,Q,R)-chains.
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(5) DependencyGraphProof (EQUIVALENT)
395.04/246.28	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 5 less nodes.
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(6)
395.04/246.28	Complex Obligation (AND)
395.04/246.28	
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(7)
395.04/246.28	Obligation:
395.04/246.28	Q DP problem:
395.04/246.28	The TRS P consists of the following rules:
395.04/246.28	
395.04/246.28	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
395.04/246.28	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
395.04/246.28	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
395.04/246.28	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
395.04/246.28	
395.04/246.28	The TRS R consists of the following rules:
395.04/246.28	
395.04/246.28	   top(go_up(x)) -> top(reduce(x))
395.04/246.28	   reduce(f(x_1)) -> check_f(redex_f(x_1))
395.04/246.28	   reduce(g(x_1)) -> check_g(redex_g(x_1))
395.04/246.28	   reduce(a) -> go_up(f(a))
395.04/246.28	   reduce(a) -> go_up(g(a))
395.04/246.28	   redex_f(f(f(x))) -> result_f(b)
395.04/246.28	   redex_g(g(g(x))) -> result_g(b)
395.04/246.28	   redex_g(f(f(g(x)))) -> result_g(b)
395.04/246.28	   redex_f(g(g(f(x)))) -> result_f(b)
395.04/246.28	   redex_g(f(g(f(x)))) -> result_g(b)
395.04/246.28	   check_f(result_f(x)) -> go_up(x)
395.04/246.28	   check_g(result_g(x)) -> go_up(x)
395.04/246.28	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
395.04/246.28	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
395.04/246.28	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
395.04/246.28	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
395.04/246.28	
395.04/246.28	The set Q consists of the following terms:
395.04/246.28	
395.04/246.28	   top(go_up(x0))
395.04/246.28	   reduce(f(x0))
395.04/246.28	   reduce(g(x0))
395.04/246.28	   reduce(a)
395.04/246.28	   redex_f(f(f(x0)))
395.04/246.28	   redex_g(g(g(x0)))
395.04/246.28	   redex_g(f(f(g(x0))))
395.04/246.28	   redex_f(g(g(f(x0))))
395.04/246.28	   redex_g(f(g(f(x0))))
395.04/246.28	   check_f(result_f(x0))
395.04/246.28	   check_g(result_g(x0))
395.04/246.28	   check_f(redex_f(x0))
395.04/246.28	   check_g(redex_g(x0))
395.04/246.28	   in_f_1(go_up(x0))
395.04/246.28	   in_g_1(go_up(x0))
395.04/246.28	
395.04/246.28	We have to consider all minimal (P,Q,R)-chains.
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(8) UsableRulesProof (EQUIVALENT)
395.04/246.28	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.
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(9)
395.04/246.28	Obligation:
395.04/246.28	Q DP problem:
395.04/246.28	The TRS P consists of the following rules:
395.04/246.28	
395.04/246.28	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
395.04/246.28	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
395.04/246.28	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
395.04/246.28	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
395.04/246.28	
395.04/246.28	The TRS R consists of the following rules:
395.04/246.28	
395.04/246.28	   redex_g(g(g(x))) -> result_g(b)
395.04/246.28	   redex_g(f(f(g(x)))) -> result_g(b)
395.04/246.28	   redex_g(f(g(f(x)))) -> result_g(b)
395.04/246.28	   redex_f(f(f(x))) -> result_f(b)
395.04/246.28	   redex_f(g(g(f(x)))) -> result_f(b)
395.04/246.28	
395.04/246.28	The set Q consists of the following terms:
395.04/246.28	
395.04/246.28	   top(go_up(x0))
395.04/246.28	   reduce(f(x0))
395.04/246.28	   reduce(g(x0))
395.04/246.28	   reduce(a)
395.04/246.28	   redex_f(f(f(x0)))
395.04/246.28	   redex_g(g(g(x0)))
395.04/246.28	   redex_g(f(f(g(x0))))
395.04/246.28	   redex_f(g(g(f(x0))))
395.04/246.28	   redex_g(f(g(f(x0))))
395.04/246.28	   check_f(result_f(x0))
395.04/246.28	   check_g(result_g(x0))
395.04/246.28	   check_f(redex_f(x0))
395.04/246.28	   check_g(redex_g(x0))
395.04/246.28	   in_f_1(go_up(x0))
395.04/246.28	   in_g_1(go_up(x0))
395.04/246.28	
395.04/246.28	We have to consider all minimal (P,Q,R)-chains.
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(10) QReductionProof (EQUIVALENT)
395.04/246.28	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
395.04/246.28	
395.04/246.28	   top(go_up(x0))
395.04/246.28	   reduce(f(x0))
395.04/246.28	   reduce(g(x0))
395.04/246.28	   reduce(a)
395.04/246.28	   check_f(result_f(x0))
395.04/246.28	   check_g(result_g(x0))
395.04/246.28	   check_f(redex_f(x0))
395.04/246.28	   check_g(redex_g(x0))
395.04/246.28	   in_f_1(go_up(x0))
395.04/246.28	   in_g_1(go_up(x0))
395.04/246.28	
395.04/246.28	
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(11)
395.04/246.28	Obligation:
395.04/246.28	Q DP problem:
395.04/246.28	The TRS P consists of the following rules:
395.04/246.28	
395.04/246.28	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
395.04/246.28	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
395.04/246.28	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
395.04/246.28	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
395.04/246.28	
395.04/246.28	The TRS R consists of the following rules:
395.04/246.28	
395.04/246.28	   redex_g(g(g(x))) -> result_g(b)
395.04/246.28	   redex_g(f(f(g(x)))) -> result_g(b)
395.04/246.28	   redex_g(f(g(f(x)))) -> result_g(b)
395.04/246.28	   redex_f(f(f(x))) -> result_f(b)
395.04/246.28	   redex_f(g(g(f(x)))) -> result_f(b)
395.04/246.28	
395.04/246.28	The set Q consists of the following terms:
395.04/246.28	
395.04/246.28	   redex_f(f(f(x0)))
395.04/246.28	   redex_g(g(g(x0)))
395.04/246.28	   redex_g(f(f(g(x0))))
395.04/246.28	   redex_f(g(g(f(x0))))
395.04/246.28	   redex_g(f(g(f(x0))))
395.04/246.28	
395.04/246.28	We have to consider all minimal (P,Q,R)-chains.
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(12) UsableRulesReductionPairsProof (EQUIVALENT)
395.04/246.28	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.
395.04/246.28	
395.04/246.28	The following dependency pairs can be deleted:
395.04/246.28	
395.04/246.28	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
395.04/246.28	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
395.04/246.28	The following rules are removed from R:
395.04/246.28	
395.04/246.28	   redex_g(g(g(x))) -> result_g(b)
395.04/246.28	   redex_g(f(f(g(x)))) -> result_g(b)
395.04/246.28	   redex_g(f(g(f(x)))) -> result_g(b)
395.04/246.28	   redex_f(f(f(x))) -> result_f(b)
395.04/246.28	   redex_f(g(g(f(x)))) -> result_f(b)
395.04/246.28	Used ordering: POLO with Polynomial interpretation [POLO]:
395.04/246.28	
395.04/246.28	   POL(CHECK_F(x_1)) = 2*x_1
395.04/246.28	   POL(CHECK_G(x_1)) = x_1
395.04/246.28	   POL(REDUCE(x_1)) = 2*x_1
395.04/246.28	   POL(b) = 0
395.04/246.28	   POL(f(x_1)) = 2*x_1
395.04/246.28	   POL(g(x_1)) = 2*x_1
395.04/246.28	   POL(redex_f(x_1)) = x_1
395.04/246.28	   POL(redex_g(x_1)) = 2*x_1
395.04/246.28	   POL(result_f(x_1)) = 2*x_1
395.04/246.28	   POL(result_g(x_1)) = x_1
395.04/246.28	
395.04/246.28	
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(13)
395.04/246.28	Obligation:
395.04/246.28	Q DP problem:
395.04/246.28	The TRS P consists of the following rules:
395.04/246.28	
395.04/246.28	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
395.04/246.28	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
395.04/246.28	
395.04/246.28	R is empty.
395.04/246.28	The set Q consists of the following terms:
395.04/246.28	
395.04/246.28	   redex_f(f(f(x0)))
395.04/246.28	   redex_g(g(g(x0)))
395.04/246.28	   redex_g(f(f(g(x0))))
395.04/246.28	   redex_f(g(g(f(x0))))
395.04/246.28	   redex_g(f(g(f(x0))))
395.04/246.28	
395.04/246.28	We have to consider all minimal (P,Q,R)-chains.
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(14) DependencyGraphProof (EQUIVALENT)
395.04/246.28	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(15)
395.04/246.28	TRUE
395.04/246.28	
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(16)
395.04/246.28	Obligation:
395.04/246.28	Q DP problem:
395.04/246.28	The TRS P consists of the following rules:
395.04/246.28	
395.04/246.28	   TOP(go_up(x)) -> TOP(reduce(x))
395.04/246.28	
395.04/246.28	The TRS R consists of the following rules:
395.04/246.28	
395.04/246.28	   top(go_up(x)) -> top(reduce(x))
395.04/246.28	   reduce(f(x_1)) -> check_f(redex_f(x_1))
395.04/246.28	   reduce(g(x_1)) -> check_g(redex_g(x_1))
395.04/246.28	   reduce(a) -> go_up(f(a))
395.04/246.28	   reduce(a) -> go_up(g(a))
395.04/246.28	   redex_f(f(f(x))) -> result_f(b)
395.04/246.28	   redex_g(g(g(x))) -> result_g(b)
395.04/246.28	   redex_g(f(f(g(x)))) -> result_g(b)
395.04/246.28	   redex_f(g(g(f(x)))) -> result_f(b)
395.04/246.28	   redex_g(f(g(f(x)))) -> result_g(b)
395.04/246.28	   check_f(result_f(x)) -> go_up(x)
395.04/246.28	   check_g(result_g(x)) -> go_up(x)
395.04/246.28	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
395.04/246.28	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
395.04/246.28	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
395.04/246.28	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
395.04/246.28	
395.04/246.28	The set Q consists of the following terms:
395.04/246.28	
395.04/246.28	   top(go_up(x0))
395.04/246.28	   reduce(f(x0))
395.04/246.28	   reduce(g(x0))
395.04/246.28	   reduce(a)
395.04/246.28	   redex_f(f(f(x0)))
395.04/246.28	   redex_g(g(g(x0)))
395.04/246.28	   redex_g(f(f(g(x0))))
395.04/246.28	   redex_f(g(g(f(x0))))
395.04/246.28	   redex_g(f(g(f(x0))))
395.04/246.28	   check_f(result_f(x0))
395.04/246.28	   check_g(result_g(x0))
395.04/246.28	   check_f(redex_f(x0))
395.04/246.28	   check_g(redex_g(x0))
395.04/246.28	   in_f_1(go_up(x0))
395.04/246.28	   in_g_1(go_up(x0))
395.04/246.28	
395.04/246.28	We have to consider all minimal (P,Q,R)-chains.
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(17) UsableRulesProof (EQUIVALENT)
395.04/246.28	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.
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(18)
395.04/246.28	Obligation:
395.04/246.28	Q DP problem:
395.04/246.28	The TRS P consists of the following rules:
395.04/246.28	
395.04/246.28	   TOP(go_up(x)) -> TOP(reduce(x))
395.04/246.28	
395.04/246.28	The TRS R consists of the following rules:
395.04/246.28	
395.04/246.28	   reduce(f(x_1)) -> check_f(redex_f(x_1))
395.04/246.28	   reduce(g(x_1)) -> check_g(redex_g(x_1))
395.04/246.28	   reduce(a) -> go_up(f(a))
395.04/246.28	   reduce(a) -> go_up(g(a))
395.04/246.28	   redex_g(g(g(x))) -> result_g(b)
395.04/246.28	   redex_g(f(f(g(x)))) -> result_g(b)
395.04/246.28	   redex_g(f(g(f(x)))) -> result_g(b)
395.04/246.28	   check_g(result_g(x)) -> go_up(x)
395.04/246.28	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
395.04/246.28	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
395.04/246.28	   redex_f(f(f(x))) -> result_f(b)
395.04/246.28	   redex_f(g(g(f(x)))) -> result_f(b)
395.04/246.28	   check_f(result_f(x)) -> go_up(x)
395.04/246.28	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
395.04/246.28	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
395.04/246.28	
395.04/246.28	The set Q consists of the following terms:
395.04/246.28	
395.04/246.28	   top(go_up(x0))
395.04/246.28	   reduce(f(x0))
395.04/246.28	   reduce(g(x0))
395.04/246.28	   reduce(a)
395.04/246.28	   redex_f(f(f(x0)))
395.04/246.28	   redex_g(g(g(x0)))
395.04/246.28	   redex_g(f(f(g(x0))))
395.04/246.28	   redex_f(g(g(f(x0))))
395.04/246.28	   redex_g(f(g(f(x0))))
395.04/246.28	   check_f(result_f(x0))
395.04/246.28	   check_g(result_g(x0))
395.04/246.28	   check_f(redex_f(x0))
395.04/246.28	   check_g(redex_g(x0))
395.04/246.28	   in_f_1(go_up(x0))
395.04/246.28	   in_g_1(go_up(x0))
395.04/246.28	
395.04/246.28	We have to consider all minimal (P,Q,R)-chains.
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(19) QReductionProof (EQUIVALENT)
395.04/246.28	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
395.04/246.28	
395.04/246.28	   top(go_up(x0))
395.04/246.28	
395.04/246.28	
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(20)
395.04/246.28	Obligation:
395.04/246.28	Q DP problem:
395.04/246.28	The TRS P consists of the following rules:
395.04/246.28	
395.04/246.28	   TOP(go_up(x)) -> TOP(reduce(x))
395.04/246.28	
395.04/246.28	The TRS R consists of the following rules:
395.04/246.28	
395.04/246.28	   reduce(f(x_1)) -> check_f(redex_f(x_1))
395.04/246.28	   reduce(g(x_1)) -> check_g(redex_g(x_1))
395.04/246.28	   reduce(a) -> go_up(f(a))
395.04/246.28	   reduce(a) -> go_up(g(a))
395.04/246.28	   redex_g(g(g(x))) -> result_g(b)
395.04/246.28	   redex_g(f(f(g(x)))) -> result_g(b)
395.04/246.28	   redex_g(f(g(f(x)))) -> result_g(b)
395.04/246.28	   check_g(result_g(x)) -> go_up(x)
395.04/246.28	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
395.04/246.28	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
395.04/246.28	   redex_f(f(f(x))) -> result_f(b)
395.04/246.28	   redex_f(g(g(f(x)))) -> result_f(b)
395.04/246.28	   check_f(result_f(x)) -> go_up(x)
395.04/246.28	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
395.04/246.28	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
395.04/246.28	
395.04/246.28	The set Q consists of the following terms:
395.04/246.28	
395.04/246.28	   reduce(f(x0))
395.04/246.28	   reduce(g(x0))
395.04/246.28	   reduce(a)
395.04/246.28	   redex_f(f(f(x0)))
395.04/246.28	   redex_g(g(g(x0)))
395.04/246.28	   redex_g(f(f(g(x0))))
395.04/246.28	   redex_f(g(g(f(x0))))
395.04/246.28	   redex_g(f(g(f(x0))))
395.04/246.28	   check_f(result_f(x0))
395.04/246.28	   check_g(result_g(x0))
395.04/246.28	   check_f(redex_f(x0))
395.04/246.28	   check_g(redex_g(x0))
395.04/246.28	   in_f_1(go_up(x0))
395.04/246.28	   in_g_1(go_up(x0))
395.04/246.28	
395.04/246.28	We have to consider all minimal (P,Q,R)-chains.
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(21) TransformationProof (EQUIVALENT)
395.04/246.28	By narrowing [LPAR04] the rule TOP(go_up(x)) -> TOP(reduce(x)) at position [0] we obtained the following new rules [LPAR04]:
395.04/246.28	
395.04/246.28	   (TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))),TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))))
395.04/246.28	   (TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))),TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))))
395.04/246.28	   (TOP(go_up(a)) -> TOP(go_up(f(a))),TOP(go_up(a)) -> TOP(go_up(f(a))))
395.04/246.28	   (TOP(go_up(a)) -> TOP(go_up(g(a))),TOP(go_up(a)) -> TOP(go_up(g(a))))
395.04/246.28	
395.04/246.28	
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(22)
395.04/246.28	Obligation:
395.04/246.28	Q DP problem:
395.04/246.28	The TRS P consists of the following rules:
395.04/246.28	
395.04/246.28	   TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0)))
395.04/246.28	   TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0)))
395.04/246.28	   TOP(go_up(a)) -> TOP(go_up(f(a)))
395.04/246.28	   TOP(go_up(a)) -> TOP(go_up(g(a)))
395.04/246.28	
395.04/246.28	The TRS R consists of the following rules:
395.04/246.28	
395.04/246.28	   reduce(f(x_1)) -> check_f(redex_f(x_1))
395.04/246.28	   reduce(g(x_1)) -> check_g(redex_g(x_1))
395.04/246.28	   reduce(a) -> go_up(f(a))
395.04/246.28	   reduce(a) -> go_up(g(a))
395.04/246.28	   redex_g(g(g(x))) -> result_g(b)
395.04/246.28	   redex_g(f(f(g(x)))) -> result_g(b)
395.04/246.28	   redex_g(f(g(f(x)))) -> result_g(b)
395.04/246.28	   check_g(result_g(x)) -> go_up(x)
395.04/246.28	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
395.04/246.28	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
395.04/246.28	   redex_f(f(f(x))) -> result_f(b)
395.04/246.28	   redex_f(g(g(f(x)))) -> result_f(b)
395.04/246.28	   check_f(result_f(x)) -> go_up(x)
395.04/246.28	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
395.04/246.28	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
395.04/246.28	
395.04/246.28	The set Q consists of the following terms:
395.04/246.28	
395.04/246.28	   reduce(f(x0))
395.04/246.28	   reduce(g(x0))
395.04/246.28	   reduce(a)
395.04/246.28	   redex_f(f(f(x0)))
395.04/246.28	   redex_g(g(g(x0)))
395.04/246.28	   redex_g(f(f(g(x0))))
395.04/246.28	   redex_f(g(g(f(x0))))
395.04/246.28	   redex_g(f(g(f(x0))))
395.04/246.28	   check_f(result_f(x0))
395.04/246.28	   check_g(result_g(x0))
395.04/246.28	   check_f(redex_f(x0))
395.04/246.28	   check_g(redex_g(x0))
395.04/246.28	   in_f_1(go_up(x0))
395.04/246.28	   in_g_1(go_up(x0))
395.04/246.28	
395.04/246.28	We have to consider all minimal (P,Q,R)-chains.
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(23) QDPOrderProof (EQUIVALENT)
395.04/246.28	We use the reduction pair processor [LPAR04,JAR06].
395.04/246.28	
395.04/246.28	
395.04/246.28	The following pairs can be oriented strictly and are deleted.
395.04/246.28	
395.04/246.28	   TOP(go_up(a)) -> TOP(go_up(f(a)))
395.04/246.28	   TOP(go_up(a)) -> TOP(go_up(g(a)))
395.04/246.28	The remaining pairs can at least be oriented weakly.
395.04/246.28	Used ordering:  Polynomial interpretation [POLO]:
395.04/246.28	
395.04/246.28	   POL(TOP(x_1)) = x_1
395.04/246.28	   POL(a) = 1
395.04/246.28	   POL(b) = 0
395.04/246.28	   POL(check_f(x_1)) = x_1
395.04/246.28	   POL(check_g(x_1)) = x_1
395.04/246.28	   POL(f(x_1)) = 0
395.04/246.28	   POL(g(x_1)) = 0
395.04/246.28	   POL(go_up(x_1)) = x_1
395.04/246.28	   POL(in_f_1(x_1)) = 0
395.04/246.28	   POL(in_g_1(x_1)) = 0
395.04/246.28	   POL(redex_f(x_1)) = 0
395.04/246.28	   POL(redex_g(x_1)) = 0
395.04/246.28	   POL(reduce(x_1)) = 0
395.04/246.28	   POL(result_f(x_1)) = x_1
395.04/246.28	   POL(result_g(x_1)) = x_1
395.04/246.28	
395.04/246.28	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
395.04/246.28	
395.04/246.28	   redex_f(f(f(x))) -> result_f(b)
395.04/246.28	   redex_f(g(g(f(x)))) -> result_f(b)
395.04/246.28	   check_f(result_f(x)) -> go_up(x)
395.04/246.28	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
395.04/246.28	   redex_g(g(g(x))) -> result_g(b)
395.04/246.28	   redex_g(f(f(g(x)))) -> result_g(b)
395.04/246.28	   redex_g(f(g(f(x)))) -> result_g(b)
395.04/246.28	   check_g(result_g(x)) -> go_up(x)
395.04/246.28	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
395.04/246.28	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
395.04/246.28	   reduce(f(x_1)) -> check_f(redex_f(x_1))
395.04/246.28	   reduce(g(x_1)) -> check_g(redex_g(x_1))
395.04/246.28	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
395.04/246.28	
395.04/246.28	
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(24)
395.04/246.28	Obligation:
395.04/246.28	Q DP problem:
395.04/246.28	The TRS P consists of the following rules:
395.04/246.28	
395.04/246.28	   TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0)))
395.04/246.28	   TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0)))
395.04/246.28	
395.04/246.28	The TRS R consists of the following rules:
395.04/246.28	
395.04/246.28	   reduce(f(x_1)) -> check_f(redex_f(x_1))
395.04/246.28	   reduce(g(x_1)) -> check_g(redex_g(x_1))
395.04/246.28	   reduce(a) -> go_up(f(a))
395.04/246.28	   reduce(a) -> go_up(g(a))
395.04/246.28	   redex_g(g(g(x))) -> result_g(b)
395.04/246.28	   redex_g(f(f(g(x)))) -> result_g(b)
395.04/246.28	   redex_g(f(g(f(x)))) -> result_g(b)
395.04/246.28	   check_g(result_g(x)) -> go_up(x)
395.04/246.28	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
395.04/246.28	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
395.04/246.28	   redex_f(f(f(x))) -> result_f(b)
395.04/246.28	   redex_f(g(g(f(x)))) -> result_f(b)
395.04/246.28	   check_f(result_f(x)) -> go_up(x)
395.04/246.28	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
395.04/246.28	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
395.04/246.28	
395.04/246.28	The set Q consists of the following terms:
395.04/246.28	
395.04/246.28	   reduce(f(x0))
395.04/246.28	   reduce(g(x0))
395.04/246.28	   reduce(a)
395.04/246.28	   redex_f(f(f(x0)))
395.04/246.28	   redex_g(g(g(x0)))
395.04/246.28	   redex_g(f(f(g(x0))))
395.04/246.28	   redex_f(g(g(f(x0))))
395.04/246.28	   redex_g(f(g(f(x0))))
395.04/246.28	   check_f(result_f(x0))
395.04/246.28	   check_g(result_g(x0))
395.04/246.28	   check_f(redex_f(x0))
395.04/246.28	   check_g(redex_g(x0))
395.04/246.28	   in_f_1(go_up(x0))
395.04/246.28	   in_g_1(go_up(x0))
395.04/246.28	
395.04/246.28	We have to consider all minimal (P,Q,R)-chains.
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(25) UsableRulesProof (EQUIVALENT)
395.04/246.28	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.
395.04/246.28	----------------------------------------
395.04/246.28	
395.04/246.28	(26)
395.04/246.28	Obligation:
395.04/246.28	Q DP problem:
395.04/246.28	The TRS P consists of the following rules:
395.04/246.28	
395.04/246.28	   TOP(go_up(x)) -> TOP(reduce(x))
395.04/246.28	
395.04/246.28	The TRS R consists of the following rules:
395.04/246.28	
395.04/246.28	   reduce(f(x_1)) -> check_f(redex_f(x_1))
395.04/246.28	   reduce(g(x_1)) -> check_g(redex_g(x_1))
395.04/246.28	   reduce(a) -> go_up(f(a))
395.04/246.28	   reduce(a) -> go_up(g(a))
395.04/246.28	   redex_g(g(g(x))) -> result_g(b)
395.04/246.28	   redex_g(f(f(g(x)))) -> result_g(b)
395.04/246.28	   redex_g(f(g(f(x)))) -> result_g(b)
395.04/246.28	   check_g(result_g(x)) -> go_up(x)
395.04/246.28	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
395.04/246.28	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
395.04/246.28	   redex_f(f(f(x))) -> result_f(b)
395.04/246.28	   redex_f(g(g(f(x)))) -> result_f(b)
395.04/246.29	   check_f(result_f(x)) -> go_up(x)
395.04/246.29	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
395.04/246.29	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
395.04/246.29	
395.04/246.29	The set Q consists of the following terms:
395.04/246.29	
395.04/246.29	   top(go_up(x0))
395.04/246.29	   reduce(f(x0))
395.04/246.29	   reduce(g(x0))
395.04/246.29	   reduce(a)
395.04/246.29	   redex_f(f(f(x0)))
395.04/246.29	   redex_g(g(g(x0)))
395.04/246.29	   redex_g(f(f(g(x0))))
395.04/246.29	   redex_f(g(g(f(x0))))
395.04/246.29	   redex_g(f(g(f(x0))))
395.04/246.29	   check_f(result_f(x0))
395.04/246.29	   check_g(result_g(x0))
395.04/246.29	   check_f(redex_f(x0))
395.04/246.29	   check_g(redex_g(x0))
395.04/246.29	   in_f_1(go_up(x0))
395.04/246.29	   in_g_1(go_up(x0))
395.04/246.29	
395.04/246.29	We have to consider all minimal (P,Q,R)-chains.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(27) QReductionProof (EQUIVALENT)
395.04/246.29	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
395.04/246.29	
395.04/246.29	   top(go_up(x0))
395.04/246.29	
395.04/246.29	
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(28)
395.04/246.29	Obligation:
395.04/246.29	Q DP problem:
395.04/246.29	The TRS P consists of the following rules:
395.04/246.29	
395.04/246.29	   TOP(go_up(x)) -> TOP(reduce(x))
395.04/246.29	
395.04/246.29	The TRS R consists of the following rules:
395.04/246.29	
395.04/246.29	   reduce(f(x_1)) -> check_f(redex_f(x_1))
395.04/246.29	   reduce(g(x_1)) -> check_g(redex_g(x_1))
395.04/246.29	   reduce(a) -> go_up(f(a))
395.04/246.29	   reduce(a) -> go_up(g(a))
395.04/246.29	   redex_g(g(g(x))) -> result_g(b)
395.04/246.29	   redex_g(f(f(g(x)))) -> result_g(b)
395.04/246.29	   redex_g(f(g(f(x)))) -> result_g(b)
395.04/246.29	   check_g(result_g(x)) -> go_up(x)
395.04/246.29	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
395.04/246.29	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
395.04/246.29	   redex_f(f(f(x))) -> result_f(b)
395.04/246.29	   redex_f(g(g(f(x)))) -> result_f(b)
395.04/246.29	   check_f(result_f(x)) -> go_up(x)
395.04/246.29	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
395.04/246.29	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
395.04/246.29	
395.04/246.29	The set Q consists of the following terms:
395.04/246.29	
395.04/246.29	   reduce(f(x0))
395.04/246.29	   reduce(g(x0))
395.04/246.29	   reduce(a)
395.04/246.29	   redex_f(f(f(x0)))
395.04/246.29	   redex_g(g(g(x0)))
395.04/246.29	   redex_g(f(f(g(x0))))
395.04/246.29	   redex_f(g(g(f(x0))))
395.04/246.29	   redex_g(f(g(f(x0))))
395.04/246.29	   check_f(result_f(x0))
395.04/246.29	   check_g(result_g(x0))
395.04/246.29	   check_f(redex_f(x0))
395.04/246.29	   check_g(redex_g(x0))
395.04/246.29	   in_f_1(go_up(x0))
395.04/246.29	   in_g_1(go_up(x0))
395.04/246.29	
395.04/246.29	We have to consider all minimal (P,Q,R)-chains.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(29) Trivial-Transformation (SOUND)
395.04/246.29	We applied the Trivial transformation  to transform the outermost TRS to a standard TRS.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(30)
395.04/246.29	Obligation:
395.04/246.29	Q restricted rewrite system:
395.04/246.29	The TRS R consists of the following rules:
395.04/246.29	
395.04/246.29	   a -> f(a)
395.04/246.29	   a -> g(a)
395.04/246.29	   f(f(f(x))) -> b
395.04/246.29	   g(g(g(x))) -> b
395.04/246.29	   g(f(f(g(x)))) -> b
395.04/246.29	   f(g(g(f(x)))) -> b
395.04/246.29	   g(f(g(f(x)))) -> b
395.04/246.29	
395.04/246.29	Q is empty.
395.04/246.29	
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(31) DependencyPairsProof (EQUIVALENT)
395.04/246.29	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(32)
395.04/246.29	Obligation:
395.04/246.29	Q DP problem:
395.04/246.29	The TRS P consists of the following rules:
395.04/246.29	
395.04/246.29	   A -> F(a)
395.04/246.29	   A -> A
395.04/246.29	   A -> G(a)
395.04/246.29	
395.04/246.29	The TRS R consists of the following rules:
395.04/246.29	
395.04/246.29	   a -> f(a)
395.04/246.29	   a -> g(a)
395.04/246.29	   f(f(f(x))) -> b
395.04/246.29	   g(g(g(x))) -> b
395.04/246.29	   g(f(f(g(x)))) -> b
395.04/246.29	   f(g(g(f(x)))) -> b
395.04/246.29	   g(f(g(f(x)))) -> b
395.04/246.29	
395.04/246.29	Q is empty.
395.04/246.29	We have to consider all minimal (P,Q,R)-chains.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(33) DependencyGraphProof (EQUIVALENT)
395.04/246.29	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(34)
395.04/246.29	Obligation:
395.04/246.29	Q DP problem:
395.04/246.29	The TRS P consists of the following rules:
395.04/246.29	
395.04/246.29	   A -> A
395.04/246.29	
395.04/246.29	The TRS R consists of the following rules:
395.04/246.29	
395.04/246.29	   a -> f(a)
395.04/246.29	   a -> g(a)
395.04/246.29	   f(f(f(x))) -> b
395.04/246.29	   g(g(g(x))) -> b
395.04/246.29	   g(f(f(g(x)))) -> b
395.04/246.29	   f(g(g(f(x)))) -> b
395.04/246.29	   g(f(g(f(x)))) -> b
395.04/246.29	
395.04/246.29	Q is empty.
395.04/246.29	We have to consider all minimal (P,Q,R)-chains.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(35) UsableRulesProof (EQUIVALENT)
395.04/246.29	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.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(36)
395.04/246.29	Obligation:
395.04/246.29	Q DP problem:
395.04/246.29	The TRS P consists of the following rules:
395.04/246.29	
395.04/246.29	   A -> A
395.04/246.29	
395.04/246.29	R is empty.
395.04/246.29	Q is empty.
395.04/246.29	We have to consider all minimal (P,Q,R)-chains.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(37) NonTerminationLoopProof (COMPLETE)
395.04/246.29	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
395.04/246.29	Found a loop by semiunifying a rule from P directly.
395.04/246.29	
395.04/246.29	s = A evaluates to  t =A
395.04/246.29	
395.04/246.29	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
395.04/246.29	* Matcher: [ ]
395.04/246.29	* Semiunifier: [ ]
395.04/246.29	
395.04/246.29	--------------------------------------------------------------------------------
395.04/246.29	Rewriting sequence
395.04/246.29	
395.04/246.29	The DP semiunifies directly so there is only one rewrite step from A to A.
395.04/246.29	
395.04/246.29	
395.04/246.29	
395.04/246.29	
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(38)
395.04/246.29	NO
395.04/246.29	
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(39) Raffelsieper-Zantema-Transformation (SOUND)
395.04/246.29	We applied the Raffelsieper-Zantema transformation  to transform the outermost TRS to a standard TRS.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(40)
395.04/246.29	Obligation:
395.04/246.29	Q restricted rewrite system:
395.04/246.29	The TRS R consists of the following rules:
395.04/246.29	
395.04/246.29	   down(a) -> up(f(a))
395.04/246.29	   down(a) -> up(g(a))
395.04/246.29	   down(f(f(f(x)))) -> up(b)
395.04/246.29	   down(g(g(g(x)))) -> up(b)
395.04/246.29	   down(g(f(f(g(x))))) -> up(b)
395.04/246.29	   down(f(g(g(f(x))))) -> up(b)
395.04/246.29	   down(g(f(g(f(x))))) -> up(b)
395.04/246.29	   top(up(x)) -> top(down(x))
395.04/246.29	   down(f(a)) -> f_flat(down(a))
395.04/246.29	   down(f(b)) -> f_flat(down(b))
395.04/246.29	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.04/246.29	   down(g(a)) -> g_flat(down(a))
395.04/246.29	   down(g(b)) -> g_flat(down(b))
395.04/246.29	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.04/246.29	   down(f(f(a))) -> f_flat(down(f(a)))
395.04/246.29	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
395.04/246.29	   down(f(f(b))) -> f_flat(down(f(b)))
395.04/246.29	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.04/246.29	   down(f(g(a))) -> f_flat(down(g(a)))
395.04/246.29	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.04/246.29	   down(f(g(b))) -> f_flat(down(g(b)))
395.04/246.29	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.04/246.29	   down(g(f(a))) -> g_flat(down(f(a)))
395.04/246.29	   down(g(f(b))) -> g_flat(down(f(b)))
395.04/246.29	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.04/246.29	   down(g(g(a))) -> g_flat(down(g(a)))
395.04/246.29	   down(g(g(f(y18)))) -> g_flat(down(g(f(y18))))
395.04/246.29	   down(g(g(b))) -> g_flat(down(g(b)))
395.04/246.29	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.04/246.29	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.04/246.29	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.04/246.29	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.04/246.29	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.04/246.29	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.04/246.29	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.04/246.29	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.04/246.29	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.04/246.29	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.04/246.29	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.04/246.29	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.04/246.29	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.04/246.29	   f_flat(up(x_1)) -> up(f(x_1))
395.04/246.29	   g_flat(up(x_1)) -> up(g(x_1))
395.04/246.29	
395.04/246.29	Q is empty.
395.04/246.29	
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(41) DependencyPairsProof (EQUIVALENT)
395.04/246.29	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(42)
395.04/246.29	Obligation:
395.04/246.29	Q DP problem:
395.04/246.29	The TRS P consists of the following rules:
395.04/246.29	
395.04/246.29	   TOP(up(x)) -> TOP(down(x))
395.04/246.29	   TOP(up(x)) -> DOWN(x)
395.04/246.29	   DOWN(f(a)) -> F_FLAT(down(a))
395.04/246.29	   DOWN(f(a)) -> DOWN(a)
395.04/246.29	   DOWN(f(b)) -> F_FLAT(down(b))
395.04/246.29	   DOWN(f(b)) -> DOWN(b)
395.04/246.29	   DOWN(f(fresh_constant)) -> F_FLAT(down(fresh_constant))
395.04/246.29	   DOWN(f(fresh_constant)) -> DOWN(fresh_constant)
395.04/246.29	   DOWN(g(a)) -> G_FLAT(down(a))
395.04/246.29	   DOWN(g(a)) -> DOWN(a)
395.04/246.29	   DOWN(g(b)) -> G_FLAT(down(b))
395.04/246.29	   DOWN(g(b)) -> DOWN(b)
395.04/246.29	   DOWN(g(fresh_constant)) -> G_FLAT(down(fresh_constant))
395.04/246.29	   DOWN(g(fresh_constant)) -> DOWN(fresh_constant)
395.04/246.29	   DOWN(f(f(a))) -> F_FLAT(down(f(a)))
395.04/246.29	   DOWN(f(f(a))) -> DOWN(f(a))
395.04/246.29	   DOWN(f(f(g(y10)))) -> F_FLAT(down(f(g(y10))))
395.04/246.29	   DOWN(f(f(g(y10)))) -> DOWN(f(g(y10)))
395.04/246.29	   DOWN(f(f(b))) -> F_FLAT(down(f(b)))
395.04/246.29	   DOWN(f(f(b))) -> DOWN(f(b))
395.04/246.29	   DOWN(f(f(fresh_constant))) -> F_FLAT(down(f(fresh_constant)))
395.04/246.29	   DOWN(f(f(fresh_constant))) -> DOWN(f(fresh_constant))
395.04/246.29	   DOWN(f(g(a))) -> F_FLAT(down(g(a)))
395.04/246.29	   DOWN(f(g(a))) -> DOWN(g(a))
395.04/246.29	   DOWN(f(g(f(y12)))) -> F_FLAT(down(g(f(y12))))
395.04/246.29	   DOWN(f(g(f(y12)))) -> DOWN(g(f(y12)))
395.04/246.29	   DOWN(f(g(b))) -> F_FLAT(down(g(b)))
395.04/246.29	   DOWN(f(g(b))) -> DOWN(g(b))
395.04/246.29	   DOWN(f(g(fresh_constant))) -> F_FLAT(down(g(fresh_constant)))
395.04/246.29	   DOWN(f(g(fresh_constant))) -> DOWN(g(fresh_constant))
395.04/246.29	   DOWN(g(f(a))) -> G_FLAT(down(f(a)))
395.04/246.29	   DOWN(g(f(a))) -> DOWN(f(a))
395.04/246.29	   DOWN(g(f(b))) -> G_FLAT(down(f(b)))
395.04/246.29	   DOWN(g(f(b))) -> DOWN(f(b))
395.04/246.29	   DOWN(g(f(fresh_constant))) -> G_FLAT(down(f(fresh_constant)))
395.04/246.29	   DOWN(g(f(fresh_constant))) -> DOWN(f(fresh_constant))
395.04/246.29	   DOWN(g(g(a))) -> G_FLAT(down(g(a)))
395.04/246.29	   DOWN(g(g(a))) -> DOWN(g(a))
395.04/246.29	   DOWN(g(g(f(y18)))) -> G_FLAT(down(g(f(y18))))
395.04/246.29	   DOWN(g(g(f(y18)))) -> DOWN(g(f(y18)))
395.04/246.29	   DOWN(g(g(b))) -> G_FLAT(down(g(b)))
395.04/246.29	   DOWN(g(g(b))) -> DOWN(g(b))
395.04/246.29	   DOWN(g(g(fresh_constant))) -> G_FLAT(down(g(fresh_constant)))
395.04/246.29	   DOWN(g(g(fresh_constant))) -> DOWN(g(fresh_constant))
395.04/246.29	   DOWN(f(g(g(a)))) -> F_FLAT(down(g(g(a))))
395.04/246.29	   DOWN(f(g(g(a)))) -> DOWN(g(g(a)))
395.04/246.29	   DOWN(f(g(g(g(y22))))) -> F_FLAT(down(g(g(g(y22)))))
395.04/246.29	   DOWN(f(g(g(g(y22))))) -> DOWN(g(g(g(y22))))
395.04/246.29	   DOWN(f(g(g(b)))) -> F_FLAT(down(g(g(b))))
395.04/246.29	   DOWN(f(g(g(b)))) -> DOWN(g(g(b)))
395.04/246.29	   DOWN(f(g(g(fresh_constant)))) -> F_FLAT(down(g(g(fresh_constant))))
395.04/246.29	   DOWN(f(g(g(fresh_constant)))) -> DOWN(g(g(fresh_constant)))
395.04/246.29	   DOWN(g(f(f(a)))) -> G_FLAT(down(f(f(a))))
395.04/246.29	   DOWN(g(f(f(a)))) -> DOWN(f(f(a)))
395.04/246.29	   DOWN(g(f(f(f(y24))))) -> G_FLAT(down(f(f(f(y24)))))
395.04/246.29	   DOWN(g(f(f(f(y24))))) -> DOWN(f(f(f(y24))))
395.04/246.29	   DOWN(g(f(f(b)))) -> G_FLAT(down(f(f(b))))
395.04/246.29	   DOWN(g(f(f(b)))) -> DOWN(f(f(b)))
395.04/246.29	   DOWN(g(f(f(fresh_constant)))) -> G_FLAT(down(f(f(fresh_constant))))
395.04/246.29	   DOWN(g(f(f(fresh_constant)))) -> DOWN(f(f(fresh_constant)))
395.04/246.29	   DOWN(g(f(g(a)))) -> G_FLAT(down(f(g(a))))
395.04/246.29	   DOWN(g(f(g(a)))) -> DOWN(f(g(a)))
395.04/246.29	   DOWN(g(f(g(g(y28))))) -> G_FLAT(down(f(g(g(y28)))))
395.04/246.29	   DOWN(g(f(g(g(y28))))) -> DOWN(f(g(g(y28))))
395.04/246.29	   DOWN(g(f(g(b)))) -> G_FLAT(down(f(g(b))))
395.04/246.29	   DOWN(g(f(g(b)))) -> DOWN(f(g(b)))
395.04/246.29	   DOWN(g(f(g(fresh_constant)))) -> G_FLAT(down(f(g(fresh_constant))))
395.04/246.29	   DOWN(g(f(g(fresh_constant)))) -> DOWN(f(g(fresh_constant)))
395.04/246.29	
395.04/246.29	The TRS R consists of the following rules:
395.04/246.29	
395.04/246.29	   down(a) -> up(f(a))
395.04/246.29	   down(a) -> up(g(a))
395.04/246.29	   down(f(f(f(x)))) -> up(b)
395.04/246.29	   down(g(g(g(x)))) -> up(b)
395.04/246.29	   down(g(f(f(g(x))))) -> up(b)
395.04/246.29	   down(f(g(g(f(x))))) -> up(b)
395.04/246.29	   down(g(f(g(f(x))))) -> up(b)
395.04/246.29	   top(up(x)) -> top(down(x))
395.04/246.29	   down(f(a)) -> f_flat(down(a))
395.04/246.29	   down(f(b)) -> f_flat(down(b))
395.04/246.29	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.04/246.29	   down(g(a)) -> g_flat(down(a))
395.04/246.29	   down(g(b)) -> g_flat(down(b))
395.04/246.29	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.04/246.29	   down(f(f(a))) -> f_flat(down(f(a)))
395.04/246.29	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
395.04/246.29	   down(f(f(b))) -> f_flat(down(f(b)))
395.04/246.29	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.04/246.29	   down(f(g(a))) -> f_flat(down(g(a)))
395.04/246.29	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.04/246.29	   down(f(g(b))) -> f_flat(down(g(b)))
395.04/246.29	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.04/246.29	   down(g(f(a))) -> g_flat(down(f(a)))
395.04/246.29	   down(g(f(b))) -> g_flat(down(f(b)))
395.04/246.29	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.04/246.29	   down(g(g(a))) -> g_flat(down(g(a)))
395.04/246.29	   down(g(g(f(y18)))) -> g_flat(down(g(f(y18))))
395.04/246.29	   down(g(g(b))) -> g_flat(down(g(b)))
395.04/246.29	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.04/246.29	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.04/246.29	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.04/246.29	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.04/246.29	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.04/246.29	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.04/246.29	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.04/246.29	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.04/246.29	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.04/246.29	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.04/246.29	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.04/246.29	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.04/246.29	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.04/246.29	   f_flat(up(x_1)) -> up(f(x_1))
395.04/246.29	   g_flat(up(x_1)) -> up(g(x_1))
395.04/246.29	
395.04/246.29	Q is empty.
395.04/246.29	We have to consider all minimal (P,Q,R)-chains.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(43) DependencyGraphProof (EQUIVALENT)
395.04/246.29	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 67 less nodes.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(44)
395.04/246.29	Obligation:
395.04/246.29	Q DP problem:
395.04/246.29	The TRS P consists of the following rules:
395.04/246.29	
395.04/246.29	   TOP(up(x)) -> TOP(down(x))
395.04/246.29	
395.04/246.29	The TRS R consists of the following rules:
395.04/246.29	
395.04/246.29	   down(a) -> up(f(a))
395.04/246.29	   down(a) -> up(g(a))
395.04/246.29	   down(f(f(f(x)))) -> up(b)
395.04/246.29	   down(g(g(g(x)))) -> up(b)
395.04/246.29	   down(g(f(f(g(x))))) -> up(b)
395.04/246.29	   down(f(g(g(f(x))))) -> up(b)
395.04/246.29	   down(g(f(g(f(x))))) -> up(b)
395.04/246.29	   top(up(x)) -> top(down(x))
395.04/246.29	   down(f(a)) -> f_flat(down(a))
395.04/246.29	   down(f(b)) -> f_flat(down(b))
395.04/246.29	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.04/246.29	   down(g(a)) -> g_flat(down(a))
395.04/246.29	   down(g(b)) -> g_flat(down(b))
395.04/246.29	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.04/246.29	   down(f(f(a))) -> f_flat(down(f(a)))
395.04/246.29	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
395.04/246.29	   down(f(f(b))) -> f_flat(down(f(b)))
395.04/246.29	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.04/246.29	   down(f(g(a))) -> f_flat(down(g(a)))
395.04/246.29	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.04/246.29	   down(f(g(b))) -> f_flat(down(g(b)))
395.04/246.29	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.04/246.29	   down(g(f(a))) -> g_flat(down(f(a)))
395.04/246.29	   down(g(f(b))) -> g_flat(down(f(b)))
395.04/246.29	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.04/246.29	   down(g(g(a))) -> g_flat(down(g(a)))
395.04/246.29	   down(g(g(f(y18)))) -> g_flat(down(g(f(y18))))
395.04/246.29	   down(g(g(b))) -> g_flat(down(g(b)))
395.04/246.29	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.04/246.29	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.04/246.29	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.04/246.29	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.04/246.29	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.04/246.29	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.04/246.29	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.04/246.29	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.04/246.29	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.04/246.29	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.04/246.29	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.04/246.29	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.04/246.29	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.04/246.29	   f_flat(up(x_1)) -> up(f(x_1))
395.04/246.29	   g_flat(up(x_1)) -> up(g(x_1))
395.04/246.29	
395.04/246.29	Q is empty.
395.04/246.29	We have to consider all minimal (P,Q,R)-chains.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(45) UsableRulesProof (EQUIVALENT)
395.04/246.29	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.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(46)
395.04/246.29	Obligation:
395.04/246.29	Q DP problem:
395.04/246.29	The TRS P consists of the following rules:
395.04/246.29	
395.04/246.29	   TOP(up(x)) -> TOP(down(x))
395.04/246.29	
395.04/246.29	The TRS R consists of the following rules:
395.04/246.29	
395.04/246.29	   down(a) -> up(f(a))
395.04/246.29	   down(a) -> up(g(a))
395.04/246.29	   down(f(f(f(x)))) -> up(b)
395.04/246.29	   down(g(g(g(x)))) -> up(b)
395.04/246.29	   down(g(f(f(g(x))))) -> up(b)
395.04/246.29	   down(f(g(g(f(x))))) -> up(b)
395.04/246.29	   down(g(f(g(f(x))))) -> up(b)
395.04/246.29	   down(f(a)) -> f_flat(down(a))
395.04/246.29	   down(f(b)) -> f_flat(down(b))
395.04/246.29	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.04/246.29	   down(g(a)) -> g_flat(down(a))
395.04/246.29	   down(g(b)) -> g_flat(down(b))
395.04/246.29	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.04/246.29	   down(f(f(a))) -> f_flat(down(f(a)))
395.04/246.29	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
395.04/246.29	   down(f(f(b))) -> f_flat(down(f(b)))
395.04/246.29	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.04/246.29	   down(f(g(a))) -> f_flat(down(g(a)))
395.04/246.29	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.04/246.29	   down(f(g(b))) -> f_flat(down(g(b)))
395.04/246.29	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.04/246.29	   down(g(f(a))) -> g_flat(down(f(a)))
395.04/246.29	   down(g(f(b))) -> g_flat(down(f(b)))
395.04/246.29	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.04/246.29	   down(g(g(a))) -> g_flat(down(g(a)))
395.04/246.29	   down(g(g(f(y18)))) -> g_flat(down(g(f(y18))))
395.04/246.29	   down(g(g(b))) -> g_flat(down(g(b)))
395.04/246.29	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.04/246.29	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.04/246.29	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.04/246.29	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.04/246.29	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.04/246.29	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.04/246.29	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.04/246.29	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.04/246.29	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.04/246.29	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.04/246.29	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.04/246.29	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.04/246.29	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.04/246.29	   g_flat(up(x_1)) -> up(g(x_1))
395.04/246.29	   f_flat(up(x_1)) -> up(f(x_1))
395.04/246.29	
395.04/246.29	Q is empty.
395.04/246.29	We have to consider all minimal (P,Q,R)-chains.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(47) TransformationProof (EQUIVALENT)
395.04/246.29	By narrowing [LPAR04] the rule TOP(up(x)) -> TOP(down(x)) at position [0] we obtained the following new rules [LPAR04]:
395.04/246.29	
395.04/246.29	   (TOP(up(a)) -> TOP(up(f(a))),TOP(up(a)) -> TOP(up(f(a))))
395.04/246.29	   (TOP(up(a)) -> TOP(up(g(a))),TOP(up(a)) -> TOP(up(g(a))))
395.04/246.29	   (TOP(up(f(f(f(x0))))) -> TOP(up(b)),TOP(up(f(f(f(x0))))) -> TOP(up(b)))
395.04/246.29	   (TOP(up(g(g(g(x0))))) -> TOP(up(b)),TOP(up(g(g(g(x0))))) -> TOP(up(b)))
395.04/246.29	   (TOP(up(g(f(f(g(x0)))))) -> TOP(up(b)),TOP(up(g(f(f(g(x0)))))) -> TOP(up(b)))
395.04/246.29	   (TOP(up(f(g(g(f(x0)))))) -> TOP(up(b)),TOP(up(f(g(g(f(x0)))))) -> TOP(up(b)))
395.04/246.29	   (TOP(up(g(f(g(f(x0)))))) -> TOP(up(b)),TOP(up(g(f(g(f(x0)))))) -> TOP(up(b)))
395.04/246.29	   (TOP(up(f(a))) -> TOP(f_flat(down(a))),TOP(up(f(a))) -> TOP(f_flat(down(a))))
395.04/246.29	   (TOP(up(f(b))) -> TOP(f_flat(down(b))),TOP(up(f(b))) -> TOP(f_flat(down(b))))
395.04/246.29	   (TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))),TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))))
395.04/246.29	   (TOP(up(g(a))) -> TOP(g_flat(down(a))),TOP(up(g(a))) -> TOP(g_flat(down(a))))
395.04/246.29	   (TOP(up(g(b))) -> TOP(g_flat(down(b))),TOP(up(g(b))) -> TOP(g_flat(down(b))))
395.04/246.29	   (TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))),TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))))
395.04/246.29	   (TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))),TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))))
395.04/246.29	   (TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))),TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))))
395.04/246.29	   (TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))),TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))))
395.04/246.29	   (TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))),TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))))
395.04/246.29	   (TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a)))),TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a)))))
395.04/246.29	   (TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0))))),TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0))))))
395.04/246.29	   (TOP(up(f(g(b)))) -> TOP(f_flat(down(g(b)))),TOP(up(f(g(b)))) -> TOP(f_flat(down(g(b)))))
395.04/246.29	   (TOP(up(f(g(fresh_constant)))) -> TOP(f_flat(down(g(fresh_constant)))),TOP(up(f(g(fresh_constant)))) -> TOP(f_flat(down(g(fresh_constant)))))
395.04/246.29	   (TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a)))),TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a)))))
395.04/246.29	   (TOP(up(g(f(b)))) -> TOP(g_flat(down(f(b)))),TOP(up(g(f(b)))) -> TOP(g_flat(down(f(b)))))
395.04/246.29	   (TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant)))),TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant)))))
395.04/246.29	   (TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))),TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))))
395.04/246.29	   (TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))),TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))))
395.04/246.29	   (TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))),TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))))
395.04/246.29	   (TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))),TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))))
395.04/246.29	   (TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a))))),TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a))))))
395.04/246.29	   (TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0)))))),TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0)))))))
395.04/246.29	   (TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b))))),TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b))))))
395.04/246.29	   (TOP(up(f(g(g(fresh_constant))))) -> TOP(f_flat(down(g(g(fresh_constant))))),TOP(up(f(g(g(fresh_constant))))) -> TOP(f_flat(down(g(g(fresh_constant))))))
395.04/246.29	   (TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a))))),TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a))))))
395.04/246.29	   (TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0)))))),TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0)))))))
395.04/246.29	   (TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b))))),TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b))))))
395.04/246.29	   (TOP(up(g(f(f(fresh_constant))))) -> TOP(g_flat(down(f(f(fresh_constant))))),TOP(up(g(f(f(fresh_constant))))) -> TOP(g_flat(down(f(f(fresh_constant))))))
395.04/246.29	   (TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a))))),TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a))))))
395.04/246.29	   (TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0)))))),TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0)))))))
395.04/246.29	   (TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b))))),TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b))))))
395.04/246.29	   (TOP(up(g(f(g(fresh_constant))))) -> TOP(g_flat(down(f(g(fresh_constant))))),TOP(up(g(f(g(fresh_constant))))) -> TOP(g_flat(down(f(g(fresh_constant))))))
395.04/246.29	
395.04/246.29	
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(48)
395.04/246.29	Obligation:
395.04/246.29	Q DP problem:
395.04/246.29	The TRS P consists of the following rules:
395.04/246.29	
395.04/246.29	   TOP(up(a)) -> TOP(up(f(a)))
395.04/246.29	   TOP(up(a)) -> TOP(up(g(a)))
395.04/246.29	   TOP(up(f(f(f(x0))))) -> TOP(up(b))
395.04/246.29	   TOP(up(g(g(g(x0))))) -> TOP(up(b))
395.04/246.29	   TOP(up(g(f(f(g(x0)))))) -> TOP(up(b))
395.04/246.29	   TOP(up(f(g(g(f(x0)))))) -> TOP(up(b))
395.04/246.29	   TOP(up(g(f(g(f(x0)))))) -> TOP(up(b))
395.04/246.29	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
395.04/246.29	   TOP(up(f(b))) -> TOP(f_flat(down(b)))
395.04/246.29	   TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant)))
395.04/246.29	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
395.04/246.29	   TOP(up(g(b))) -> TOP(g_flat(down(b)))
395.04/246.29	   TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant)))
395.04/246.29	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.04/246.29	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.04/246.29	   TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b))))
395.04/246.29	   TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant))))
395.04/246.29	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.04/246.29	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.04/246.29	   TOP(up(f(g(b)))) -> TOP(f_flat(down(g(b))))
395.04/246.29	   TOP(up(f(g(fresh_constant)))) -> TOP(f_flat(down(g(fresh_constant))))
395.04/246.29	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.04/246.29	   TOP(up(g(f(b)))) -> TOP(g_flat(down(f(b))))
395.04/246.29	   TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant))))
395.04/246.29	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.04/246.29	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.04/246.29	   TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b))))
395.04/246.29	   TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant))))
395.04/246.29	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.04/246.29	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.04/246.29	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.04/246.29	   TOP(up(f(g(g(fresh_constant))))) -> TOP(f_flat(down(g(g(fresh_constant)))))
395.04/246.29	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.04/246.29	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.04/246.29	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.04/246.29	   TOP(up(g(f(f(fresh_constant))))) -> TOP(g_flat(down(f(f(fresh_constant)))))
395.04/246.29	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.04/246.29	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.04/246.29	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.04/246.29	   TOP(up(g(f(g(fresh_constant))))) -> TOP(g_flat(down(f(g(fresh_constant)))))
395.04/246.29	
395.04/246.29	The TRS R consists of the following rules:
395.04/246.29	
395.04/246.29	   down(a) -> up(f(a))
395.04/246.29	   down(a) -> up(g(a))
395.04/246.29	   down(f(f(f(x)))) -> up(b)
395.04/246.29	   down(g(g(g(x)))) -> up(b)
395.04/246.29	   down(g(f(f(g(x))))) -> up(b)
395.04/246.29	   down(f(g(g(f(x))))) -> up(b)
395.04/246.29	   down(g(f(g(f(x))))) -> up(b)
395.04/246.29	   down(f(a)) -> f_flat(down(a))
395.04/246.29	   down(f(b)) -> f_flat(down(b))
395.04/246.29	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.04/246.29	   down(g(a)) -> g_flat(down(a))
395.04/246.29	   down(g(b)) -> g_flat(down(b))
395.04/246.29	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.04/246.29	   down(f(f(a))) -> f_flat(down(f(a)))
395.04/246.29	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
395.04/246.29	   down(f(f(b))) -> f_flat(down(f(b)))
395.04/246.29	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.04/246.29	   down(f(g(a))) -> f_flat(down(g(a)))
395.04/246.29	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.04/246.29	   down(f(g(b))) -> f_flat(down(g(b)))
395.04/246.29	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.04/246.29	   down(g(f(a))) -> g_flat(down(f(a)))
395.04/246.29	   down(g(f(b))) -> g_flat(down(f(b)))
395.04/246.29	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.04/246.29	   down(g(g(a))) -> g_flat(down(g(a)))
395.04/246.29	   down(g(g(f(y18)))) -> g_flat(down(g(f(y18))))
395.04/246.29	   down(g(g(b))) -> g_flat(down(g(b)))
395.04/246.29	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.04/246.29	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.04/246.29	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.04/246.29	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.04/246.29	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.04/246.29	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.04/246.29	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.04/246.29	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.04/246.29	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.04/246.29	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.04/246.29	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.04/246.29	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.04/246.29	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.04/246.29	   g_flat(up(x_1)) -> up(g(x_1))
395.04/246.29	   f_flat(up(x_1)) -> up(f(x_1))
395.04/246.29	
395.04/246.29	Q is empty.
395.04/246.29	We have to consider all minimal (P,Q,R)-chains.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(49) DependencyGraphProof (EQUIVALENT)
395.04/246.29	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 11 less nodes.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(50)
395.04/246.29	Obligation:
395.04/246.29	Q DP problem:
395.04/246.29	The TRS P consists of the following rules:
395.04/246.29	
395.04/246.29	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
395.04/246.29	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
395.04/246.29	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.04/246.29	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.04/246.29	   TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b))))
395.04/246.29	   TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant))))
395.04/246.29	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.04/246.29	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.04/246.29	   TOP(up(f(g(b)))) -> TOP(f_flat(down(g(b))))
395.04/246.29	   TOP(up(f(g(fresh_constant)))) -> TOP(f_flat(down(g(fresh_constant))))
395.04/246.29	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.04/246.29	   TOP(up(g(f(b)))) -> TOP(g_flat(down(f(b))))
395.04/246.29	   TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant))))
395.04/246.29	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.04/246.29	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.04/246.29	   TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b))))
395.04/246.29	   TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant))))
395.04/246.29	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.04/246.29	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.04/246.29	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.04/246.29	   TOP(up(f(g(g(fresh_constant))))) -> TOP(f_flat(down(g(g(fresh_constant)))))
395.04/246.29	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.04/246.29	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.04/246.29	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.04/246.29	   TOP(up(g(f(f(fresh_constant))))) -> TOP(g_flat(down(f(f(fresh_constant)))))
395.04/246.29	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.04/246.29	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.04/246.29	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.04/246.29	   TOP(up(g(f(g(fresh_constant))))) -> TOP(g_flat(down(f(g(fresh_constant)))))
395.04/246.29	
395.04/246.29	The TRS R consists of the following rules:
395.04/246.29	
395.04/246.29	   down(a) -> up(f(a))
395.04/246.29	   down(a) -> up(g(a))
395.04/246.29	   down(f(f(f(x)))) -> up(b)
395.04/246.29	   down(g(g(g(x)))) -> up(b)
395.04/246.29	   down(g(f(f(g(x))))) -> up(b)
395.04/246.29	   down(f(g(g(f(x))))) -> up(b)
395.04/246.29	   down(g(f(g(f(x))))) -> up(b)
395.04/246.29	   down(f(a)) -> f_flat(down(a))
395.04/246.29	   down(f(b)) -> f_flat(down(b))
395.04/246.29	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.04/246.29	   down(g(a)) -> g_flat(down(a))
395.04/246.29	   down(g(b)) -> g_flat(down(b))
395.04/246.29	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.04/246.29	   down(f(f(a))) -> f_flat(down(f(a)))
395.04/246.29	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
395.04/246.29	   down(f(f(b))) -> f_flat(down(f(b)))
395.04/246.29	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.04/246.29	   down(f(g(a))) -> f_flat(down(g(a)))
395.04/246.29	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.04/246.29	   down(f(g(b))) -> f_flat(down(g(b)))
395.04/246.29	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.04/246.29	   down(g(f(a))) -> g_flat(down(f(a)))
395.04/246.29	   down(g(f(b))) -> g_flat(down(f(b)))
395.04/246.29	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.04/246.29	   down(g(g(a))) -> g_flat(down(g(a)))
395.04/246.29	   down(g(g(f(y18)))) -> g_flat(down(g(f(y18))))
395.04/246.29	   down(g(g(b))) -> g_flat(down(g(b)))
395.04/246.29	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.04/246.29	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.04/246.29	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.04/246.29	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.04/246.29	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.04/246.29	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.04/246.29	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.04/246.29	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.04/246.29	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.04/246.29	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.04/246.29	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.04/246.29	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.04/246.29	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.04/246.29	   g_flat(up(x_1)) -> up(g(x_1))
395.04/246.29	   f_flat(up(x_1)) -> up(f(x_1))
395.04/246.29	
395.04/246.29	Q is empty.
395.04/246.29	We have to consider all minimal (P,Q,R)-chains.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(51) UsableRulesProof (EQUIVALENT)
395.04/246.29	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.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(52)
395.04/246.29	Obligation:
395.04/246.29	Q DP problem:
395.04/246.29	The TRS P consists of the following rules:
395.04/246.29	
395.04/246.29	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
395.04/246.29	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
395.04/246.29	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.04/246.29	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.04/246.29	   TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b))))
395.04/246.29	   TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant))))
395.04/246.29	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.04/246.29	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.04/246.29	   TOP(up(f(g(b)))) -> TOP(f_flat(down(g(b))))
395.04/246.29	   TOP(up(f(g(fresh_constant)))) -> TOP(f_flat(down(g(fresh_constant))))
395.04/246.29	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.04/246.29	   TOP(up(g(f(b)))) -> TOP(g_flat(down(f(b))))
395.04/246.29	   TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant))))
395.04/246.29	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.04/246.29	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.04/246.29	   TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b))))
395.04/246.29	   TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant))))
395.04/246.29	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.04/246.29	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.04/246.29	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.04/246.29	   TOP(up(f(g(g(fresh_constant))))) -> TOP(f_flat(down(g(g(fresh_constant)))))
395.04/246.29	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.04/246.29	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.04/246.29	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.04/246.29	   TOP(up(g(f(f(fresh_constant))))) -> TOP(g_flat(down(f(f(fresh_constant)))))
395.04/246.29	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.04/246.29	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.04/246.29	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.04/246.29	   TOP(up(g(f(g(fresh_constant))))) -> TOP(g_flat(down(f(g(fresh_constant)))))
395.04/246.29	
395.04/246.29	The TRS R consists of the following rules:
395.04/246.29	
395.04/246.29	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.04/246.29	   g_flat(up(x_1)) -> up(g(x_1))
395.04/246.29	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.04/246.29	   f_flat(up(x_1)) -> up(f(x_1))
395.04/246.29	   down(f(g(b))) -> f_flat(down(g(b)))
395.04/246.29	   down(g(b)) -> g_flat(down(b))
395.04/246.29	   down(f(g(g(f(x))))) -> up(b)
395.04/246.29	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.04/246.29	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.04/246.29	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.04/246.29	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.04/246.29	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.04/246.29	   down(g(g(b))) -> g_flat(down(g(b)))
395.04/246.29	   down(g(g(g(x)))) -> up(b)
395.04/246.29	   down(g(g(a))) -> g_flat(down(g(a)))
395.04/246.29	   down(g(a)) -> g_flat(down(a))
395.04/246.29	   down(a) -> up(f(a))
395.04/246.29	   down(a) -> up(g(a))
395.04/246.29	   down(f(g(a))) -> f_flat(down(g(a)))
395.04/246.29	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.04/246.29	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.04/246.29	   down(f(f(b))) -> f_flat(down(f(b)))
395.04/246.29	   down(f(b)) -> f_flat(down(b))
395.04/246.29	   down(f(f(f(x)))) -> up(b)
395.04/246.29	   down(f(f(a))) -> f_flat(down(f(a)))
395.04/246.29	   down(f(a)) -> f_flat(down(a))
395.04/246.29	   down(g(f(f(g(x))))) -> up(b)
395.04/246.29	   down(g(f(g(f(x))))) -> up(b)
395.04/246.29	   down(g(f(a))) -> g_flat(down(f(a)))
395.04/246.29	   down(g(f(b))) -> g_flat(down(f(b)))
395.04/246.29	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.04/246.29	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.04/246.29	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.04/246.29	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.04/246.29	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.04/246.29	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.04/246.29	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.04/246.29	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.04/246.29	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.04/246.29	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.04/246.29	
395.04/246.29	Q is empty.
395.04/246.29	We have to consider all minimal (P,Q,R)-chains.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(53) TransformationProof (EQUIVALENT)
395.04/246.29	By narrowing [LPAR04] the rule TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) at position [0] we obtained the following new rules [LPAR04]:
395.04/246.29	
395.04/246.29	   (TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))),TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))))
395.04/246.29	
395.04/246.29	
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(54)
395.04/246.29	Obligation:
395.04/246.29	Q DP problem:
395.04/246.29	The TRS P consists of the following rules:
395.04/246.29	
395.04/246.29	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
395.04/246.29	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
395.04/246.29	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.04/246.29	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.04/246.29	   TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant))))
395.04/246.29	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.04/246.29	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.04/246.29	   TOP(up(f(g(b)))) -> TOP(f_flat(down(g(b))))
395.04/246.29	   TOP(up(f(g(fresh_constant)))) -> TOP(f_flat(down(g(fresh_constant))))
395.04/246.29	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.04/246.29	   TOP(up(g(f(b)))) -> TOP(g_flat(down(f(b))))
395.04/246.29	   TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant))))
395.04/246.29	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.04/246.29	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.04/246.29	   TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b))))
395.04/246.29	   TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant))))
395.04/246.29	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.04/246.29	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.04/246.29	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.04/246.29	   TOP(up(f(g(g(fresh_constant))))) -> TOP(f_flat(down(g(g(fresh_constant)))))
395.04/246.29	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.04/246.29	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.04/246.29	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.04/246.29	   TOP(up(g(f(f(fresh_constant))))) -> TOP(g_flat(down(f(f(fresh_constant)))))
395.04/246.29	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.04/246.29	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.04/246.29	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.04/246.29	   TOP(up(g(f(g(fresh_constant))))) -> TOP(g_flat(down(f(g(fresh_constant)))))
395.04/246.29	   TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b))))
395.04/246.29	
395.04/246.29	The TRS R consists of the following rules:
395.04/246.29	
395.04/246.29	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.04/246.29	   g_flat(up(x_1)) -> up(g(x_1))
395.04/246.29	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.04/246.29	   f_flat(up(x_1)) -> up(f(x_1))
395.04/246.29	   down(f(g(b))) -> f_flat(down(g(b)))
395.04/246.29	   down(g(b)) -> g_flat(down(b))
395.04/246.29	   down(f(g(g(f(x))))) -> up(b)
395.04/246.29	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.04/246.29	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.04/246.29	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.04/246.29	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.04/246.29	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.04/246.29	   down(g(g(b))) -> g_flat(down(g(b)))
395.04/246.29	   down(g(g(g(x)))) -> up(b)
395.04/246.29	   down(g(g(a))) -> g_flat(down(g(a)))
395.04/246.29	   down(g(a)) -> g_flat(down(a))
395.04/246.29	   down(a) -> up(f(a))
395.04/246.29	   down(a) -> up(g(a))
395.04/246.29	   down(f(g(a))) -> f_flat(down(g(a)))
395.04/246.29	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.04/246.29	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.04/246.29	   down(f(f(b))) -> f_flat(down(f(b)))
395.04/246.29	   down(f(b)) -> f_flat(down(b))
395.04/246.29	   down(f(f(f(x)))) -> up(b)
395.04/246.29	   down(f(f(a))) -> f_flat(down(f(a)))
395.04/246.29	   down(f(a)) -> f_flat(down(a))
395.04/246.29	   down(g(f(f(g(x))))) -> up(b)
395.04/246.29	   down(g(f(g(f(x))))) -> up(b)
395.04/246.29	   down(g(f(a))) -> g_flat(down(f(a)))
395.04/246.29	   down(g(f(b))) -> g_flat(down(f(b)))
395.04/246.29	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.04/246.29	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.04/246.29	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.04/246.29	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.04/246.29	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.04/246.29	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.04/246.29	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.04/246.29	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.04/246.29	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.04/246.29	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.04/246.29	
395.04/246.29	Q is empty.
395.04/246.29	We have to consider all minimal (P,Q,R)-chains.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(55) DependencyGraphProof (EQUIVALENT)
395.04/246.29	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(56)
395.04/246.29	Obligation:
395.04/246.29	Q DP problem:
395.04/246.29	The TRS P consists of the following rules:
395.04/246.29	
395.04/246.29	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
395.04/246.29	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
395.04/246.29	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.04/246.29	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.04/246.29	   TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant))))
395.04/246.29	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.04/246.29	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.04/246.29	   TOP(up(f(g(b)))) -> TOP(f_flat(down(g(b))))
395.04/246.29	   TOP(up(f(g(fresh_constant)))) -> TOP(f_flat(down(g(fresh_constant))))
395.04/246.29	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.04/246.29	   TOP(up(g(f(b)))) -> TOP(g_flat(down(f(b))))
395.04/246.29	   TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant))))
395.04/246.29	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.04/246.29	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.04/246.29	   TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b))))
395.04/246.29	   TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant))))
395.04/246.29	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.04/246.29	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.04/246.29	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.04/246.29	   TOP(up(f(g(g(fresh_constant))))) -> TOP(f_flat(down(g(g(fresh_constant)))))
395.04/246.29	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.04/246.29	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.04/246.29	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.04/246.29	   TOP(up(g(f(f(fresh_constant))))) -> TOP(g_flat(down(f(f(fresh_constant)))))
395.04/246.29	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.04/246.29	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.04/246.29	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.04/246.29	   TOP(up(g(f(g(fresh_constant))))) -> TOP(g_flat(down(f(g(fresh_constant)))))
395.04/246.29	
395.04/246.29	The TRS R consists of the following rules:
395.04/246.29	
395.04/246.29	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.04/246.29	   g_flat(up(x_1)) -> up(g(x_1))
395.04/246.29	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.04/246.29	   f_flat(up(x_1)) -> up(f(x_1))
395.04/246.29	   down(f(g(b))) -> f_flat(down(g(b)))
395.04/246.29	   down(g(b)) -> g_flat(down(b))
395.04/246.29	   down(f(g(g(f(x))))) -> up(b)
395.04/246.29	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.04/246.29	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.04/246.29	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.04/246.29	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.04/246.29	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.04/246.29	   down(g(g(b))) -> g_flat(down(g(b)))
395.04/246.29	   down(g(g(g(x)))) -> up(b)
395.04/246.29	   down(g(g(a))) -> g_flat(down(g(a)))
395.04/246.29	   down(g(a)) -> g_flat(down(a))
395.04/246.29	   down(a) -> up(f(a))
395.04/246.29	   down(a) -> up(g(a))
395.04/246.29	   down(f(g(a))) -> f_flat(down(g(a)))
395.04/246.29	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.04/246.29	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.04/246.29	   down(f(f(b))) -> f_flat(down(f(b)))
395.04/246.29	   down(f(b)) -> f_flat(down(b))
395.04/246.29	   down(f(f(f(x)))) -> up(b)
395.04/246.29	   down(f(f(a))) -> f_flat(down(f(a)))
395.04/246.29	   down(f(a)) -> f_flat(down(a))
395.04/246.29	   down(g(f(f(g(x))))) -> up(b)
395.04/246.29	   down(g(f(g(f(x))))) -> up(b)
395.04/246.29	   down(g(f(a))) -> g_flat(down(f(a)))
395.04/246.29	   down(g(f(b))) -> g_flat(down(f(b)))
395.04/246.29	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.04/246.29	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.04/246.29	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.04/246.29	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.04/246.29	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.04/246.29	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.04/246.29	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.04/246.29	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.04/246.29	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.04/246.29	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.04/246.29	
395.04/246.29	Q is empty.
395.04/246.29	We have to consider all minimal (P,Q,R)-chains.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(57) TransformationProof (EQUIVALENT)
395.04/246.29	By narrowing [LPAR04] the rule TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) at position [0] we obtained the following new rules [LPAR04]:
395.04/246.29	
395.04/246.29	   (TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(f_flat(down(fresh_constant)))),TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(f_flat(down(fresh_constant)))))
395.04/246.29	
395.04/246.29	
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(58)
395.04/246.29	Obligation:
395.04/246.29	Q DP problem:
395.04/246.29	The TRS P consists of the following rules:
395.04/246.29	
395.04/246.29	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
395.04/246.29	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
395.04/246.29	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.04/246.29	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.04/246.29	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.04/246.29	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.04/246.29	   TOP(up(f(g(b)))) -> TOP(f_flat(down(g(b))))
395.04/246.29	   TOP(up(f(g(fresh_constant)))) -> TOP(f_flat(down(g(fresh_constant))))
395.04/246.29	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.04/246.29	   TOP(up(g(f(b)))) -> TOP(g_flat(down(f(b))))
395.04/246.29	   TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant))))
395.04/246.29	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.04/246.29	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.04/246.29	   TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b))))
395.04/246.29	   TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant))))
395.04/246.29	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.04/246.29	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.04/246.29	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.04/246.29	   TOP(up(f(g(g(fresh_constant))))) -> TOP(f_flat(down(g(g(fresh_constant)))))
395.04/246.29	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.04/246.29	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.04/246.29	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.04/246.29	   TOP(up(g(f(f(fresh_constant))))) -> TOP(g_flat(down(f(f(fresh_constant)))))
395.04/246.29	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.04/246.29	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.04/246.29	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.04/246.29	   TOP(up(g(f(g(fresh_constant))))) -> TOP(g_flat(down(f(g(fresh_constant)))))
395.04/246.29	   TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(f_flat(down(fresh_constant))))
395.04/246.29	
395.04/246.29	The TRS R consists of the following rules:
395.04/246.29	
395.04/246.29	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.04/246.29	   g_flat(up(x_1)) -> up(g(x_1))
395.04/246.29	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.04/246.29	   f_flat(up(x_1)) -> up(f(x_1))
395.04/246.29	   down(f(g(b))) -> f_flat(down(g(b)))
395.04/246.29	   down(g(b)) -> g_flat(down(b))
395.04/246.29	   down(f(g(g(f(x))))) -> up(b)
395.04/246.29	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.04/246.29	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.04/246.29	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.04/246.29	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.04/246.29	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.04/246.29	   down(g(g(b))) -> g_flat(down(g(b)))
395.04/246.29	   down(g(g(g(x)))) -> up(b)
395.04/246.29	   down(g(g(a))) -> g_flat(down(g(a)))
395.04/246.29	   down(g(a)) -> g_flat(down(a))
395.04/246.29	   down(a) -> up(f(a))
395.04/246.29	   down(a) -> up(g(a))
395.04/246.29	   down(f(g(a))) -> f_flat(down(g(a)))
395.04/246.29	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.04/246.29	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.04/246.29	   down(f(f(b))) -> f_flat(down(f(b)))
395.04/246.29	   down(f(b)) -> f_flat(down(b))
395.04/246.29	   down(f(f(f(x)))) -> up(b)
395.04/246.29	   down(f(f(a))) -> f_flat(down(f(a)))
395.04/246.29	   down(f(a)) -> f_flat(down(a))
395.04/246.29	   down(g(f(f(g(x))))) -> up(b)
395.04/246.29	   down(g(f(g(f(x))))) -> up(b)
395.04/246.29	   down(g(f(a))) -> g_flat(down(f(a)))
395.04/246.29	   down(g(f(b))) -> g_flat(down(f(b)))
395.04/246.29	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.04/246.29	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.04/246.29	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.04/246.29	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.04/246.29	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.04/246.29	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.04/246.29	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.04/246.29	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.04/246.29	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.04/246.29	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.04/246.29	
395.04/246.29	Q is empty.
395.04/246.29	We have to consider all minimal (P,Q,R)-chains.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(59) DependencyGraphProof (EQUIVALENT)
395.04/246.29	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(60)
395.04/246.29	Obligation:
395.04/246.29	Q DP problem:
395.04/246.29	The TRS P consists of the following rules:
395.04/246.29	
395.04/246.29	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
395.04/246.29	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
395.04/246.29	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.04/246.29	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.04/246.29	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.04/246.29	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.04/246.29	   TOP(up(f(g(b)))) -> TOP(f_flat(down(g(b))))
395.04/246.29	   TOP(up(f(g(fresh_constant)))) -> TOP(f_flat(down(g(fresh_constant))))
395.04/246.29	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.04/246.29	   TOP(up(g(f(b)))) -> TOP(g_flat(down(f(b))))
395.04/246.29	   TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant))))
395.04/246.29	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.04/246.29	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.04/246.29	   TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b))))
395.04/246.29	   TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant))))
395.04/246.29	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.04/246.29	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.04/246.29	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.04/246.29	   TOP(up(f(g(g(fresh_constant))))) -> TOP(f_flat(down(g(g(fresh_constant)))))
395.04/246.29	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.04/246.29	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.04/246.29	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.04/246.29	   TOP(up(g(f(f(fresh_constant))))) -> TOP(g_flat(down(f(f(fresh_constant)))))
395.04/246.29	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.04/246.29	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.04/246.29	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.04/246.29	   TOP(up(g(f(g(fresh_constant))))) -> TOP(g_flat(down(f(g(fresh_constant)))))
395.04/246.29	
395.04/246.29	The TRS R consists of the following rules:
395.04/246.29	
395.04/246.29	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.04/246.29	   g_flat(up(x_1)) -> up(g(x_1))
395.04/246.29	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.04/246.29	   f_flat(up(x_1)) -> up(f(x_1))
395.04/246.29	   down(f(g(b))) -> f_flat(down(g(b)))
395.04/246.29	   down(g(b)) -> g_flat(down(b))
395.04/246.29	   down(f(g(g(f(x))))) -> up(b)
395.04/246.29	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.04/246.29	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.04/246.29	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.04/246.29	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.04/246.29	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.04/246.29	   down(g(g(b))) -> g_flat(down(g(b)))
395.04/246.29	   down(g(g(g(x)))) -> up(b)
395.04/246.29	   down(g(g(a))) -> g_flat(down(g(a)))
395.04/246.29	   down(g(a)) -> g_flat(down(a))
395.04/246.29	   down(a) -> up(f(a))
395.04/246.29	   down(a) -> up(g(a))
395.04/246.29	   down(f(g(a))) -> f_flat(down(g(a)))
395.04/246.29	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.04/246.29	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.04/246.29	   down(f(f(b))) -> f_flat(down(f(b)))
395.04/246.29	   down(f(b)) -> f_flat(down(b))
395.04/246.29	   down(f(f(f(x)))) -> up(b)
395.04/246.29	   down(f(f(a))) -> f_flat(down(f(a)))
395.04/246.29	   down(f(a)) -> f_flat(down(a))
395.04/246.29	   down(g(f(f(g(x))))) -> up(b)
395.04/246.29	   down(g(f(g(f(x))))) -> up(b)
395.04/246.29	   down(g(f(a))) -> g_flat(down(f(a)))
395.04/246.29	   down(g(f(b))) -> g_flat(down(f(b)))
395.04/246.29	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.04/246.29	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.04/246.29	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.04/246.29	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.04/246.29	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.04/246.29	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.04/246.29	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.04/246.29	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.04/246.29	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.04/246.29	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.04/246.29	
395.04/246.29	Q is empty.
395.04/246.29	We have to consider all minimal (P,Q,R)-chains.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(61) TransformationProof (EQUIVALENT)
395.04/246.29	By narrowing [LPAR04] the rule TOP(up(f(g(b)))) -> TOP(f_flat(down(g(b)))) at position [0] we obtained the following new rules [LPAR04]:
395.04/246.29	
395.04/246.29	   (TOP(up(f(g(b)))) -> TOP(f_flat(g_flat(down(b)))),TOP(up(f(g(b)))) -> TOP(f_flat(g_flat(down(b)))))
395.04/246.29	
395.04/246.29	
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(62)
395.04/246.29	Obligation:
395.04/246.29	Q DP problem:
395.04/246.29	The TRS P consists of the following rules:
395.04/246.29	
395.04/246.29	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
395.04/246.29	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
395.04/246.29	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.04/246.29	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.04/246.29	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.04/246.29	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.04/246.29	   TOP(up(f(g(fresh_constant)))) -> TOP(f_flat(down(g(fresh_constant))))
395.04/246.29	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.04/246.29	   TOP(up(g(f(b)))) -> TOP(g_flat(down(f(b))))
395.04/246.29	   TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant))))
395.04/246.29	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.04/246.29	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.04/246.29	   TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b))))
395.04/246.29	   TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant))))
395.04/246.29	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.04/246.29	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.04/246.29	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.04/246.29	   TOP(up(f(g(g(fresh_constant))))) -> TOP(f_flat(down(g(g(fresh_constant)))))
395.04/246.29	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.04/246.29	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.04/246.29	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.04/246.29	   TOP(up(g(f(f(fresh_constant))))) -> TOP(g_flat(down(f(f(fresh_constant)))))
395.04/246.29	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.04/246.29	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.04/246.29	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.04/246.29	   TOP(up(g(f(g(fresh_constant))))) -> TOP(g_flat(down(f(g(fresh_constant)))))
395.04/246.29	   TOP(up(f(g(b)))) -> TOP(f_flat(g_flat(down(b))))
395.04/246.29	
395.04/246.29	The TRS R consists of the following rules:
395.04/246.29	
395.04/246.29	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.04/246.29	   g_flat(up(x_1)) -> up(g(x_1))
395.04/246.29	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.04/246.29	   f_flat(up(x_1)) -> up(f(x_1))
395.04/246.29	   down(f(g(b))) -> f_flat(down(g(b)))
395.04/246.29	   down(g(b)) -> g_flat(down(b))
395.04/246.29	   down(f(g(g(f(x))))) -> up(b)
395.04/246.29	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.04/246.29	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.04/246.29	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.04/246.29	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.04/246.29	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.04/246.29	   down(g(g(b))) -> g_flat(down(g(b)))
395.04/246.29	   down(g(g(g(x)))) -> up(b)
395.04/246.29	   down(g(g(a))) -> g_flat(down(g(a)))
395.04/246.29	   down(g(a)) -> g_flat(down(a))
395.04/246.29	   down(a) -> up(f(a))
395.04/246.29	   down(a) -> up(g(a))
395.04/246.29	   down(f(g(a))) -> f_flat(down(g(a)))
395.04/246.29	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.04/246.29	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.04/246.29	   down(f(f(b))) -> f_flat(down(f(b)))
395.04/246.29	   down(f(b)) -> f_flat(down(b))
395.04/246.29	   down(f(f(f(x)))) -> up(b)
395.04/246.29	   down(f(f(a))) -> f_flat(down(f(a)))
395.04/246.29	   down(f(a)) -> f_flat(down(a))
395.04/246.29	   down(g(f(f(g(x))))) -> up(b)
395.04/246.29	   down(g(f(g(f(x))))) -> up(b)
395.04/246.29	   down(g(f(a))) -> g_flat(down(f(a)))
395.04/246.29	   down(g(f(b))) -> g_flat(down(f(b)))
395.04/246.29	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.04/246.29	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.04/246.29	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.04/246.29	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.04/246.29	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.04/246.29	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.04/246.29	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.04/246.29	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.04/246.29	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.04/246.29	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.04/246.29	
395.04/246.29	Q is empty.
395.04/246.29	We have to consider all minimal (P,Q,R)-chains.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(63) DependencyGraphProof (EQUIVALENT)
395.04/246.29	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(64)
395.04/246.29	Obligation:
395.04/246.29	Q DP problem:
395.04/246.29	The TRS P consists of the following rules:
395.04/246.29	
395.04/246.29	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
395.04/246.29	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
395.04/246.29	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.04/246.29	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.04/246.29	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.04/246.29	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.04/246.29	   TOP(up(f(g(fresh_constant)))) -> TOP(f_flat(down(g(fresh_constant))))
395.04/246.29	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.04/246.29	   TOP(up(g(f(b)))) -> TOP(g_flat(down(f(b))))
395.04/246.29	   TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant))))
395.04/246.29	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.04/246.29	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.04/246.29	   TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b))))
395.04/246.29	   TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant))))
395.04/246.29	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.04/246.29	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.04/246.29	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.04/246.29	   TOP(up(f(g(g(fresh_constant))))) -> TOP(f_flat(down(g(g(fresh_constant)))))
395.04/246.29	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.04/246.29	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.04/246.29	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.04/246.29	   TOP(up(g(f(f(fresh_constant))))) -> TOP(g_flat(down(f(f(fresh_constant)))))
395.04/246.29	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.04/246.29	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.04/246.29	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.04/246.29	   TOP(up(g(f(g(fresh_constant))))) -> TOP(g_flat(down(f(g(fresh_constant)))))
395.04/246.29	
395.04/246.29	The TRS R consists of the following rules:
395.04/246.29	
395.04/246.29	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.04/246.29	   g_flat(up(x_1)) -> up(g(x_1))
395.04/246.29	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.04/246.29	   f_flat(up(x_1)) -> up(f(x_1))
395.04/246.29	   down(f(g(b))) -> f_flat(down(g(b)))
395.04/246.29	   down(g(b)) -> g_flat(down(b))
395.04/246.29	   down(f(g(g(f(x))))) -> up(b)
395.04/246.29	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.04/246.29	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.04/246.29	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.04/246.29	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.04/246.29	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.04/246.29	   down(g(g(b))) -> g_flat(down(g(b)))
395.04/246.29	   down(g(g(g(x)))) -> up(b)
395.04/246.29	   down(g(g(a))) -> g_flat(down(g(a)))
395.04/246.29	   down(g(a)) -> g_flat(down(a))
395.04/246.29	   down(a) -> up(f(a))
395.04/246.29	   down(a) -> up(g(a))
395.04/246.29	   down(f(g(a))) -> f_flat(down(g(a)))
395.04/246.29	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.04/246.29	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.04/246.29	   down(f(f(b))) -> f_flat(down(f(b)))
395.04/246.29	   down(f(b)) -> f_flat(down(b))
395.04/246.29	   down(f(f(f(x)))) -> up(b)
395.04/246.29	   down(f(f(a))) -> f_flat(down(f(a)))
395.04/246.29	   down(f(a)) -> f_flat(down(a))
395.04/246.29	   down(g(f(f(g(x))))) -> up(b)
395.04/246.29	   down(g(f(g(f(x))))) -> up(b)
395.04/246.29	   down(g(f(a))) -> g_flat(down(f(a)))
395.04/246.29	   down(g(f(b))) -> g_flat(down(f(b)))
395.04/246.29	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.04/246.29	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.04/246.29	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.04/246.29	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.04/246.29	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.04/246.29	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.04/246.29	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.04/246.29	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.04/246.29	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.04/246.29	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.04/246.29	
395.04/246.29	Q is empty.
395.04/246.29	We have to consider all minimal (P,Q,R)-chains.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(65) TransformationProof (EQUIVALENT)
395.04/246.29	By narrowing [LPAR04] the rule TOP(up(f(g(fresh_constant)))) -> TOP(f_flat(down(g(fresh_constant)))) at position [0] we obtained the following new rules [LPAR04]:
395.04/246.29	
395.04/246.29	   (TOP(up(f(g(fresh_constant)))) -> TOP(f_flat(g_flat(down(fresh_constant)))),TOP(up(f(g(fresh_constant)))) -> TOP(f_flat(g_flat(down(fresh_constant)))))
395.04/246.29	
395.04/246.29	
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(66)
395.04/246.29	Obligation:
395.04/246.29	Q DP problem:
395.04/246.29	The TRS P consists of the following rules:
395.04/246.29	
395.04/246.29	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
395.04/246.29	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
395.04/246.29	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.04/246.29	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.04/246.29	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.04/246.29	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.04/246.29	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.04/246.29	   TOP(up(g(f(b)))) -> TOP(g_flat(down(f(b))))
395.04/246.29	   TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant))))
395.04/246.29	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.04/246.29	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.04/246.29	   TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b))))
395.04/246.29	   TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant))))
395.04/246.29	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.04/246.29	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.04/246.29	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.04/246.29	   TOP(up(f(g(g(fresh_constant))))) -> TOP(f_flat(down(g(g(fresh_constant)))))
395.04/246.29	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.04/246.29	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.04/246.29	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.04/246.29	   TOP(up(g(f(f(fresh_constant))))) -> TOP(g_flat(down(f(f(fresh_constant)))))
395.04/246.29	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.04/246.29	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.04/246.29	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.04/246.29	   TOP(up(g(f(g(fresh_constant))))) -> TOP(g_flat(down(f(g(fresh_constant)))))
395.04/246.29	   TOP(up(f(g(fresh_constant)))) -> TOP(f_flat(g_flat(down(fresh_constant))))
395.04/246.29	
395.04/246.29	The TRS R consists of the following rules:
395.04/246.29	
395.04/246.29	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.04/246.29	   g_flat(up(x_1)) -> up(g(x_1))
395.04/246.29	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.04/246.29	   f_flat(up(x_1)) -> up(f(x_1))
395.04/246.29	   down(f(g(b))) -> f_flat(down(g(b)))
395.04/246.29	   down(g(b)) -> g_flat(down(b))
395.04/246.29	   down(f(g(g(f(x))))) -> up(b)
395.04/246.29	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.04/246.29	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.04/246.29	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.04/246.29	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.04/246.29	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.04/246.29	   down(g(g(b))) -> g_flat(down(g(b)))
395.04/246.29	   down(g(g(g(x)))) -> up(b)
395.04/246.29	   down(g(g(a))) -> g_flat(down(g(a)))
395.04/246.29	   down(g(a)) -> g_flat(down(a))
395.04/246.29	   down(a) -> up(f(a))
395.04/246.29	   down(a) -> up(g(a))
395.04/246.29	   down(f(g(a))) -> f_flat(down(g(a)))
395.04/246.29	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.04/246.29	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.04/246.29	   down(f(f(b))) -> f_flat(down(f(b)))
395.04/246.29	   down(f(b)) -> f_flat(down(b))
395.04/246.29	   down(f(f(f(x)))) -> up(b)
395.04/246.29	   down(f(f(a))) -> f_flat(down(f(a)))
395.04/246.29	   down(f(a)) -> f_flat(down(a))
395.04/246.29	   down(g(f(f(g(x))))) -> up(b)
395.04/246.29	   down(g(f(g(f(x))))) -> up(b)
395.04/246.29	   down(g(f(a))) -> g_flat(down(f(a)))
395.04/246.29	   down(g(f(b))) -> g_flat(down(f(b)))
395.04/246.29	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.04/246.29	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.04/246.29	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.04/246.29	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.04/246.29	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.04/246.29	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.04/246.29	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.04/246.29	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.04/246.29	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.04/246.29	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.04/246.29	
395.04/246.29	Q is empty.
395.04/246.29	We have to consider all minimal (P,Q,R)-chains.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(67) DependencyGraphProof (EQUIVALENT)
395.04/246.29	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
395.04/246.29	----------------------------------------
395.04/246.29	
395.04/246.29	(68)
395.04/246.29	Obligation:
395.04/246.29	Q DP problem:
395.04/246.29	The TRS P consists of the following rules:
395.04/246.29	
395.04/246.29	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
395.04/246.29	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
395.04/246.29	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.04/246.29	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.04/246.29	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.04/246.29	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.04/246.29	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.04/246.29	   TOP(up(g(f(b)))) -> TOP(g_flat(down(f(b))))
395.04/246.29	   TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant))))
395.04/246.29	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.04/246.29	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.04/246.29	   TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b))))
395.04/246.29	   TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant))))
395.04/246.29	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.04/246.29	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.04/246.29	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.04/246.29	   TOP(up(f(g(g(fresh_constant))))) -> TOP(f_flat(down(g(g(fresh_constant)))))
395.04/246.29	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.04/246.29	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.04/246.29	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.04/246.29	   TOP(up(g(f(f(fresh_constant))))) -> TOP(g_flat(down(f(f(fresh_constant)))))
395.04/246.29	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.04/246.29	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.04/246.29	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.04/246.29	   TOP(up(g(f(g(fresh_constant))))) -> TOP(g_flat(down(f(g(fresh_constant)))))
395.04/246.29	
395.04/246.29	The TRS R consists of the following rules:
395.04/246.29	
395.04/246.29	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.04/246.29	   g_flat(up(x_1)) -> up(g(x_1))
395.04/246.29	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.04/246.29	   f_flat(up(x_1)) -> up(f(x_1))
395.04/246.29	   down(f(g(b))) -> f_flat(down(g(b)))
395.04/246.29	   down(g(b)) -> g_flat(down(b))
395.04/246.29	   down(f(g(g(f(x))))) -> up(b)
395.04/246.29	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.04/246.29	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.04/246.29	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.04/246.29	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.04/246.29	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.04/246.29	   down(g(g(b))) -> g_flat(down(g(b)))
395.04/246.29	   down(g(g(g(x)))) -> up(b)
395.04/246.29	   down(g(g(a))) -> g_flat(down(g(a)))
395.04/246.29	   down(g(a)) -> g_flat(down(a))
395.04/246.29	   down(a) -> up(f(a))
395.04/246.29	   down(a) -> up(g(a))
395.04/246.29	   down(f(g(a))) -> f_flat(down(g(a)))
395.04/246.29	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.04/246.29	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.04/246.29	   down(f(f(b))) -> f_flat(down(f(b)))
395.04/246.29	   down(f(b)) -> f_flat(down(b))
395.04/246.29	   down(f(f(f(x)))) -> up(b)
395.04/246.29	   down(f(f(a))) -> f_flat(down(f(a)))
395.04/246.29	   down(f(a)) -> f_flat(down(a))
395.04/246.29	   down(g(f(f(g(x))))) -> up(b)
395.04/246.29	   down(g(f(g(f(x))))) -> up(b)
395.04/246.29	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.29	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.29	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.29	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.29	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.29	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.29	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.29	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.29	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.29	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.29	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.29	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.29	
395.09/246.29	Q is empty.
395.09/246.29	We have to consider all minimal (P,Q,R)-chains.
395.09/246.29	----------------------------------------
395.09/246.29	
395.09/246.29	(69) TransformationProof (EQUIVALENT)
395.09/246.29	By narrowing [LPAR04] the rule TOP(up(g(f(b)))) -> TOP(g_flat(down(f(b)))) at position [0] we obtained the following new rules [LPAR04]:
395.09/246.29	
395.09/246.29	   (TOP(up(g(f(b)))) -> TOP(g_flat(f_flat(down(b)))),TOP(up(g(f(b)))) -> TOP(g_flat(f_flat(down(b)))))
395.09/246.29	
395.09/246.29	
395.09/246.29	----------------------------------------
395.09/246.29	
395.09/246.29	(70)
395.09/246.29	Obligation:
395.09/246.29	Q DP problem:
395.09/246.29	The TRS P consists of the following rules:
395.09/246.29	
395.09/246.29	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
395.09/246.29	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
395.09/246.29	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.09/246.29	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.29	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.09/246.29	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.29	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.09/246.29	   TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant))))
395.09/246.29	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.09/246.29	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.29	   TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b))))
395.09/246.29	   TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant))))
395.09/246.29	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.29	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.29	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.09/246.29	   TOP(up(f(g(g(fresh_constant))))) -> TOP(f_flat(down(g(g(fresh_constant)))))
395.09/246.29	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.29	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.29	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.09/246.29	   TOP(up(g(f(f(fresh_constant))))) -> TOP(g_flat(down(f(f(fresh_constant)))))
395.09/246.29	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.29	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.29	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.29	   TOP(up(g(f(g(fresh_constant))))) -> TOP(g_flat(down(f(g(fresh_constant)))))
395.09/246.29	   TOP(up(g(f(b)))) -> TOP(g_flat(f_flat(down(b))))
395.09/246.29	
395.09/246.29	The TRS R consists of the following rules:
395.09/246.29	
395.09/246.29	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.29	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.29	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.09/246.29	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.29	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.29	   down(g(b)) -> g_flat(down(b))
395.09/246.29	   down(f(g(g(f(x))))) -> up(b)
395.09/246.29	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.29	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.29	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.29	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.29	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.29	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.29	   down(g(g(g(x)))) -> up(b)
395.09/246.29	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.29	   down(g(a)) -> g_flat(down(a))
395.09/246.29	   down(a) -> up(f(a))
395.09/246.29	   down(a) -> up(g(a))
395.09/246.29	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.29	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.29	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.09/246.29	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.29	   down(f(b)) -> f_flat(down(b))
395.09/246.29	   down(f(f(f(x)))) -> up(b)
395.09/246.29	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.29	   down(f(a)) -> f_flat(down(a))
395.09/246.29	   down(g(f(f(g(x))))) -> up(b)
395.09/246.29	   down(g(f(g(f(x))))) -> up(b)
395.09/246.29	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.29	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.29	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.29	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.29	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.29	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.29	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.29	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.29	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.29	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.29	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.29	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.29	
395.09/246.29	Q is empty.
395.09/246.29	We have to consider all minimal (P,Q,R)-chains.
395.09/246.29	----------------------------------------
395.09/246.29	
395.09/246.29	(71) DependencyGraphProof (EQUIVALENT)
395.09/246.29	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
395.09/246.29	----------------------------------------
395.09/246.29	
395.09/246.29	(72)
395.09/246.29	Obligation:
395.09/246.29	Q DP problem:
395.09/246.29	The TRS P consists of the following rules:
395.09/246.29	
395.09/246.29	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
395.09/246.29	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
395.09/246.29	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.09/246.29	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.29	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.09/246.29	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.29	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.09/246.29	   TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant))))
395.09/246.29	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.09/246.29	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.29	   TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b))))
395.09/246.29	   TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant))))
395.09/246.29	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.29	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.29	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.09/246.29	   TOP(up(f(g(g(fresh_constant))))) -> TOP(f_flat(down(g(g(fresh_constant)))))
395.09/246.29	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.29	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.29	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.09/246.29	   TOP(up(g(f(f(fresh_constant))))) -> TOP(g_flat(down(f(f(fresh_constant)))))
395.09/246.29	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.29	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.29	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.29	   TOP(up(g(f(g(fresh_constant))))) -> TOP(g_flat(down(f(g(fresh_constant)))))
395.09/246.29	
395.09/246.29	The TRS R consists of the following rules:
395.09/246.29	
395.09/246.29	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.30	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.30	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.09/246.30	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.30	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.30	   down(g(b)) -> g_flat(down(b))
395.09/246.30	   down(f(g(g(f(x))))) -> up(b)
395.09/246.30	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.30	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.30	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.30	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.30	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.30	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.30	   down(g(g(g(x)))) -> up(b)
395.09/246.30	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.30	   down(g(a)) -> g_flat(down(a))
395.09/246.30	   down(a) -> up(f(a))
395.09/246.30	   down(a) -> up(g(a))
395.09/246.30	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.30	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.30	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.09/246.30	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.30	   down(f(b)) -> f_flat(down(b))
395.09/246.30	   down(f(f(f(x)))) -> up(b)
395.09/246.30	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.30	   down(f(a)) -> f_flat(down(a))
395.09/246.30	   down(g(f(f(g(x))))) -> up(b)
395.09/246.30	   down(g(f(g(f(x))))) -> up(b)
395.09/246.30	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.30	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.30	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.30	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.30	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.30	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.30	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.30	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.30	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.30	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.30	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.30	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.30	
395.09/246.30	Q is empty.
395.09/246.30	We have to consider all minimal (P,Q,R)-chains.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(73) TransformationProof (EQUIVALENT)
395.09/246.30	By narrowing [LPAR04] the rule TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant)))) at position [0] we obtained the following new rules [LPAR04]:
395.09/246.30	
395.09/246.30	   (TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(f_flat(down(fresh_constant)))),TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(f_flat(down(fresh_constant)))))
395.09/246.30	
395.09/246.30	
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(74)
395.09/246.30	Obligation:
395.09/246.30	Q DP problem:
395.09/246.30	The TRS P consists of the following rules:
395.09/246.30	
395.09/246.30	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
395.09/246.30	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
395.09/246.30	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.09/246.30	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.30	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.09/246.30	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.09/246.30	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.09/246.30	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b))))
395.09/246.30	   TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant))))
395.09/246.30	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.30	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.30	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.09/246.30	   TOP(up(f(g(g(fresh_constant))))) -> TOP(f_flat(down(g(g(fresh_constant)))))
395.09/246.30	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.30	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.30	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.09/246.30	   TOP(up(g(f(f(fresh_constant))))) -> TOP(g_flat(down(f(f(fresh_constant)))))
395.09/246.30	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.30	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.30	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.30	   TOP(up(g(f(g(fresh_constant))))) -> TOP(g_flat(down(f(g(fresh_constant)))))
395.09/246.30	   TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(f_flat(down(fresh_constant))))
395.09/246.30	
395.09/246.30	The TRS R consists of the following rules:
395.09/246.30	
395.09/246.30	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.30	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.30	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.09/246.30	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.30	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.30	   down(g(b)) -> g_flat(down(b))
395.09/246.30	   down(f(g(g(f(x))))) -> up(b)
395.09/246.30	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.30	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.30	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.30	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.30	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.30	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.30	   down(g(g(g(x)))) -> up(b)
395.09/246.30	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.30	   down(g(a)) -> g_flat(down(a))
395.09/246.30	   down(a) -> up(f(a))
395.09/246.30	   down(a) -> up(g(a))
395.09/246.30	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.30	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.30	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.09/246.30	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.30	   down(f(b)) -> f_flat(down(b))
395.09/246.30	   down(f(f(f(x)))) -> up(b)
395.09/246.30	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.30	   down(f(a)) -> f_flat(down(a))
395.09/246.30	   down(g(f(f(g(x))))) -> up(b)
395.09/246.30	   down(g(f(g(f(x))))) -> up(b)
395.09/246.30	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.30	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.30	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.30	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.30	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.30	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.30	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.30	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.30	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.30	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.30	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.30	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.30	
395.09/246.30	Q is empty.
395.09/246.30	We have to consider all minimal (P,Q,R)-chains.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(75) DependencyGraphProof (EQUIVALENT)
395.09/246.30	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(76)
395.09/246.30	Obligation:
395.09/246.30	Q DP problem:
395.09/246.30	The TRS P consists of the following rules:
395.09/246.30	
395.09/246.30	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
395.09/246.30	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
395.09/246.30	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.09/246.30	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.30	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.09/246.30	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.09/246.30	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.09/246.30	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b))))
395.09/246.30	   TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant))))
395.09/246.30	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.30	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.30	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.09/246.30	   TOP(up(f(g(g(fresh_constant))))) -> TOP(f_flat(down(g(g(fresh_constant)))))
395.09/246.30	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.30	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.30	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.09/246.30	   TOP(up(g(f(f(fresh_constant))))) -> TOP(g_flat(down(f(f(fresh_constant)))))
395.09/246.30	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.30	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.30	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.30	   TOP(up(g(f(g(fresh_constant))))) -> TOP(g_flat(down(f(g(fresh_constant)))))
395.09/246.30	
395.09/246.30	The TRS R consists of the following rules:
395.09/246.30	
395.09/246.30	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.30	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.30	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.09/246.30	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.30	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.30	   down(g(b)) -> g_flat(down(b))
395.09/246.30	   down(f(g(g(f(x))))) -> up(b)
395.09/246.30	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.30	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.30	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.30	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.30	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.30	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.30	   down(g(g(g(x)))) -> up(b)
395.09/246.30	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.30	   down(g(a)) -> g_flat(down(a))
395.09/246.30	   down(a) -> up(f(a))
395.09/246.30	   down(a) -> up(g(a))
395.09/246.30	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.30	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.30	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.09/246.30	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.30	   down(f(b)) -> f_flat(down(b))
395.09/246.30	   down(f(f(f(x)))) -> up(b)
395.09/246.30	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.30	   down(f(a)) -> f_flat(down(a))
395.09/246.30	   down(g(f(f(g(x))))) -> up(b)
395.09/246.30	   down(g(f(g(f(x))))) -> up(b)
395.09/246.30	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.30	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.30	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.30	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.30	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.30	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.30	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.30	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.30	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.30	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.30	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.30	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.30	
395.09/246.30	Q is empty.
395.09/246.30	We have to consider all minimal (P,Q,R)-chains.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(77) TransformationProof (EQUIVALENT)
395.09/246.30	By narrowing [LPAR04] the rule TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) at position [0] we obtained the following new rules [LPAR04]:
395.09/246.30	
395.09/246.30	   (TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))),TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))))
395.09/246.30	
395.09/246.30	
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(78)
395.09/246.30	Obligation:
395.09/246.30	Q DP problem:
395.09/246.30	The TRS P consists of the following rules:
395.09/246.30	
395.09/246.30	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
395.09/246.30	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
395.09/246.30	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.09/246.30	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.30	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.09/246.30	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.09/246.30	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.09/246.30	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant))))
395.09/246.30	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.30	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.30	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.09/246.30	   TOP(up(f(g(g(fresh_constant))))) -> TOP(f_flat(down(g(g(fresh_constant)))))
395.09/246.30	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.30	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.30	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.09/246.30	   TOP(up(g(f(f(fresh_constant))))) -> TOP(g_flat(down(f(f(fresh_constant)))))
395.09/246.30	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.30	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.30	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.30	   TOP(up(g(f(g(fresh_constant))))) -> TOP(g_flat(down(f(g(fresh_constant)))))
395.09/246.30	   TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b))))
395.09/246.30	
395.09/246.30	The TRS R consists of the following rules:
395.09/246.30	
395.09/246.30	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.30	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.30	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.09/246.30	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.30	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.30	   down(g(b)) -> g_flat(down(b))
395.09/246.30	   down(f(g(g(f(x))))) -> up(b)
395.09/246.30	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.30	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.30	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.30	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.30	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.30	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.30	   down(g(g(g(x)))) -> up(b)
395.09/246.30	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.30	   down(g(a)) -> g_flat(down(a))
395.09/246.30	   down(a) -> up(f(a))
395.09/246.30	   down(a) -> up(g(a))
395.09/246.30	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.30	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.30	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.09/246.30	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.30	   down(f(b)) -> f_flat(down(b))
395.09/246.30	   down(f(f(f(x)))) -> up(b)
395.09/246.30	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.30	   down(f(a)) -> f_flat(down(a))
395.09/246.30	   down(g(f(f(g(x))))) -> up(b)
395.09/246.30	   down(g(f(g(f(x))))) -> up(b)
395.09/246.30	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.30	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.30	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.30	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.30	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.30	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.30	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.30	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.30	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.30	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.30	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.30	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.30	
395.09/246.30	Q is empty.
395.09/246.30	We have to consider all minimal (P,Q,R)-chains.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(79) DependencyGraphProof (EQUIVALENT)
395.09/246.30	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(80)
395.09/246.30	Obligation:
395.09/246.30	Q DP problem:
395.09/246.30	The TRS P consists of the following rules:
395.09/246.30	
395.09/246.30	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
395.09/246.30	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
395.09/246.30	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.09/246.30	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.30	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.09/246.30	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.09/246.30	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.09/246.30	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant))))
395.09/246.30	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.30	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.30	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.09/246.30	   TOP(up(f(g(g(fresh_constant))))) -> TOP(f_flat(down(g(g(fresh_constant)))))
395.09/246.30	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.30	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.30	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.09/246.30	   TOP(up(g(f(f(fresh_constant))))) -> TOP(g_flat(down(f(f(fresh_constant)))))
395.09/246.30	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.30	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.30	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.30	   TOP(up(g(f(g(fresh_constant))))) -> TOP(g_flat(down(f(g(fresh_constant)))))
395.09/246.30	
395.09/246.30	The TRS R consists of the following rules:
395.09/246.30	
395.09/246.30	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.30	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.30	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.09/246.30	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.30	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.30	   down(g(b)) -> g_flat(down(b))
395.09/246.30	   down(f(g(g(f(x))))) -> up(b)
395.09/246.30	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.30	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.30	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.30	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.30	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.30	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.30	   down(g(g(g(x)))) -> up(b)
395.09/246.30	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.30	   down(g(a)) -> g_flat(down(a))
395.09/246.30	   down(a) -> up(f(a))
395.09/246.30	   down(a) -> up(g(a))
395.09/246.30	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.30	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.30	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.09/246.30	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.30	   down(f(b)) -> f_flat(down(b))
395.09/246.30	   down(f(f(f(x)))) -> up(b)
395.09/246.30	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.30	   down(f(a)) -> f_flat(down(a))
395.09/246.30	   down(g(f(f(g(x))))) -> up(b)
395.09/246.30	   down(g(f(g(f(x))))) -> up(b)
395.09/246.30	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.30	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.30	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.30	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.30	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.30	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.30	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.30	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.30	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.30	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.30	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.30	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.30	
395.09/246.30	Q is empty.
395.09/246.30	We have to consider all minimal (P,Q,R)-chains.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(81) TransformationProof (EQUIVALENT)
395.09/246.30	By narrowing [LPAR04] the rule TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) at position [0] we obtained the following new rules [LPAR04]:
395.09/246.30	
395.09/246.30	   (TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(g_flat(down(fresh_constant)))),TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(g_flat(down(fresh_constant)))))
395.09/246.30	
395.09/246.30	
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(82)
395.09/246.30	Obligation:
395.09/246.30	Q DP problem:
395.09/246.30	The TRS P consists of the following rules:
395.09/246.30	
395.09/246.30	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
395.09/246.30	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
395.09/246.30	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.09/246.30	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.30	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.09/246.30	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.09/246.30	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.09/246.30	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.30	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.30	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.09/246.30	   TOP(up(f(g(g(fresh_constant))))) -> TOP(f_flat(down(g(g(fresh_constant)))))
395.09/246.30	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.30	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.30	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.09/246.30	   TOP(up(g(f(f(fresh_constant))))) -> TOP(g_flat(down(f(f(fresh_constant)))))
395.09/246.30	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.30	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.30	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.30	   TOP(up(g(f(g(fresh_constant))))) -> TOP(g_flat(down(f(g(fresh_constant)))))
395.09/246.30	   TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(g_flat(down(fresh_constant))))
395.09/246.30	
395.09/246.30	The TRS R consists of the following rules:
395.09/246.30	
395.09/246.30	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.30	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.30	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.09/246.30	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.30	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.30	   down(g(b)) -> g_flat(down(b))
395.09/246.30	   down(f(g(g(f(x))))) -> up(b)
395.09/246.30	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.30	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.30	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.30	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.30	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.30	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.30	   down(g(g(g(x)))) -> up(b)
395.09/246.30	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.30	   down(g(a)) -> g_flat(down(a))
395.09/246.30	   down(a) -> up(f(a))
395.09/246.30	   down(a) -> up(g(a))
395.09/246.30	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.30	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.30	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.09/246.30	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.30	   down(f(b)) -> f_flat(down(b))
395.09/246.30	   down(f(f(f(x)))) -> up(b)
395.09/246.30	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.30	   down(f(a)) -> f_flat(down(a))
395.09/246.30	   down(g(f(f(g(x))))) -> up(b)
395.09/246.30	   down(g(f(g(f(x))))) -> up(b)
395.09/246.30	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.30	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.30	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.30	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.30	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.30	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.30	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.30	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.30	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.30	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.30	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.30	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.30	
395.09/246.30	Q is empty.
395.09/246.30	We have to consider all minimal (P,Q,R)-chains.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(83) DependencyGraphProof (EQUIVALENT)
395.09/246.30	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(84)
395.09/246.30	Obligation:
395.09/246.30	Q DP problem:
395.09/246.30	The TRS P consists of the following rules:
395.09/246.30	
395.09/246.30	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
395.09/246.30	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
395.09/246.30	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.09/246.30	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.30	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.09/246.30	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.09/246.30	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.09/246.30	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.30	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.30	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.09/246.30	   TOP(up(f(g(g(fresh_constant))))) -> TOP(f_flat(down(g(g(fresh_constant)))))
395.09/246.30	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.30	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.30	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.09/246.30	   TOP(up(g(f(f(fresh_constant))))) -> TOP(g_flat(down(f(f(fresh_constant)))))
395.09/246.30	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.30	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.30	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.30	   TOP(up(g(f(g(fresh_constant))))) -> TOP(g_flat(down(f(g(fresh_constant)))))
395.09/246.30	
395.09/246.30	The TRS R consists of the following rules:
395.09/246.30	
395.09/246.30	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.30	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.30	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.09/246.30	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.30	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.30	   down(g(b)) -> g_flat(down(b))
395.09/246.30	   down(f(g(g(f(x))))) -> up(b)
395.09/246.30	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.30	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.30	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.30	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.30	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.30	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.30	   down(g(g(g(x)))) -> up(b)
395.09/246.30	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.30	   down(g(a)) -> g_flat(down(a))
395.09/246.30	   down(a) -> up(f(a))
395.09/246.30	   down(a) -> up(g(a))
395.09/246.30	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.30	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.30	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.09/246.30	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.30	   down(f(b)) -> f_flat(down(b))
395.09/246.30	   down(f(f(f(x)))) -> up(b)
395.09/246.30	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.30	   down(f(a)) -> f_flat(down(a))
395.09/246.30	   down(g(f(f(g(x))))) -> up(b)
395.09/246.30	   down(g(f(g(f(x))))) -> up(b)
395.09/246.30	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.30	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.30	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.30	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.30	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.30	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.30	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.30	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.30	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.30	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.30	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.30	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.30	
395.09/246.30	Q is empty.
395.09/246.30	We have to consider all minimal (P,Q,R)-chains.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(85) MRRProof (EQUIVALENT)
395.09/246.30	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.
395.09/246.30	
395.09/246.30	Strictly oriented dependency pairs:
395.09/246.30	
395.09/246.30	   TOP(up(f(g(g(fresh_constant))))) -> TOP(f_flat(down(g(g(fresh_constant)))))
395.09/246.30	   TOP(up(g(f(f(fresh_constant))))) -> TOP(g_flat(down(f(f(fresh_constant)))))
395.09/246.30	   TOP(up(g(f(g(fresh_constant))))) -> TOP(g_flat(down(f(g(fresh_constant)))))
395.09/246.30	
395.09/246.30	
395.09/246.30	Used ordering: Polynomial interpretation [POLO]:
395.09/246.30	
395.09/246.30	   POL(TOP(x_1)) = x_1
395.09/246.30	   POL(a) = 0
395.09/246.30	   POL(b) = 0
395.09/246.30	   POL(down(x_1)) = x_1
395.09/246.30	   POL(f(x_1)) = x_1
395.09/246.30	   POL(f_flat(x_1)) = x_1
395.09/246.30	   POL(fresh_constant) = 1
395.09/246.30	   POL(g(x_1)) = 2*x_1
395.09/246.30	   POL(g_flat(x_1)) = 2*x_1
395.09/246.30	   POL(up(x_1)) = 2*x_1
395.09/246.30	
395.09/246.30	
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(86)
395.09/246.30	Obligation:
395.09/246.30	Q DP problem:
395.09/246.30	The TRS P consists of the following rules:
395.09/246.30	
395.09/246.30	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
395.09/246.30	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
395.09/246.30	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.09/246.30	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.30	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.09/246.30	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.09/246.30	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.09/246.30	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.30	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.30	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.09/246.30	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.30	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.30	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.09/246.30	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.30	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.30	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.30	
395.09/246.30	The TRS R consists of the following rules:
395.09/246.30	
395.09/246.30	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.30	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.30	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.09/246.30	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.30	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.30	   down(g(b)) -> g_flat(down(b))
395.09/246.30	   down(f(g(g(f(x))))) -> up(b)
395.09/246.30	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.30	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.30	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.30	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.30	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.30	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.30	   down(g(g(g(x)))) -> up(b)
395.09/246.30	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.30	   down(g(a)) -> g_flat(down(a))
395.09/246.30	   down(a) -> up(f(a))
395.09/246.30	   down(a) -> up(g(a))
395.09/246.30	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.30	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.30	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.09/246.30	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.30	   down(f(b)) -> f_flat(down(b))
395.09/246.30	   down(f(f(f(x)))) -> up(b)
395.09/246.30	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.30	   down(f(a)) -> f_flat(down(a))
395.09/246.30	   down(g(f(f(g(x))))) -> up(b)
395.09/246.30	   down(g(f(g(f(x))))) -> up(b)
395.09/246.30	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.30	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.30	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.30	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.30	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.30	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.30	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.30	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.30	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.30	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.30	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.30	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.30	
395.09/246.30	Q is empty.
395.09/246.30	We have to consider all minimal (P,Q,R)-chains.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(87) QDPOrderProof (EQUIVALENT)
395.09/246.30	We use the reduction pair processor [LPAR04,JAR06].
395.09/246.30	
395.09/246.30	
395.09/246.30	The following pairs can be oriented strictly and are deleted.
395.09/246.30	
395.09/246.30	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
395.09/246.30	The remaining pairs can at least be oriented weakly.
395.09/246.30	Used ordering:  Matrix interpretation [MATRO]:
395.09/246.30	
395.09/246.30	Non-tuple symbols: 
395.09/246.30	<<<
395.09/246.30	 M( a ) =  	[[0], [1]]
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( b ) =  	[[0], [0]]
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( down_1(x_1) ) =  	[[1], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( f_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 1], [0, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( fresh_constant ) =  	[[0], [0]]
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( up_1(x_1) ) =  	[[0], [0]] 	+ 	[[1, 0], [0, 1]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( f_flat_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 1], [0, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( g_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( g_flat_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	Tuple symbols: 
395.09/246.30	<<<
395.09/246.30	 M( TOP_1(x_1) ) =  	[[0]] 	+ 	[[1, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	
395.09/246.30	
395.09/246.30	Matrix type: 
395.09/246.30	
395.09/246.30	We used a basic matrix type which is not further parametrizeable.
395.09/246.30	
395.09/246.30	
395.09/246.30	
395.09/246.30	
395.09/246.30	
395.09/246.30	As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
395.09/246.30	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
395.09/246.30	
395.09/246.30	   down(a) -> up(f(a))
395.09/246.30	   down(a) -> up(g(a))
395.09/246.30	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.30	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.30	   down(f(a)) -> f_flat(down(a))
395.09/246.30	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.30	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.30	   down(f(g(g(f(x))))) -> up(b)
395.09/246.30	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.30	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.30	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.30	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.30	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.30	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.30	   down(g(a)) -> g_flat(down(a))
395.09/246.30	   down(g(f(f(g(x))))) -> up(b)
395.09/246.30	   down(g(f(g(f(x))))) -> up(b)
395.09/246.30	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.30	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.30	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.30	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.30	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.30	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.30	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.30	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.30	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.30	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.30	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.30	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.30	   down(g(g(g(x)))) -> up(b)
395.09/246.30	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.30	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.09/246.30	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.30	   down(g(b)) -> g_flat(down(b))
395.09/246.30	
395.09/246.30	
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(88)
395.09/246.30	Obligation:
395.09/246.30	Q DP problem:
395.09/246.30	The TRS P consists of the following rules:
395.09/246.30	
395.09/246.30	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
395.09/246.30	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.09/246.30	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.30	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.09/246.30	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.09/246.30	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.09/246.30	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.30	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.30	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.09/246.30	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.30	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.30	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.09/246.30	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.30	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.30	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.30	
395.09/246.30	The TRS R consists of the following rules:
395.09/246.30	
395.09/246.30	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.30	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.30	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.09/246.30	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.30	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.30	   down(g(b)) -> g_flat(down(b))
395.09/246.30	   down(f(g(g(f(x))))) -> up(b)
395.09/246.30	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.30	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.30	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.30	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.30	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.30	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.30	   down(g(g(g(x)))) -> up(b)
395.09/246.30	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.30	   down(g(a)) -> g_flat(down(a))
395.09/246.30	   down(a) -> up(f(a))
395.09/246.30	   down(a) -> up(g(a))
395.09/246.30	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.30	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.30	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.09/246.30	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.30	   down(f(b)) -> f_flat(down(b))
395.09/246.30	   down(f(f(f(x)))) -> up(b)
395.09/246.30	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.30	   down(f(a)) -> f_flat(down(a))
395.09/246.30	   down(g(f(f(g(x))))) -> up(b)
395.09/246.30	   down(g(f(g(f(x))))) -> up(b)
395.09/246.30	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.30	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.30	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.30	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.30	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.30	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.30	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.30	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.30	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.30	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.30	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.30	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.30	
395.09/246.30	Q is empty.
395.09/246.30	We have to consider all minimal (P,Q,R)-chains.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(89) QDPOrderProof (EQUIVALENT)
395.09/246.30	We use the reduction pair processor [LPAR04,JAR06].
395.09/246.30	
395.09/246.30	
395.09/246.30	The following pairs can be oriented strictly and are deleted.
395.09/246.30	
395.09/246.30	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
395.09/246.30	The remaining pairs can at least be oriented weakly.
395.09/246.30	Used ordering:  Matrix interpretation [MATRO]:
395.09/246.30	
395.09/246.30	Non-tuple symbols: 
395.09/246.30	<<<
395.09/246.30	 M( a ) =  	[[1], [0]]
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( b ) =  	[[0], [0]]
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( down_1(x_1) ) =  	[[1], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( f_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( fresh_constant ) =  	[[0], [0]]
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( up_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 1], [1, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( f_flat_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( g_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [1, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( g_flat_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 1], [0, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	Tuple symbols: 
395.09/246.30	<<<
395.09/246.30	 M( TOP_1(x_1) ) =  	[[0]] 	+ 	[[1, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	
395.09/246.30	
395.09/246.30	Matrix type: 
395.09/246.30	
395.09/246.30	We used a basic matrix type which is not further parametrizeable.
395.09/246.30	
395.09/246.30	
395.09/246.30	
395.09/246.30	
395.09/246.30	
395.09/246.30	As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
395.09/246.30	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
395.09/246.30	
395.09/246.30	   down(a) -> up(f(a))
395.09/246.30	   down(a) -> up(g(a))
395.09/246.30	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.30	   down(f(a)) -> f_flat(down(a))
395.09/246.30	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.30	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.30	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.30	   down(f(g(g(f(x))))) -> up(b)
395.09/246.30	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.30	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.30	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.30	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.30	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.30	   down(g(a)) -> g_flat(down(a))
395.09/246.30	   down(g(f(f(g(x))))) -> up(b)
395.09/246.30	   down(g(f(g(f(x))))) -> up(b)
395.09/246.30	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.30	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.30	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.30	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.30	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.30	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.30	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.30	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.30	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.30	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.30	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.30	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.30	   down(f(f(f(x)))) -> up(b)
395.09/246.30	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.30	   down(f(b)) -> f_flat(down(b))
395.09/246.30	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.09/246.30	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.30	
395.09/246.30	
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(90)
395.09/246.30	Obligation:
395.09/246.30	Q DP problem:
395.09/246.30	The TRS P consists of the following rules:
395.09/246.30	
395.09/246.30	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.09/246.30	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.30	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.09/246.30	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.09/246.30	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.09/246.30	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.30	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.30	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.09/246.30	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.30	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.30	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.09/246.30	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.30	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.30	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.30	
395.09/246.30	The TRS R consists of the following rules:
395.09/246.30	
395.09/246.30	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.30	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.30	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.09/246.30	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.30	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.30	   down(g(b)) -> g_flat(down(b))
395.09/246.30	   down(f(g(g(f(x))))) -> up(b)
395.09/246.30	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.30	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.30	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.30	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.30	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.30	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.30	   down(g(g(g(x)))) -> up(b)
395.09/246.30	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.30	   down(g(a)) -> g_flat(down(a))
395.09/246.30	   down(a) -> up(f(a))
395.09/246.30	   down(a) -> up(g(a))
395.09/246.30	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.30	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.30	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.09/246.30	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.30	   down(f(b)) -> f_flat(down(b))
395.09/246.30	   down(f(f(f(x)))) -> up(b)
395.09/246.30	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.30	   down(f(a)) -> f_flat(down(a))
395.09/246.30	   down(g(f(f(g(x))))) -> up(b)
395.09/246.30	   down(g(f(g(f(x))))) -> up(b)
395.09/246.30	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.30	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.30	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.30	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.30	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.30	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.30	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.30	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.30	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.30	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.30	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.30	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.30	
395.09/246.30	Q is empty.
395.09/246.30	We have to consider all minimal (P,Q,R)-chains.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(91) QDPOrderProof (EQUIVALENT)
395.09/246.30	We use the reduction pair processor [LPAR04,JAR06].
395.09/246.30	
395.09/246.30	
395.09/246.30	The following pairs can be oriented strictly and are deleted.
395.09/246.30	
395.09/246.30	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
395.09/246.30	The remaining pairs can at least be oriented weakly.
395.09/246.30	Used ordering:  Matrix interpretation [MATRO]:
395.09/246.30	
395.09/246.30	Non-tuple symbols: 
395.09/246.30	<<<
395.09/246.30	 M( a ) =  	[[1], [0]]
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( b ) =  	[[0], [0]]
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( down_1(x_1) ) =  	[[0], [0]] 	+ 	[[1, 0], [0, 1]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( f_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 1], [1, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( fresh_constant ) =  	[[0], [0]]
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( up_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 1], [1, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( f_flat_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 1], [1, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( g_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( g_flat_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	Tuple symbols: 
395.09/246.30	<<<
395.09/246.30	 M( TOP_1(x_1) ) =  	[[0]] 	+ 	[[0, 1]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	
395.09/246.30	
395.09/246.30	Matrix type: 
395.09/246.30	
395.09/246.30	We used a basic matrix type which is not further parametrizeable.
395.09/246.30	
395.09/246.30	
395.09/246.30	
395.09/246.30	
395.09/246.30	
395.09/246.30	As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
395.09/246.30	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
395.09/246.30	
395.09/246.30	   down(f(a)) -> f_flat(down(a))
395.09/246.30	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.30	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.30	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.30	   down(f(g(g(f(x))))) -> up(b)
395.09/246.30	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.30	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.30	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.30	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.30	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.30	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.30	   down(g(a)) -> g_flat(down(a))
395.09/246.30	   down(g(f(f(g(x))))) -> up(b)
395.09/246.30	   down(g(f(g(f(x))))) -> up(b)
395.09/246.30	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.30	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.30	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.30	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.30	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.30	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.30	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.30	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.30	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.30	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.30	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.30	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.30	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.30	   down(g(g(g(x)))) -> up(b)
395.09/246.30	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.30	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.09/246.30	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.30	   down(g(b)) -> g_flat(down(b))
395.09/246.30	   down(a) -> up(f(a))
395.09/246.30	   down(a) -> up(g(a))
395.09/246.30	
395.09/246.30	
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(92)
395.09/246.30	Obligation:
395.09/246.30	Q DP problem:
395.09/246.30	The TRS P consists of the following rules:
395.09/246.30	
395.09/246.30	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.30	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.09/246.30	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.09/246.30	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.09/246.30	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.30	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.30	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.09/246.30	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.30	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.30	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.09/246.30	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.30	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.30	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.30	
395.09/246.30	The TRS R consists of the following rules:
395.09/246.30	
395.09/246.30	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.30	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.30	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.09/246.30	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.30	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.30	   down(g(b)) -> g_flat(down(b))
395.09/246.30	   down(f(g(g(f(x))))) -> up(b)
395.09/246.30	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.30	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.30	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.30	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.30	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.30	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.30	   down(g(g(g(x)))) -> up(b)
395.09/246.30	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.30	   down(g(a)) -> g_flat(down(a))
395.09/246.30	   down(a) -> up(f(a))
395.09/246.30	   down(a) -> up(g(a))
395.09/246.30	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.30	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.30	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.09/246.30	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.30	   down(f(b)) -> f_flat(down(b))
395.09/246.30	   down(f(f(f(x)))) -> up(b)
395.09/246.30	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.30	   down(f(a)) -> f_flat(down(a))
395.09/246.30	   down(g(f(f(g(x))))) -> up(b)
395.09/246.30	   down(g(f(g(f(x))))) -> up(b)
395.09/246.30	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.30	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.30	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.30	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.30	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.30	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.30	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.30	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.30	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.30	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.30	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.30	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.30	
395.09/246.30	Q is empty.
395.09/246.30	We have to consider all minimal (P,Q,R)-chains.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(93) QDPOrderProof (EQUIVALENT)
395.09/246.30	We use the reduction pair processor [LPAR04,JAR06].
395.09/246.30	
395.09/246.30	
395.09/246.30	The following pairs can be oriented strictly and are deleted.
395.09/246.30	
395.09/246.30	   TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))
395.09/246.30	The remaining pairs can at least be oriented weakly.
395.09/246.30	Used ordering:  Matrix interpretation [MATRO]:
395.09/246.30	
395.09/246.30	Non-tuple symbols: 
395.09/246.30	<<<
395.09/246.30	 M( a ) =  	[[1], [0]]
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( b ) =  	[[0], [0]]
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( down_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 1], [1, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( f_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( fresh_constant ) =  	[[0], [0]]
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( up_1(x_1) ) =  	[[0], [0]] 	+ 	[[1, 0], [0, 1]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( f_flat_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( g_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 1], [1, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	<<<
395.09/246.30	 M( g_flat_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 1], [1, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	Tuple symbols: 
395.09/246.30	<<<
395.09/246.30	 M( TOP_1(x_1) ) =  	[[0]] 	+ 	[[1, 0]] 	* 	x_1
395.09/246.30	>>>
395.09/246.30	
395.09/246.30	
395.09/246.30	
395.09/246.30	Matrix type: 
395.09/246.30	
395.09/246.30	We used a basic matrix type which is not further parametrizeable.
395.09/246.30	
395.09/246.30	
395.09/246.30	
395.09/246.30	
395.09/246.30	
395.09/246.30	As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
395.09/246.30	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
395.09/246.30	
395.09/246.30	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.30	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.30	   down(f(g(g(f(x))))) -> up(b)
395.09/246.30	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.30	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.30	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.30	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.30	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.30	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.30	   down(g(a)) -> g_flat(down(a))
395.09/246.30	   down(g(f(f(g(x))))) -> up(b)
395.09/246.30	   down(g(f(g(f(x))))) -> up(b)
395.09/246.30	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.30	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.30	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.30	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.30	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.30	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.30	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.30	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.30	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.30	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.30	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.30	   down(f(a)) -> f_flat(down(a))
395.09/246.30	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.30	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.30	   down(f(f(f(x)))) -> up(b)
395.09/246.30	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.30	   down(a) -> up(f(a))
395.09/246.30	   down(a) -> up(g(a))
395.09/246.30	   down(f(b)) -> f_flat(down(b))
395.09/246.30	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.09/246.30	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.30	
395.09/246.30	
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(94)
395.09/246.30	Obligation:
395.09/246.30	Q DP problem:
395.09/246.30	The TRS P consists of the following rules:
395.09/246.30	
395.09/246.30	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.30	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.09/246.30	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.09/246.30	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.30	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.30	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.09/246.30	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.30	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.30	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.09/246.30	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.30	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.30	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.30	
395.09/246.30	The TRS R consists of the following rules:
395.09/246.30	
395.09/246.30	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.30	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.30	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.09/246.30	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.30	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.30	   down(g(b)) -> g_flat(down(b))
395.09/246.30	   down(f(g(g(f(x))))) -> up(b)
395.09/246.30	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.30	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.30	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.30	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.30	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.30	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.30	   down(g(g(g(x)))) -> up(b)
395.09/246.30	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.30	   down(g(a)) -> g_flat(down(a))
395.09/246.30	   down(a) -> up(f(a))
395.09/246.30	   down(a) -> up(g(a))
395.09/246.30	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.30	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.30	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.09/246.30	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.30	   down(f(b)) -> f_flat(down(b))
395.09/246.30	   down(f(f(f(x)))) -> up(b)
395.09/246.30	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.30	   down(f(a)) -> f_flat(down(a))
395.09/246.30	   down(g(f(f(g(x))))) -> up(b)
395.09/246.30	   down(g(f(g(f(x))))) -> up(b)
395.09/246.30	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.30	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.30	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.30	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.30	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.30	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.30	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.30	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.30	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.30	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.30	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.30	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.30	
395.09/246.30	Q is empty.
395.09/246.30	We have to consider all minimal (P,Q,R)-chains.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(95) SplitQDPProof (EQUIVALENT)
395.09/246.30	We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem
395.09/246.30	
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(96)
395.09/246.30	Complex Obligation (AND)
395.09/246.30	
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(97)
395.09/246.30	Obligation:
395.09/246.30	Q DP problem:
395.09/246.30	The TRS P consists of the following rules:
395.09/246.30	
395.09/246.30	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.30	   TOP(up(f(g(a)))) -> TOP(f_flat(down(g(a))))
395.09/246.30	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(g(f(a)))) -> TOP(g_flat(down(f(a))))
395.09/246.30	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.30	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.30	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.09/246.30	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.30	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.30	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.09/246.30	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.30	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.30	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.30	
395.09/246.30	The TRS R consists of the following rules:
395.09/246.30	
395.09/246.30	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.30	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.30	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.09/246.30	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.30	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.30	   down(g(b)) -> g_flat(down(b))
395.09/246.30	   down(f(g(g(f(x))))) -> up(b)
395.09/246.30	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.30	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.30	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.30	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.30	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.30	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.30	   down(g(g(g(x)))) -> up(b)
395.09/246.30	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.30	   down(g(a)) -> g_flat(down(a))
395.09/246.30	   down(a) -> up(f(a))
395.09/246.30	   down(a) -> up(g(a))
395.09/246.30	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.30	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.30	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.09/246.30	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.30	   down(f(b)) -> f_flat(down(b))
395.09/246.30	   down(f(f(f(x)))) -> up(b)
395.09/246.30	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.30	   down(f(a)) -> f_flat(down(a))
395.09/246.30	   down(g(f(f(g(x))))) -> up(b)
395.09/246.30	   down(g(f(g(f(x))))) -> up(b)
395.09/246.30	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.30	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.30	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.30	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.30	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.30	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.30	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.30	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.30	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.30	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.30	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.30	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.30	
395.09/246.30	Q is empty.
395.09/246.30	We have to consider all minimal (P,Q,R)-chains.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(98) SemLabProof (SOUND)
395.09/246.30	We found the following model for the rules of the TRSs R and P.
395.09/246.30	Interpretation over the domain with elements from 0 to 1.
395.09/246.30	a: 1
395.09/246.30	b: 0
395.09/246.30	down: 0
395.09/246.30	f: 0
395.09/246.30	fresh_constant: 0
395.09/246.30	up: 0
395.09/246.30	f_flat: 0
395.09/246.30	TOP: 0
395.09/246.30	g_flat: 0
395.09/246.30	g: 0
395.09/246.30	By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(99)
395.09/246.30	Obligation:
395.09/246.30	Q DP problem:
395.09/246.30	The TRS P consists of the following rules:
395.09/246.30	
395.09/246.30	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(g.1(a.)))) -> TOP.0(f_flat.0(down.0(g.1(a.))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.30	   TOP.0(up.0(g.0(f.1(a.)))) -> TOP.0(g_flat.0(down.0(f.1(a.))))
395.09/246.30	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.30	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(g.1(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.1(a.)))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(g.0(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.1(x0))))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(g.0(b.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(b.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(f.1(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(a.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(f.0(f.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.1(x0))))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(f.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(b.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.1(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.1(a.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.0(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.1(x0))))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(b.)))))
395.09/246.30	
395.09/246.30	The TRS R consists of the following rules:
395.09/246.30	
395.09/246.30	   down.0(f.0(g.0(fresh_constant.))) -> f_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.30	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.30	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
395.09/246.30	   down.0(g.0(fresh_constant.)) -> g_flat.0(down.0(fresh_constant.))
395.09/246.30	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.30	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
395.09/246.30	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.30	   down.0(g.0(b.)) -> g_flat.0(down.0(b.))
395.09/246.30	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(f.0(g.0(g.0(f.1(x))))) -> up.0(b.)
395.09/246.30	   down.0(f.0(g.0(g.1(a.)))) -> f_flat.0(down.0(g.0(g.1(a.))))
395.09/246.30	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.30	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.30	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.30	   down.0(f.0(g.0(g.0(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.0(fresh_constant.))))
395.09/246.30	   down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.30	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.30	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.30	   down.0(g.0(g.0(g.1(x)))) -> up.0(b.)
395.09/246.30	   down.0(g.0(g.1(a.))) -> g_flat.0(down.0(g.1(a.)))
395.09/246.30	   down.0(g.1(a.)) -> g_flat.0(down.1(a.))
395.09/246.30	   down.1(a.) -> up.0(f.1(a.))
395.09/246.30	   down.1(a.) -> up.0(g.1(a.))
395.09/246.30	   down.0(f.0(g.1(a.))) -> f_flat.0(down.0(g.1(a.)))
395.09/246.30	   down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.30	   down.0(f.0(fresh_constant.)) -> f_flat.0(down.0(fresh_constant.))
395.09/246.30	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.30	   down.0(f.0(b.)) -> f_flat.0(down.0(b.))
395.09/246.30	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.30	   down.0(f.0(f.0(f.1(x)))) -> up.0(b.)
395.09/246.30	   down.0(f.0(f.1(a.))) -> f_flat.0(down.0(f.1(a.)))
395.09/246.30	   down.0(f.1(a.)) -> f_flat.0(down.1(a.))
395.09/246.30	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(g.0(f.0(f.0(g.1(x))))) -> up.0(b.)
395.09/246.30	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(g.0(f.0(g.0(f.1(x))))) -> up.0(b.)
395.09/246.30	   down.0(g.0(f.1(a.))) -> g_flat.0(down.0(f.1(a.)))
395.09/246.30	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.30	   down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.30	   down.0(g.0(f.0(f.1(a.)))) -> g_flat.0(down.0(f.0(f.1(a.))))
395.09/246.30	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.30	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.30	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.30	   down.0(g.0(f.0(f.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.0(fresh_constant.))))
395.09/246.30	   down.0(g.0(f.0(g.1(a.)))) -> g_flat.0(down.0(f.0(g.1(a.))))
395.09/246.30	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.30	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.30	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.30	   down.0(g.0(f.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.0(fresh_constant.))))
395.09/246.30	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.30	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.30	
395.09/246.30	Q is empty.
395.09/246.30	We have to consider all minimal (P,Q,R)-chains.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(100) UsableRulesReductionPairsProof (EQUIVALENT)
395.09/246.30	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.
395.09/246.30	
395.09/246.30	No dependency pairs are removed.
395.09/246.30	
395.09/246.30	The following rules are removed from R:
395.09/246.30	
395.09/246.30	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
395.09/246.30	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
395.09/246.30	Used ordering: POLO with Polynomial interpretation [POLO]:
395.09/246.30	
395.09/246.30	   POL(TOP.0(x_1)) = x_1
395.09/246.30	   POL(a.) = 0
395.09/246.30	   POL(b.) = 0
395.09/246.30	   POL(down.0(x_1)) = 1 + x_1
395.09/246.30	   POL(down.1(x_1)) = 1 + x_1
395.09/246.30	   POL(f.0(x_1)) = x_1
395.09/246.30	   POL(f.1(x_1)) = x_1
395.09/246.30	   POL(f_flat.0(x_1)) = x_1
395.09/246.30	   POL(fresh_constant.) = 0
395.09/246.30	   POL(g.0(x_1)) = x_1
395.09/246.30	   POL(g.1(x_1)) = x_1
395.09/246.30	   POL(g_flat.0(x_1)) = x_1
395.09/246.30	   POL(up.0(x_1)) = 1 + x_1
395.09/246.30	   POL(up.1(x_1)) = 1 + x_1
395.09/246.30	
395.09/246.30	
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(101)
395.09/246.30	Obligation:
395.09/246.30	Q DP problem:
395.09/246.30	The TRS P consists of the following rules:
395.09/246.30	
395.09/246.30	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(g.1(a.)))) -> TOP.0(f_flat.0(down.0(g.1(a.))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.30	   TOP.0(up.0(g.0(f.1(a.)))) -> TOP.0(g_flat.0(down.0(f.1(a.))))
395.09/246.30	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.30	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(g.1(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.1(a.)))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(g.0(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.1(x0))))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(g.0(b.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(b.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(f.1(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(a.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(f.0(f.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.1(x0))))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(f.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(b.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.1(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.1(a.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.0(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.1(x0))))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(b.)))))
395.09/246.30	
395.09/246.30	The TRS R consists of the following rules:
395.09/246.30	
395.09/246.30	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.30	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.30	   down.0(g.0(b.)) -> g_flat.0(down.0(b.))
395.09/246.30	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.30	   down.0(f.0(g.0(g.1(a.)))) -> f_flat.0(down.0(g.0(g.1(a.))))
395.09/246.30	   down.0(g.0(g.1(a.))) -> g_flat.0(down.0(g.1(a.)))
395.09/246.30	   down.0(g.1(a.)) -> g_flat.0(down.1(a.))
395.09/246.30	   down.1(a.) -> up.0(f.1(a.))
395.09/246.30	   down.1(a.) -> up.0(g.1(a.))
395.09/246.30	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(f.0(g.0(g.0(f.1(x))))) -> up.0(b.)
395.09/246.30	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.30	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.30	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.30	   down.0(f.0(g.0(g.0(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.0(fresh_constant.))))
395.09/246.30	   down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.30	   down.0(g.0(fresh_constant.)) -> g_flat.0(down.0(fresh_constant.))
395.09/246.30	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.30	   down.0(g.0(g.0(g.1(x)))) -> up.0(b.)
395.09/246.30	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.30	   down.0(f.0(g.1(a.))) -> f_flat.0(down.0(g.1(a.)))
395.09/246.30	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.30	   down.0(f.0(b.)) -> f_flat.0(down.0(b.))
395.09/246.30	   down.0(f.0(f.0(f.1(x)))) -> up.0(b.)
395.09/246.30	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.30	   down.0(f.0(f.1(a.))) -> f_flat.0(down.0(f.1(a.)))
395.09/246.30	   down.0(f.1(a.)) -> f_flat.0(down.1(a.))
395.09/246.30	   down.0(g.0(f.1(a.))) -> g_flat.0(down.0(f.1(a.)))
395.09/246.30	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(g.0(f.0(f.0(g.1(x))))) -> up.0(b.)
395.09/246.30	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(g.0(f.0(g.0(f.1(x))))) -> up.0(b.)
395.09/246.30	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.30	   down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.30	   down.0(g.0(f.0(f.1(a.)))) -> g_flat.0(down.0(f.0(f.1(a.))))
395.09/246.30	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.30	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.30	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.30	   down.0(g.0(f.0(f.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.0(fresh_constant.))))
395.09/246.30	   down.0(g.0(f.0(g.1(a.)))) -> g_flat.0(down.0(f.0(g.1(a.))))
395.09/246.30	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.30	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.30	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.30	   down.0(g.0(f.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.0(fresh_constant.))))
395.09/246.30	   down.0(f.0(g.0(fresh_constant.))) -> f_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.30	   down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.30	   down.0(f.0(fresh_constant.)) -> f_flat.0(down.0(fresh_constant.))
395.09/246.30	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.30	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.30	
395.09/246.30	Q is empty.
395.09/246.30	We have to consider all minimal (P,Q,R)-chains.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(102) MRRProof (EQUIVALENT)
395.09/246.30	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.
395.09/246.30	
395.09/246.30	
395.09/246.30	Strictly oriented rules of the TRS R:
395.09/246.30	
395.09/246.30	   down.0(f.0(g.0(g.0(f.1(x))))) -> up.0(b.)
395.09/246.30	   down.0(g.0(g.0(g.1(x)))) -> up.0(b.)
395.09/246.30	   down.0(f.0(f.0(f.1(x)))) -> up.0(b.)
395.09/246.30	   down.0(g.0(f.0(f.0(g.1(x))))) -> up.0(b.)
395.09/246.30	   down.0(g.0(f.0(g.0(f.1(x))))) -> up.0(b.)
395.09/246.30	
395.09/246.30	Used ordering: Polynomial interpretation [POLO]:
395.09/246.30	
395.09/246.30	   POL(TOP.0(x_1)) = x_1
395.09/246.30	   POL(a.) = 0
395.09/246.30	   POL(b.) = 0
395.09/246.30	   POL(down.0(x_1)) = x_1
395.09/246.30	   POL(down.1(x_1)) = 1 + x_1
395.09/246.30	   POL(f.0(x_1)) = x_1
395.09/246.30	   POL(f.1(x_1)) = 1 + x_1
395.09/246.30	   POL(f_flat.0(x_1)) = x_1
395.09/246.30	   POL(fresh_constant.) = 0
395.09/246.30	   POL(g.0(x_1)) = x_1
395.09/246.30	   POL(g.1(x_1)) = 1 + x_1
395.09/246.30	   POL(g_flat.0(x_1)) = x_1
395.09/246.30	   POL(up.0(x_1)) = x_1
395.09/246.30	
395.09/246.30	
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(103)
395.09/246.30	Obligation:
395.09/246.30	Q DP problem:
395.09/246.30	The TRS P consists of the following rules:
395.09/246.30	
395.09/246.30	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(g.1(a.)))) -> TOP.0(f_flat.0(down.0(g.1(a.))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.30	   TOP.0(up.0(g.0(f.1(a.)))) -> TOP.0(g_flat.0(down.0(f.1(a.))))
395.09/246.30	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.30	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(g.1(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.1(a.)))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(g.0(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.1(x0))))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(g.0(b.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(b.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(f.1(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(a.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(f.0(f.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.1(x0))))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(f.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(b.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.1(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.1(a.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.0(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.1(x0))))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(b.)))))
395.09/246.30	
395.09/246.30	The TRS R consists of the following rules:
395.09/246.30	
395.09/246.30	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.30	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.30	   down.0(g.0(b.)) -> g_flat.0(down.0(b.))
395.09/246.30	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.30	   down.0(f.0(g.0(g.1(a.)))) -> f_flat.0(down.0(g.0(g.1(a.))))
395.09/246.30	   down.0(g.0(g.1(a.))) -> g_flat.0(down.0(g.1(a.)))
395.09/246.30	   down.0(g.1(a.)) -> g_flat.0(down.1(a.))
395.09/246.30	   down.1(a.) -> up.0(f.1(a.))
395.09/246.30	   down.1(a.) -> up.0(g.1(a.))
395.09/246.30	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.30	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.30	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.30	   down.0(f.0(g.0(g.0(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.0(fresh_constant.))))
395.09/246.30	   down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.30	   down.0(g.0(fresh_constant.)) -> g_flat.0(down.0(fresh_constant.))
395.09/246.30	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.30	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.30	   down.0(f.0(g.1(a.))) -> f_flat.0(down.0(g.1(a.)))
395.09/246.30	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.30	   down.0(f.0(b.)) -> f_flat.0(down.0(b.))
395.09/246.30	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.30	   down.0(f.0(f.1(a.))) -> f_flat.0(down.0(f.1(a.)))
395.09/246.30	   down.0(f.1(a.)) -> f_flat.0(down.1(a.))
395.09/246.30	   down.0(g.0(f.1(a.))) -> g_flat.0(down.0(f.1(a.)))
395.09/246.30	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.30	   down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.30	   down.0(g.0(f.0(f.1(a.)))) -> g_flat.0(down.0(f.0(f.1(a.))))
395.09/246.30	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.30	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.30	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.30	   down.0(g.0(f.0(f.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.0(fresh_constant.))))
395.09/246.30	   down.0(g.0(f.0(g.1(a.)))) -> g_flat.0(down.0(f.0(g.1(a.))))
395.09/246.30	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.30	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.30	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.30	   down.0(g.0(f.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.0(fresh_constant.))))
395.09/246.30	   down.0(f.0(g.0(fresh_constant.))) -> f_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.30	   down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.30	   down.0(f.0(fresh_constant.)) -> f_flat.0(down.0(fresh_constant.))
395.09/246.30	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.30	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.30	
395.09/246.30	Q is empty.
395.09/246.30	We have to consider all minimal (P,Q,R)-chains.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(104) DependencyGraphProof (EQUIVALENT)
395.09/246.30	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(105)
395.09/246.30	Obligation:
395.09/246.30	Q DP problem:
395.09/246.30	The TRS P consists of the following rules:
395.09/246.30	
395.09/246.30	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(g.1(a.)))) -> TOP.0(f_flat.0(down.0(g.1(a.))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.30	   TOP.0(up.0(g.0(f.1(a.)))) -> TOP.0(g_flat.0(down.0(f.1(a.))))
395.09/246.30	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.30	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(g.1(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.1(a.)))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(g.0(b.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(b.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(f.1(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(a.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(f.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(b.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.1(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.1(a.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.0(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.1(x0))))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(b.)))))
395.09/246.30	
395.09/246.30	The TRS R consists of the following rules:
395.09/246.30	
395.09/246.30	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.30	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.30	   down.0(g.0(b.)) -> g_flat.0(down.0(b.))
395.09/246.30	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.30	   down.0(f.0(g.0(g.1(a.)))) -> f_flat.0(down.0(g.0(g.1(a.))))
395.09/246.30	   down.0(g.0(g.1(a.))) -> g_flat.0(down.0(g.1(a.)))
395.09/246.30	   down.0(g.1(a.)) -> g_flat.0(down.1(a.))
395.09/246.30	   down.1(a.) -> up.0(f.1(a.))
395.09/246.30	   down.1(a.) -> up.0(g.1(a.))
395.09/246.30	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.30	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.30	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.30	   down.0(f.0(g.0(g.0(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.0(fresh_constant.))))
395.09/246.30	   down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.30	   down.0(g.0(fresh_constant.)) -> g_flat.0(down.0(fresh_constant.))
395.09/246.30	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.30	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.30	   down.0(f.0(g.1(a.))) -> f_flat.0(down.0(g.1(a.)))
395.09/246.30	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.30	   down.0(f.0(b.)) -> f_flat.0(down.0(b.))
395.09/246.30	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.30	   down.0(f.0(f.1(a.))) -> f_flat.0(down.0(f.1(a.)))
395.09/246.30	   down.0(f.1(a.)) -> f_flat.0(down.1(a.))
395.09/246.30	   down.0(g.0(f.1(a.))) -> g_flat.0(down.0(f.1(a.)))
395.09/246.30	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.30	   down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.30	   down.0(g.0(f.0(f.1(a.)))) -> g_flat.0(down.0(f.0(f.1(a.))))
395.09/246.30	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.30	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.30	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.30	   down.0(g.0(f.0(f.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.0(fresh_constant.))))
395.09/246.30	   down.0(g.0(f.0(g.1(a.)))) -> g_flat.0(down.0(f.0(g.1(a.))))
395.09/246.30	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.30	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.30	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.30	   down.0(g.0(f.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.0(fresh_constant.))))
395.09/246.30	   down.0(f.0(g.0(fresh_constant.))) -> f_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.30	   down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.30	   down.0(f.0(fresh_constant.)) -> f_flat.0(down.0(fresh_constant.))
395.09/246.30	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.30	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.30	
395.09/246.30	Q is empty.
395.09/246.30	We have to consider all minimal (P,Q,R)-chains.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(106) QDPOrderProof (EQUIVALENT)
395.09/246.30	We use the reduction pair processor [LPAR04,JAR06].
395.09/246.30	
395.09/246.30	
395.09/246.30	The following pairs can be oriented strictly and are deleted.
395.09/246.30	
395.09/246.30	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(g.1(a.)))) -> TOP.0(f_flat.0(down.0(g.1(a.))))
395.09/246.30	The remaining pairs can at least be oriented weakly.
395.09/246.30	Used ordering:  Polynomial interpretation [POLO]:
395.09/246.30	
395.09/246.30	   POL(TOP.0(x_1)) = x_1
395.09/246.30	   POL(a.) = 1
395.09/246.30	   POL(b.) = 0
395.09/246.30	   POL(down.0(x_1)) = 0
395.09/246.30	   POL(down.1(x_1)) = 0
395.09/246.30	   POL(f.0(x_1)) = x_1
395.09/246.30	   POL(f.1(x_1)) = 0
395.09/246.30	   POL(f_flat.0(x_1)) = x_1
395.09/246.30	   POL(fresh_constant.) = 0
395.09/246.30	   POL(g.0(x_1)) = 0
395.09/246.30	   POL(g.1(x_1)) = 1 + x_1
395.09/246.30	   POL(g_flat.0(x_1)) = 0
395.09/246.30	   POL(up.0(x_1)) = x_1
395.09/246.30	
395.09/246.30	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
395.09/246.30	
395.09/246.30	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.30	   down.0(f.0(g.0(g.1(a.)))) -> f_flat.0(down.0(g.0(g.1(a.))))
395.09/246.30	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.30	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.30	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.30	   down.0(f.0(g.0(g.0(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.0(fresh_constant.))))
395.09/246.30	   down.0(f.0(g.0(fresh_constant.))) -> f_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.30	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.30	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.30	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.30	   down.0(f.0(g.1(a.))) -> f_flat.0(down.0(g.1(a.)))
395.09/246.30	   down.0(g.1(a.)) -> g_flat.0(down.1(a.))
395.09/246.30	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.30	   down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.30	   down.0(g.0(f.0(f.1(a.)))) -> g_flat.0(down.0(f.0(f.1(a.))))
395.09/246.30	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.30	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.30	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.30	   down.0(g.0(f.0(f.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.0(fresh_constant.))))
395.09/246.30	   down.0(g.0(f.0(g.1(a.)))) -> g_flat.0(down.0(f.0(g.1(a.))))
395.09/246.30	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.30	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.30	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.30	   down.0(g.0(f.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.0(fresh_constant.))))
395.09/246.30	   down.0(g.0(f.1(a.))) -> g_flat.0(down.0(f.1(a.)))
395.09/246.30	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.30	   down.0(g.0(g.1(a.))) -> g_flat.0(down.0(g.1(a.)))
395.09/246.30	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.30	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.30	   down.0(g.0(b.)) -> g_flat.0(down.0(b.))
395.09/246.30	   down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.30	   down.0(g.0(fresh_constant.)) -> g_flat.0(down.0(fresh_constant.))
395.09/246.30	
395.09/246.30	
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(107)
395.09/246.30	Obligation:
395.09/246.30	Q DP problem:
395.09/246.30	The TRS P consists of the following rules:
395.09/246.30	
395.09/246.30	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.30	   TOP.0(up.0(g.0(f.1(a.)))) -> TOP.0(g_flat.0(down.0(f.1(a.))))
395.09/246.30	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.30	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(g.1(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.1(a.)))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(g.0(b.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(b.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(f.1(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(a.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(f.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(b.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.1(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.1(a.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.0(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.1(x0))))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(b.)))))
395.09/246.30	
395.09/246.30	The TRS R consists of the following rules:
395.09/246.30	
395.09/246.30	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.30	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.30	   down.0(g.0(b.)) -> g_flat.0(down.0(b.))
395.09/246.30	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.30	   down.0(f.0(g.0(g.1(a.)))) -> f_flat.0(down.0(g.0(g.1(a.))))
395.09/246.30	   down.0(g.0(g.1(a.))) -> g_flat.0(down.0(g.1(a.)))
395.09/246.30	   down.0(g.1(a.)) -> g_flat.0(down.1(a.))
395.09/246.30	   down.1(a.) -> up.0(f.1(a.))
395.09/246.30	   down.1(a.) -> up.0(g.1(a.))
395.09/246.30	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.30	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.30	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.30	   down.0(f.0(g.0(g.0(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.0(fresh_constant.))))
395.09/246.30	   down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.30	   down.0(g.0(fresh_constant.)) -> g_flat.0(down.0(fresh_constant.))
395.09/246.30	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.30	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.30	   down.0(f.0(g.1(a.))) -> f_flat.0(down.0(g.1(a.)))
395.09/246.30	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.30	   down.0(f.0(b.)) -> f_flat.0(down.0(b.))
395.09/246.30	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.30	   down.0(f.0(f.1(a.))) -> f_flat.0(down.0(f.1(a.)))
395.09/246.30	   down.0(f.1(a.)) -> f_flat.0(down.1(a.))
395.09/246.30	   down.0(g.0(f.1(a.))) -> g_flat.0(down.0(f.1(a.)))
395.09/246.30	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.30	   down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.30	   down.0(g.0(f.0(f.1(a.)))) -> g_flat.0(down.0(f.0(f.1(a.))))
395.09/246.30	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.30	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.30	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.30	   down.0(g.0(f.0(f.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.0(fresh_constant.))))
395.09/246.30	   down.0(g.0(f.0(g.1(a.)))) -> g_flat.0(down.0(f.0(g.1(a.))))
395.09/246.30	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.30	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.30	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.30	   down.0(g.0(f.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.0(fresh_constant.))))
395.09/246.30	   down.0(f.0(g.0(fresh_constant.))) -> f_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.30	   down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.30	   down.0(f.0(fresh_constant.)) -> f_flat.0(down.0(fresh_constant.))
395.09/246.30	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.30	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.30	
395.09/246.30	Q is empty.
395.09/246.30	We have to consider all minimal (P,Q,R)-chains.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(108) QDPOrderProof (EQUIVALENT)
395.09/246.30	We use the reduction pair processor [LPAR04,JAR06].
395.09/246.30	
395.09/246.30	
395.09/246.30	The following pairs can be oriented strictly and are deleted.
395.09/246.30	
395.09/246.30	   TOP.0(up.0(g.0(f.1(a.)))) -> TOP.0(g_flat.0(down.0(f.1(a.))))
395.09/246.30	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.30	The remaining pairs can at least be oriented weakly.
395.09/246.30	Used ordering:  Polynomial interpretation [POLO]:
395.09/246.30	
395.09/246.30	   POL(TOP.0(x_1)) = x_1
395.09/246.30	   POL(a.) = 1
395.09/246.30	   POL(b.) = 0
395.09/246.30	   POL(down.0(x_1)) = 0
395.09/246.30	   POL(down.1(x_1)) = 0
395.09/246.30	   POL(f.0(x_1)) = 0
395.09/246.30	   POL(f.1(x_1)) = 1 + x_1
395.09/246.30	   POL(f_flat.0(x_1)) = 0
395.09/246.30	   POL(fresh_constant.) = 0
395.09/246.30	   POL(g.0(x_1)) = x_1
395.09/246.30	   POL(g.1(x_1)) = 0
395.09/246.30	   POL(g_flat.0(x_1)) = x_1
395.09/246.30	   POL(up.0(x_1)) = x_1
395.09/246.30	
395.09/246.30	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
395.09/246.30	
395.09/246.30	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.30	   down.0(f.0(g.0(g.1(a.)))) -> f_flat.0(down.0(g.0(g.1(a.))))
395.09/246.30	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.30	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.30	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.30	   down.0(f.0(g.0(g.0(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.0(fresh_constant.))))
395.09/246.30	   down.0(f.0(g.0(fresh_constant.))) -> f_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.30	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.30	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.30	   down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.30	   down.0(g.0(f.0(f.1(a.)))) -> g_flat.0(down.0(f.0(f.1(a.))))
395.09/246.30	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.30	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.30	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.30	   down.0(g.0(f.0(f.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.0(fresh_constant.))))
395.09/246.30	   down.0(g.0(f.0(g.1(a.)))) -> g_flat.0(down.0(f.0(g.1(a.))))
395.09/246.30	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.30	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.30	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.30	   down.0(g.0(f.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.0(fresh_constant.))))
395.09/246.30	   down.0(g.0(f.1(a.))) -> g_flat.0(down.0(f.1(a.)))
395.09/246.30	   down.0(f.1(a.)) -> f_flat.0(down.1(a.))
395.09/246.30	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.30	   down.0(f.0(f.1(a.))) -> f_flat.0(down.0(f.1(a.)))
395.09/246.30	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.30	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.30	   down.0(f.0(g.1(a.))) -> f_flat.0(down.0(g.1(a.)))
395.09/246.30	   down.0(f.0(b.)) -> f_flat.0(down.0(b.))
395.09/246.30	   down.0(f.0(fresh_constant.)) -> f_flat.0(down.0(fresh_constant.))
395.09/246.30	   down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.30	
395.09/246.30	
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(109)
395.09/246.30	Obligation:
395.09/246.30	Q DP problem:
395.09/246.30	The TRS P consists of the following rules:
395.09/246.30	
395.09/246.30	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.30	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(g.1(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.1(a.)))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.30	   TOP.0(up.0(f.0(g.0(g.0(b.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(b.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(f.1(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(a.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(f.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(b.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.1(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.1(a.)))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.0(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.1(x0))))))
395.09/246.30	   TOP.0(up.0(g.0(f.0(g.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(b.)))))
395.09/246.30	
395.09/246.30	The TRS R consists of the following rules:
395.09/246.30	
395.09/246.30	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.30	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.30	   down.0(g.0(b.)) -> g_flat.0(down.0(b.))
395.09/246.30	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.30	   down.0(f.0(g.0(g.1(a.)))) -> f_flat.0(down.0(g.0(g.1(a.))))
395.09/246.30	   down.0(g.0(g.1(a.))) -> g_flat.0(down.0(g.1(a.)))
395.09/246.30	   down.0(g.1(a.)) -> g_flat.0(down.1(a.))
395.09/246.30	   down.1(a.) -> up.0(f.1(a.))
395.09/246.30	   down.1(a.) -> up.0(g.1(a.))
395.09/246.30	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.30	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.30	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.30	   down.0(f.0(g.0(g.0(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.0(fresh_constant.))))
395.09/246.30	   down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.30	   down.0(g.0(fresh_constant.)) -> g_flat.0(down.0(fresh_constant.))
395.09/246.30	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.30	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.30	   down.0(f.0(g.1(a.))) -> f_flat.0(down.0(g.1(a.)))
395.09/246.30	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.30	   down.0(f.0(b.)) -> f_flat.0(down.0(b.))
395.09/246.30	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.30	   down.0(f.0(f.1(a.))) -> f_flat.0(down.0(f.1(a.)))
395.09/246.30	   down.0(f.1(a.)) -> f_flat.0(down.1(a.))
395.09/246.30	   down.0(g.0(f.1(a.))) -> g_flat.0(down.0(f.1(a.)))
395.09/246.30	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.30	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.30	   down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.30	   down.0(g.0(f.0(f.1(a.)))) -> g_flat.0(down.0(f.0(f.1(a.))))
395.09/246.30	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.30	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.30	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.30	   down.0(g.0(f.0(f.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.0(fresh_constant.))))
395.09/246.30	   down.0(g.0(f.0(g.1(a.)))) -> g_flat.0(down.0(f.0(g.1(a.))))
395.09/246.30	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.30	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.30	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.30	   down.0(g.0(f.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.0(fresh_constant.))))
395.09/246.30	   down.0(f.0(g.0(fresh_constant.))) -> f_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.30	   down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.30	   down.0(f.0(fresh_constant.)) -> f_flat.0(down.0(fresh_constant.))
395.09/246.30	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.30	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.30	
395.09/246.30	Q is empty.
395.09/246.30	We have to consider all minimal (P,Q,R)-chains.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(110) PisEmptyProof (SOUND)
395.09/246.30	The TRS P is empty. Hence, there is no (P,Q,R) chain.
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(111)
395.09/246.30	TRUE
395.09/246.30	
395.09/246.30	----------------------------------------
395.09/246.30	
395.09/246.30	(112)
395.09/246.30	Obligation:
395.09/246.30	Q DP problem:
395.09/246.30	The TRS P consists of the following rules:
395.09/246.30	
395.09/246.30	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.30	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.30	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.30	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.30	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.09/246.30	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.30	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.30	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.09/246.30	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.30	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.30	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.30	
395.09/246.30	The TRS R consists of the following rules:
395.09/246.30	
395.09/246.30	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.30	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.30	   down(g(b)) -> g_flat(down(b))
395.09/246.30	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.30	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.30	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.30	   down(g(a)) -> g_flat(down(a))
395.09/246.30	   down(a) -> up(f(a))
395.09/246.30	   down(a) -> up(g(a))
395.09/246.30	   down(f(g(g(f(x))))) -> up(b)
395.09/246.30	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.31	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.31	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.31	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.31	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.09/246.31	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.31	   down(g(g(g(x)))) -> up(b)
395.09/246.31	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.31	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.31	   down(f(b)) -> f_flat(down(b))
395.09/246.31	   down(f(f(f(x)))) -> up(b)
395.09/246.31	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.31	   down(f(a)) -> f_flat(down(a))
395.09/246.31	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.31	   down(g(f(f(g(x))))) -> up(b)
395.09/246.31	   down(g(f(g(f(x))))) -> up(b)
395.09/246.31	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.31	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.31	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.31	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.31	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.31	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.31	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.31	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.31	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.31	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.31	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.31	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.31	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.09/246.31	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.31	
395.09/246.31	Q is empty.
395.09/246.31	We have to consider all minimal (P,Q,R)-chains.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(113) SplitQDPProof (EQUIVALENT)
395.09/246.31	We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem
395.09/246.31	
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(114)
395.09/246.31	Complex Obligation (AND)
395.09/246.31	
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(115)
395.09/246.31	Obligation:
395.09/246.31	Q DP problem:
395.09/246.31	The TRS P consists of the following rules:
395.09/246.31	
395.09/246.31	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.31	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.31	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.31	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.31	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.31	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.09/246.31	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.31	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.31	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.09/246.31	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.31	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.31	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.31	
395.09/246.31	The TRS R consists of the following rules:
395.09/246.31	
395.09/246.31	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.31	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.31	   down(g(b)) -> g_flat(down(b))
395.09/246.31	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.31	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.31	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.31	   down(g(a)) -> g_flat(down(a))
395.09/246.31	   down(a) -> up(f(a))
395.09/246.31	   down(a) -> up(g(a))
395.09/246.31	   down(f(g(g(f(x))))) -> up(b)
395.09/246.31	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.31	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.31	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.31	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.31	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
395.09/246.31	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.31	   down(g(g(g(x)))) -> up(b)
395.09/246.31	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.31	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.31	   down(f(b)) -> f_flat(down(b))
395.09/246.31	   down(f(f(f(x)))) -> up(b)
395.09/246.31	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.31	   down(f(a)) -> f_flat(down(a))
395.09/246.31	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.31	   down(g(f(f(g(x))))) -> up(b)
395.09/246.31	   down(g(f(g(f(x))))) -> up(b)
395.09/246.31	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.31	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.31	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.31	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.31	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.31	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.31	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.31	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.31	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.31	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.31	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.31	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.31	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
395.09/246.31	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.31	
395.09/246.31	Q is empty.
395.09/246.31	We have to consider all minimal (P,Q,R)-chains.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(116) SemLabProof (SOUND)
395.09/246.31	We found the following model for the rules of the TRSs R and P.
395.09/246.31	Interpretation over the domain with elements from 0 to 1.
395.09/246.31	a: 0
395.09/246.31	b: 0
395.09/246.31	down: 0
395.09/246.31	f: 0
395.09/246.31	fresh_constant: 1
395.09/246.31	up: 0
395.09/246.31	f_flat: 0
395.09/246.31	TOP: 0
395.09/246.31	g_flat: 0
395.09/246.31	g: 0
395.09/246.31	By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(117)
395.09/246.31	Obligation:
395.09/246.31	Q DP problem:
395.09/246.31	The TRS P consists of the following rules:
395.09/246.31	
395.09/246.31	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.31	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.1(x0))))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(b.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(f.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.1(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.1(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(b.)))))
395.09/246.31	
395.09/246.31	The TRS R consists of the following rules:
395.09/246.31	
395.09/246.31	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.31	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.31	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
395.09/246.31	   down.0(g.0(b.)) -> g_flat.0(down.0(b.))
395.09/246.31	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.31	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
395.09/246.31	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.31	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.31	   down.0(a.) -> up.0(f.0(a.))
395.09/246.31	   down.0(a.) -> up.0(g.0(a.))
395.09/246.31	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(f.1(x))))) -> up.0(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.31	   down.0(f.0(g.0(g.1(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	   down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.))
395.09/246.31	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.31	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.31	   down.0(g.0(g.0(g.1(x)))) -> up.0(b.)
395.09/246.31	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(f.0(b.)) -> f_flat.0(down.0(b.))
395.09/246.31	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.31	   down.0(f.0(f.0(f.1(x)))) -> up.0(b.)
395.09/246.31	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.31	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(f.0(g.1(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.1(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.31	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.31	   down.0(g.0(f.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(f.0(g.1(fresh_constant.))) -> f_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   down.0(f.1(fresh_constant.)) -> f_flat.0(down.1(fresh_constant.))
395.09/246.31	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.31	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.31	
395.09/246.31	Q is empty.
395.09/246.31	We have to consider all minimal (P,Q,R)-chains.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(118) UsableRulesReductionPairsProof (EQUIVALENT)
395.09/246.31	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.
395.09/246.31	
395.09/246.31	No dependency pairs are removed.
395.09/246.31	
395.09/246.31	The following rules are removed from R:
395.09/246.31	
395.09/246.31	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
395.09/246.31	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
395.09/246.31	   down.0(f.0(g.0(g.0(f.1(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(g.0(g.1(x)))) -> up.0(b.)
395.09/246.31	   down.0(f.0(f.0(f.1(x)))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(f.0(g.1(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.1(x))))) -> up.0(b.)
395.09/246.31	Used ordering: POLO with Polynomial interpretation [POLO]:
395.09/246.31	
395.09/246.31	   POL(TOP.0(x_1)) = x_1
395.09/246.31	   POL(a.) = 0
395.09/246.31	   POL(b.) = 0
395.09/246.31	   POL(down.0(x_1)) = x_1
395.09/246.31	   POL(down.1(x_1)) = 1 + x_1
395.09/246.31	   POL(f.0(x_1)) = x_1
395.09/246.31	   POL(f.1(x_1)) = 1 + x_1
395.09/246.31	   POL(f_flat.0(x_1)) = x_1
395.09/246.31	   POL(fresh_constant.) = 0
395.09/246.31	   POL(g.0(x_1)) = x_1
395.09/246.31	   POL(g.1(x_1)) = 1 + x_1
395.09/246.31	   POL(g_flat.0(x_1)) = x_1
395.09/246.31	   POL(up.0(x_1)) = x_1
395.09/246.31	   POL(up.1(x_1)) = 1 + x_1
395.09/246.31	
395.09/246.31	
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(119)
395.09/246.31	Obligation:
395.09/246.31	Q DP problem:
395.09/246.31	The TRS P consists of the following rules:
395.09/246.31	
395.09/246.31	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.31	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.1(x0))))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(b.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(f.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.1(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.1(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(b.)))))
395.09/246.31	
395.09/246.31	The TRS R consists of the following rules:
395.09/246.31	
395.09/246.31	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.31	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.31	   down.0(g.0(b.)) -> g_flat.0(down.0(b.))
395.09/246.31	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.31	   down.0(f.0(g.0(g.1(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	   down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.))
395.09/246.31	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.31	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.31	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.31	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.31	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.31	   down.0(a.) -> up.0(f.0(a.))
395.09/246.31	   down.0(a.) -> up.0(g.0(a.))
395.09/246.31	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(f.0(b.)) -> f_flat.0(down.0(b.))
395.09/246.31	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.31	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.31	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   down.0(f.1(fresh_constant.)) -> f_flat.0(down.1(fresh_constant.))
395.09/246.31	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.31	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.31	   down.0(g.0(f.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(f.0(g.1(fresh_constant.))) -> f_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.31	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.31	
395.09/246.31	Q is empty.
395.09/246.31	We have to consider all minimal (P,Q,R)-chains.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(120) DependencyGraphProof (EQUIVALENT)
395.09/246.31	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(121)
395.09/246.31	Obligation:
395.09/246.31	Q DP problem:
395.09/246.31	The TRS P consists of the following rules:
395.09/246.31	
395.09/246.31	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.31	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(b.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.1(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(b.)))))
395.09/246.31	
395.09/246.31	The TRS R consists of the following rules:
395.09/246.31	
395.09/246.31	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.31	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.31	   down.0(g.0(b.)) -> g_flat.0(down.0(b.))
395.09/246.31	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.31	   down.0(f.0(g.0(g.1(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	   down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.))
395.09/246.31	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.31	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.31	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.31	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.31	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.31	   down.0(a.) -> up.0(f.0(a.))
395.09/246.31	   down.0(a.) -> up.0(g.0(a.))
395.09/246.31	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(f.0(b.)) -> f_flat.0(down.0(b.))
395.09/246.31	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.31	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.31	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   down.0(f.1(fresh_constant.)) -> f_flat.0(down.1(fresh_constant.))
395.09/246.31	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.31	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.31	   down.0(g.0(f.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(f.0(g.1(fresh_constant.))) -> f_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.31	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.31	
395.09/246.31	Q is empty.
395.09/246.31	We have to consider all minimal (P,Q,R)-chains.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(122) MRRProof (EQUIVALENT)
395.09/246.31	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.
395.09/246.31	
395.09/246.31	
395.09/246.31	Strictly oriented rules of the TRS R:
395.09/246.31	
395.09/246.31	   down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.))
395.09/246.31	   down.0(f.1(fresh_constant.)) -> f_flat.0(down.1(fresh_constant.))
395.09/246.31	
395.09/246.31	Used ordering: Polynomial interpretation [POLO]:
395.09/246.31	
395.09/246.31	   POL(TOP.0(x_1)) = x_1
395.09/246.31	   POL(a.) = 0
395.09/246.31	   POL(b.) = 0
395.09/246.31	   POL(down.0(x_1)) = x_1
395.09/246.31	   POL(down.1(x_1)) = x_1
395.09/246.31	   POL(f.0(x_1)) = x_1
395.09/246.31	   POL(f.1(x_1)) = 1 + x_1
395.09/246.31	   POL(f_flat.0(x_1)) = x_1
395.09/246.31	   POL(fresh_constant.) = 0
395.09/246.31	   POL(g.0(x_1)) = x_1
395.09/246.31	   POL(g.1(x_1)) = 1 + x_1
395.09/246.31	   POL(g_flat.0(x_1)) = x_1
395.09/246.31	   POL(up.0(x_1)) = x_1
395.09/246.31	
395.09/246.31	
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(123)
395.09/246.31	Obligation:
395.09/246.31	Q DP problem:
395.09/246.31	The TRS P consists of the following rules:
395.09/246.31	
395.09/246.31	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.31	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(b.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.1(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(b.)))))
395.09/246.31	
395.09/246.31	The TRS R consists of the following rules:
395.09/246.31	
395.09/246.31	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.31	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.31	   down.0(g.0(b.)) -> g_flat.0(down.0(b.))
395.09/246.31	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.31	   down.0(f.0(g.0(g.1(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.31	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.31	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.31	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.31	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.31	   down.0(a.) -> up.0(f.0(a.))
395.09/246.31	   down.0(a.) -> up.0(g.0(a.))
395.09/246.31	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(f.0(b.)) -> f_flat.0(down.0(b.))
395.09/246.31	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.31	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.31	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.31	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.31	   down.0(g.0(f.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(f.0(g.1(fresh_constant.))) -> f_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.31	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.31	
395.09/246.31	Q is empty.
395.09/246.31	We have to consider all minimal (P,Q,R)-chains.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(124) QDPOrderProof (EQUIVALENT)
395.09/246.31	We use the reduction pair processor [LPAR04,JAR06].
395.09/246.31	
395.09/246.31	
395.09/246.31	The following pairs can be oriented strictly and are deleted.
395.09/246.31	
395.09/246.31	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0)))))
395.09/246.31	The remaining pairs can at least be oriented weakly.
395.09/246.31	Used ordering:  Polynomial interpretation [POLO]:
395.09/246.31	
395.09/246.31	   POL(TOP.0(x_1)) = x_1
395.09/246.31	   POL(a.) = 0
395.09/246.31	   POL(b.) = 0
395.09/246.31	   POL(down.0(x_1)) = x_1
395.09/246.31	   POL(f.0(x_1)) = x_1
395.09/246.31	   POL(f.1(x_1)) = 0
395.09/246.31	   POL(f_flat.0(x_1)) = x_1
395.09/246.31	   POL(fresh_constant.) = 0
395.09/246.31	   POL(g.0(x_1)) = 1
395.09/246.31	   POL(g.1(x_1)) = 0
395.09/246.31	   POL(g_flat.0(x_1)) = 1
395.09/246.31	   POL(up.0(x_1)) = 1
395.09/246.31	
395.09/246.31	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
395.09/246.31	
395.09/246.31	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.31	   down.0(f.0(g.0(g.1(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.31	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.31	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.31	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.31	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.31	   down.0(f.0(g.1(fresh_constant.))) -> f_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.31	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.31	   down.0(g.0(f.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.31	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.31	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.31	   down.0(g.0(b.)) -> g_flat.0(down.0(b.))
395.09/246.31	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.31	   down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	
395.09/246.31	
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(125)
395.09/246.31	Obligation:
395.09/246.31	Q DP problem:
395.09/246.31	The TRS P consists of the following rules:
395.09/246.31	
395.09/246.31	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.31	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(b.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.1(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(b.)))))
395.09/246.31	
395.09/246.31	The TRS R consists of the following rules:
395.09/246.31	
395.09/246.31	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.31	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.31	   down.0(g.0(b.)) -> g_flat.0(down.0(b.))
395.09/246.31	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.31	   down.0(f.0(g.0(g.1(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.31	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.31	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.31	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.31	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.31	   down.0(a.) -> up.0(f.0(a.))
395.09/246.31	   down.0(a.) -> up.0(g.0(a.))
395.09/246.31	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(f.0(b.)) -> f_flat.0(down.0(b.))
395.09/246.31	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.31	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.31	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.31	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.31	   down.0(g.0(f.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(f.0(g.1(fresh_constant.))) -> f_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.31	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.31	
395.09/246.31	Q is empty.
395.09/246.31	We have to consider all minimal (P,Q,R)-chains.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(126) QDPOrderProof (EQUIVALENT)
395.09/246.31	We use the reduction pair processor [LPAR04,JAR06].
395.09/246.31	
395.09/246.31	
395.09/246.31	The following pairs can be oriented strictly and are deleted.
395.09/246.31	
395.09/246.31	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.31	The remaining pairs can at least be oriented weakly.
395.09/246.31	Used ordering:  Polynomial interpretation [POLO]:
395.09/246.31	
395.09/246.31	   POL(TOP.0(x_1)) = x_1
395.09/246.31	   POL(a.) = 0
395.09/246.31	   POL(b.) = 0
395.09/246.31	   POL(down.0(x_1)) = x_1
395.09/246.31	   POL(f.0(x_1)) = 1
395.09/246.31	   POL(f.1(x_1)) = 0
395.09/246.31	   POL(f_flat.0(x_1)) = 1
395.09/246.31	   POL(fresh_constant.) = 0
395.09/246.31	   POL(g.0(x_1)) = x_1
395.09/246.31	   POL(g.1(x_1)) = 0
395.09/246.31	   POL(g_flat.0(x_1)) = x_1
395.09/246.31	   POL(up.0(x_1)) = 1
395.09/246.31	
395.09/246.31	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
395.09/246.31	
395.09/246.31	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.31	   down.0(f.0(g.0(g.1(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.31	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.31	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.31	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.31	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.31	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.31	   down.0(g.0(f.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.31	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.31	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.31	   down.0(f.0(b.)) -> f_flat.0(down.0(b.))
395.09/246.31	   down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.1(fresh_constant.))) -> f_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	
395.09/246.31	
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(127)
395.09/246.31	Obligation:
395.09/246.31	Q DP problem:
395.09/246.31	The TRS P consists of the following rules:
395.09/246.31	
395.09/246.31	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.31	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(b.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.1(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(b.)))))
395.09/246.31	
395.09/246.31	The TRS R consists of the following rules:
395.09/246.31	
395.09/246.31	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.31	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.31	   down.0(g.0(b.)) -> g_flat.0(down.0(b.))
395.09/246.31	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.31	   down.0(f.0(g.0(g.1(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.31	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.31	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.31	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.31	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.31	   down.0(a.) -> up.0(f.0(a.))
395.09/246.31	   down.0(a.) -> up.0(g.0(a.))
395.09/246.31	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(f.0(b.)) -> f_flat.0(down.0(b.))
395.09/246.31	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.31	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.31	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.31	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.31	   down.0(g.0(f.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(f.0(g.1(fresh_constant.))) -> f_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.31	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.31	
395.09/246.31	Q is empty.
395.09/246.31	We have to consider all minimal (P,Q,R)-chains.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(128) QDPOrderProof (EQUIVALENT)
395.09/246.31	We use the reduction pair processor [LPAR04,JAR06].
395.09/246.31	
395.09/246.31	
395.09/246.31	The following pairs can be oriented strictly and are deleted.
395.09/246.31	
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.1(x0))))))
395.09/246.31	The remaining pairs can at least be oriented weakly.
395.09/246.31	Used ordering:  Polynomial interpretation [POLO]:
395.09/246.31	
395.09/246.31	   POL(TOP.0(x_1)) = x_1
395.09/246.31	   POL(a.) = 0
395.09/246.31	   POL(b.) = 0
395.09/246.31	   POL(down.0(x_1)) = 0
395.09/246.31	   POL(f.0(x_1)) = x_1
395.09/246.31	   POL(f.1(x_1)) = 0
395.09/246.31	   POL(f_flat.0(x_1)) = x_1
395.09/246.31	   POL(fresh_constant.) = 0
395.09/246.31	   POL(g.0(x_1)) = x_1
395.09/246.31	   POL(g.1(x_1)) = 1
395.09/246.31	   POL(g_flat.0(x_1)) = x_1
395.09/246.31	   POL(up.0(x_1)) = x_1
395.09/246.31	
395.09/246.31	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
395.09/246.31	
395.09/246.31	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.31	   down.0(f.0(g.0(g.1(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.31	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.31	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.31	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.31	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.31	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.31	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.31	   down.0(g.0(f.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.31	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.31	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.31	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.31	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(g.0(b.)) -> g_flat.0(down.0(b.))
395.09/246.31	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.31	   down.0(a.) -> up.0(f.0(a.))
395.09/246.31	   down.0(a.) -> up.0(g.0(a.))
395.09/246.31	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.31	   down.0(f.0(b.)) -> f_flat.0(down.0(b.))
395.09/246.31	   down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.1(fresh_constant.))) -> f_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	   down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	
395.09/246.31	
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(129)
395.09/246.31	Obligation:
395.09/246.31	Q DP problem:
395.09/246.31	The TRS P consists of the following rules:
395.09/246.31	
395.09/246.31	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.31	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(b.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(b.)))))
395.09/246.31	
395.09/246.31	The TRS R consists of the following rules:
395.09/246.31	
395.09/246.31	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.31	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.31	   down.0(g.0(b.)) -> g_flat.0(down.0(b.))
395.09/246.31	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.31	   down.0(f.0(g.0(g.1(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.31	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.31	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.31	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.31	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.31	   down.0(a.) -> up.0(f.0(a.))
395.09/246.31	   down.0(a.) -> up.0(g.0(a.))
395.09/246.31	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(f.0(b.)) -> f_flat.0(down.0(b.))
395.09/246.31	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.31	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.31	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.31	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.31	   down.0(g.0(f.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(f.0(g.1(fresh_constant.))) -> f_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.31	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.31	
395.09/246.31	Q is empty.
395.09/246.31	We have to consider all minimal (P,Q,R)-chains.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(130) QDPOrderProof (EQUIVALENT)
395.09/246.31	We use the reduction pair processor [LPAR04,JAR06].
395.09/246.31	
395.09/246.31	
395.09/246.31	The following pairs can be oriented strictly and are deleted.
395.09/246.31	
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.31	The remaining pairs can at least be oriented weakly.
395.09/246.31	Used ordering:  Polynomial interpretation [POLO]:
395.09/246.31	
395.09/246.31	   POL(TOP.0(x_1)) = x_1
395.09/246.31	   POL(a.) = 0
395.09/246.31	   POL(b.) = 0
395.09/246.31	   POL(down.0(x_1)) = 0
395.09/246.31	   POL(f.0(x_1)) = x_1
395.09/246.31	   POL(f.1(x_1)) = 1
395.09/246.31	   POL(f_flat.0(x_1)) = x_1
395.09/246.31	   POL(fresh_constant.) = 0
395.09/246.31	   POL(g.0(x_1)) = x_1
395.09/246.31	   POL(g.1(x_1)) = 0
395.09/246.31	   POL(g_flat.0(x_1)) = x_1
395.09/246.31	   POL(up.0(x_1)) = x_1
395.09/246.31	
395.09/246.31	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
395.09/246.31	
395.09/246.31	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.31	   down.0(f.0(g.0(g.1(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.31	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.31	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.31	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.31	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.31	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.31	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.31	   down.0(g.0(f.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.31	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.31	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.31	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.31	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(g.0(b.)) -> g_flat.0(down.0(b.))
395.09/246.31	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.31	   down.0(a.) -> up.0(f.0(a.))
395.09/246.31	   down.0(a.) -> up.0(g.0(a.))
395.09/246.31	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.31	   down.0(f.0(b.)) -> f_flat.0(down.0(b.))
395.09/246.31	   down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.1(fresh_constant.))) -> f_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	   down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	
395.09/246.31	
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(131)
395.09/246.31	Obligation:
395.09/246.31	Q DP problem:
395.09/246.31	The TRS P consists of the following rules:
395.09/246.31	
395.09/246.31	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(b.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(b.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(b.)))))
395.09/246.31	
395.09/246.31	The TRS R consists of the following rules:
395.09/246.31	
395.09/246.31	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.31	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.31	   down.0(g.0(b.)) -> g_flat.0(down.0(b.))
395.09/246.31	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.31	   down.0(f.0(g.0(g.1(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.31	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.31	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.31	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.31	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.31	   down.0(a.) -> up.0(f.0(a.))
395.09/246.31	   down.0(a.) -> up.0(g.0(a.))
395.09/246.31	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(f.0(b.)) -> f_flat.0(down.0(b.))
395.09/246.31	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.31	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.31	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.31	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.31	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.31	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.31	   down.0(g.0(f.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.1(fresh_constant.))))
395.09/246.31	   down.0(f.0(g.1(fresh_constant.))) -> f_flat.0(down.0(g.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.31	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.31	
395.09/246.31	Q is empty.
395.09/246.31	We have to consider all minimal (P,Q,R)-chains.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(132) PisEmptyProof (SOUND)
395.09/246.31	The TRS P is empty. Hence, there is no (P,Q,R) chain.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(133)
395.09/246.31	TRUE
395.09/246.31	
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(134)
395.09/246.31	Obligation:
395.09/246.31	Q DP problem:
395.09/246.31	The TRS P consists of the following rules:
395.09/246.31	
395.09/246.31	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.31	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.31	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.31	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.31	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.31	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.09/246.31	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.31	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.31	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.09/246.31	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.31	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.31	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.31	
395.09/246.31	The TRS R consists of the following rules:
395.09/246.31	
395.09/246.31	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.31	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.31	   down(g(b)) -> g_flat(down(b))
395.09/246.31	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.31	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.31	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.31	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.31	   down(f(g(g(f(x))))) -> up(b)
395.09/246.31	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.31	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.31	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.31	   down(g(g(g(x)))) -> up(b)
395.09/246.31	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.31	   down(g(a)) -> g_flat(down(a))
395.09/246.31	   down(a) -> up(f(a))
395.09/246.31	   down(a) -> up(g(a))
395.09/246.31	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.31	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.31	   down(f(b)) -> f_flat(down(b))
395.09/246.31	   down(f(f(f(x)))) -> up(b)
395.09/246.31	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.31	   down(f(a)) -> f_flat(down(a))
395.09/246.31	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.31	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.31	   down(g(f(f(g(x))))) -> up(b)
395.09/246.31	   down(g(f(g(f(x))))) -> up(b)
395.09/246.31	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.31	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.31	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.31	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.31	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.31	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.31	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.31	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.31	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.31	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.31	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.31	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.31	
395.09/246.31	Q is empty.
395.09/246.31	We have to consider all minimal (P,Q,R)-chains.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(135) SplitQDPProof (EQUIVALENT)
395.09/246.31	We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem
395.09/246.31	
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(136)
395.09/246.31	Complex Obligation (AND)
395.09/246.31	
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(137)
395.09/246.31	Obligation:
395.09/246.31	Q DP problem:
395.09/246.31	The TRS P consists of the following rules:
395.09/246.31	
395.09/246.31	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.31	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.31	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.31	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.31	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.31	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.09/246.31	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.31	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.31	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.09/246.31	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.31	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.31	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.31	
395.09/246.31	The TRS R consists of the following rules:
395.09/246.31	
395.09/246.31	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.31	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.31	   down(g(b)) -> g_flat(down(b))
395.09/246.31	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.31	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.31	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.31	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.31	   down(f(g(g(f(x))))) -> up(b)
395.09/246.31	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.31	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.31	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.31	   down(g(g(g(x)))) -> up(b)
395.09/246.31	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.31	   down(g(a)) -> g_flat(down(a))
395.09/246.31	   down(a) -> up(f(a))
395.09/246.31	   down(a) -> up(g(a))
395.09/246.31	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.31	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.31	   down(f(b)) -> f_flat(down(b))
395.09/246.31	   down(f(f(f(x)))) -> up(b)
395.09/246.31	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.31	   down(f(a)) -> f_flat(down(a))
395.09/246.31	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.31	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.31	   down(g(f(f(g(x))))) -> up(b)
395.09/246.31	   down(g(f(g(f(x))))) -> up(b)
395.09/246.31	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.31	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.31	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.31	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.31	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.31	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.31	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.31	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.31	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.31	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.31	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.31	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.31	
395.09/246.31	Q is empty.
395.09/246.31	We have to consider all minimal (P,Q,R)-chains.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(138) SemLabProof (SOUND)
395.09/246.31	We found the following model for the rules of the TRSs R and P.
395.09/246.31	Interpretation over the domain with elements from 0 to 1.
395.09/246.31	a: 0
395.09/246.31	b: 1
395.09/246.31	down: 0
395.09/246.31	f: 0
395.09/246.31	fresh_constant: 0
395.09/246.31	up: 0
395.09/246.31	f_flat: 0
395.09/246.31	TOP: 0
395.09/246.31	g_flat: 0
395.09/246.31	g: 0
395.09/246.31	By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(139)
395.09/246.31	Obligation:
395.09/246.31	Q DP problem:
395.09/246.31	The TRS P consists of the following rules:
395.09/246.31	
395.09/246.31	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.31	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.1(x0))))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.1(b.))))) -> TOP.0(f_flat.0(down.0(g.0(g.1(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(f.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.1(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.1(b.))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.1(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.1(b.))))) -> TOP.0(g_flat.0(down.0(f.0(g.1(b.)))))
395.09/246.31	
395.09/246.31	The TRS R consists of the following rules:
395.09/246.31	
395.09/246.31	   down.0(f.0(g.1(b.))) -> f_flat.0(down.0(g.1(b.)))
395.09/246.31	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.31	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
395.09/246.31	   down.0(g.1(b.)) -> g_flat.0(down.1(b.))
395.09/246.31	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.31	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
395.09/246.31	   down.0(f.0(g.0(g.0(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.0(fresh_constant.))))
395.09/246.31	   down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.31	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.1(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(f.1(x))))) -> up.1(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.1(b.)))) -> f_flat.0(down.0(g.0(g.1(b.))))
395.09/246.31	   down.0(g.0(g.1(b.))) -> g_flat.0(down.0(g.1(b.)))
395.09/246.31	   down.0(g.0(g.0(g.0(x)))) -> up.1(b.)
395.09/246.31	   down.0(g.0(g.0(g.1(x)))) -> up.1(b.)
395.09/246.31	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.31	   down.0(a.) -> up.0(f.0(a.))
395.09/246.31	   down.0(a.) -> up.0(g.0(a.))
395.09/246.31	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(f.0(f.1(b.))) -> f_flat.0(down.0(f.1(b.)))
395.09/246.31	   down.0(f.1(b.)) -> f_flat.0(down.1(b.))
395.09/246.31	   down.0(f.0(f.0(f.0(x)))) -> up.1(b.)
395.09/246.31	   down.0(f.0(f.0(f.1(x)))) -> up.1(b.)
395.09/246.31	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.31	   down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.31	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.0(f.0(g.1(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.1(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.1(b.))) -> g_flat.0(down.0(f.1(b.)))
395.09/246.31	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.1(b.)))) -> g_flat.0(down.0(f.0(f.1(b.))))
395.09/246.31	   down.0(g.0(f.0(f.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.0(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.1(b.)))) -> g_flat.0(down.0(f.0(g.1(b.))))
395.09/246.31	   down.0(g.0(f.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.0(fresh_constant.))))
395.09/246.31	   down.0(f.0(g.0(fresh_constant.))) -> f_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.31	   down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.31	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.31	
395.09/246.31	Q is empty.
395.09/246.31	We have to consider all minimal (P,Q,R)-chains.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(140) MRRProof (EQUIVALENT)
395.09/246.31	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.
395.09/246.31	
395.09/246.31	
395.09/246.31	Strictly oriented rules of the TRS R:
395.09/246.31	
395.09/246.31	   down.0(g.1(b.)) -> g_flat.0(down.1(b.))
395.09/246.31	   down.0(f.1(b.)) -> f_flat.0(down.1(b.))
395.09/246.31	
395.09/246.31	Used ordering: Polynomial interpretation [POLO]:
395.09/246.31	
395.09/246.31	   POL(TOP.0(x_1)) = x_1
395.09/246.31	   POL(a.) = 0
395.09/246.31	   POL(b.) = 0
395.09/246.31	   POL(down.0(x_1)) = 1 + x_1
395.09/246.31	   POL(down.1(x_1)) = x_1
395.09/246.31	   POL(f.0(x_1)) = x_1
395.09/246.31	   POL(f.1(x_1)) = x_1
395.09/246.31	   POL(f_flat.0(x_1)) = x_1
395.09/246.31	   POL(fresh_constant.) = 0
395.09/246.31	   POL(g.0(x_1)) = x_1
395.09/246.31	   POL(g.1(x_1)) = x_1
395.09/246.31	   POL(g_flat.0(x_1)) = x_1
395.09/246.31	   POL(up.0(x_1)) = 1 + x_1
395.09/246.31	   POL(up.1(x_1)) = 1 + x_1
395.09/246.31	
395.09/246.31	
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(141)
395.09/246.31	Obligation:
395.09/246.31	Q DP problem:
395.09/246.31	The TRS P consists of the following rules:
395.09/246.31	
395.09/246.31	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.31	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.1(x0))))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.1(b.))))) -> TOP.0(f_flat.0(down.0(g.0(g.1(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(f.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.1(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.1(b.))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.1(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.1(b.))))) -> TOP.0(g_flat.0(down.0(f.0(g.1(b.)))))
395.09/246.31	
395.09/246.31	The TRS R consists of the following rules:
395.09/246.31	
395.09/246.31	   down.0(f.0(g.1(b.))) -> f_flat.0(down.0(g.1(b.)))
395.09/246.31	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.31	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
395.09/246.31	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.31	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
395.09/246.31	   down.0(f.0(g.0(g.0(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.0(fresh_constant.))))
395.09/246.31	   down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.31	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.1(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(f.1(x))))) -> up.1(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.1(b.)))) -> f_flat.0(down.0(g.0(g.1(b.))))
395.09/246.31	   down.0(g.0(g.1(b.))) -> g_flat.0(down.0(g.1(b.)))
395.09/246.31	   down.0(g.0(g.0(g.0(x)))) -> up.1(b.)
395.09/246.31	   down.0(g.0(g.0(g.1(x)))) -> up.1(b.)
395.09/246.31	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.31	   down.0(a.) -> up.0(f.0(a.))
395.09/246.31	   down.0(a.) -> up.0(g.0(a.))
395.09/246.31	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(f.0(f.1(b.))) -> f_flat.0(down.0(f.1(b.)))
395.09/246.31	   down.0(f.0(f.0(f.0(x)))) -> up.1(b.)
395.09/246.31	   down.0(f.0(f.0(f.1(x)))) -> up.1(b.)
395.09/246.31	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.31	   down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.31	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.0(f.0(g.1(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.1(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.1(b.))) -> g_flat.0(down.0(f.1(b.)))
395.09/246.31	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.1(b.)))) -> g_flat.0(down.0(f.0(f.1(b.))))
395.09/246.31	   down.0(g.0(f.0(f.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.0(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.1(b.)))) -> g_flat.0(down.0(f.0(g.1(b.))))
395.09/246.31	   down.0(g.0(f.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.0(fresh_constant.))))
395.09/246.31	   down.0(f.0(g.0(fresh_constant.))) -> f_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.31	   down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.31	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.31	
395.09/246.31	Q is empty.
395.09/246.31	We have to consider all minimal (P,Q,R)-chains.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(142) QDPOrderProof (EQUIVALENT)
395.09/246.31	We use the reduction pair processor [LPAR04,JAR06].
395.09/246.31	
395.09/246.31	
395.09/246.31	The following pairs can be oriented strictly and are deleted.
395.09/246.31	
395.09/246.31	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0)))))
395.09/246.31	The remaining pairs can at least be oriented weakly.
395.09/246.31	Used ordering:  Polynomial interpretation [POLO]:
395.09/246.31	
395.09/246.31	   POL(TOP.0(x_1)) = x_1
395.09/246.31	   POL(a.) = 0
395.09/246.31	   POL(b.) = 0
395.09/246.31	   POL(down.0(x_1)) = x_1
395.09/246.31	   POL(f.0(x_1)) = x_1
395.09/246.31	   POL(f.1(x_1)) = 0
395.09/246.31	   POL(f_flat.0(x_1)) = x_1
395.09/246.31	   POL(fresh_constant.) = 0
395.09/246.31	   POL(g.0(x_1)) = 1
395.09/246.31	   POL(g.1(x_1)) = 0
395.09/246.31	   POL(g_flat.0(x_1)) = 1
395.09/246.31	   POL(up.0(x_1)) = 1
395.09/246.31	   POL(up.1(x_1)) = 1
395.09/246.31	
395.09/246.31	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
395.09/246.31	
395.09/246.31	   down.0(f.0(g.0(g.0(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.0(fresh_constant.))))
395.09/246.31	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.31	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.1(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(f.1(x))))) -> up.1(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.1(b.)))) -> f_flat.0(down.0(g.0(g.1(b.))))
395.09/246.31	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(f.0(g.0(fresh_constant.))) -> f_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.31	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.31	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.31	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
395.09/246.31	   down.0(f.0(g.1(b.))) -> f_flat.0(down.0(g.1(b.)))
395.09/246.31	   down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.31	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.0(f.0(g.1(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.1(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.1(b.)))) -> g_flat.0(down.0(f.0(f.1(b.))))
395.09/246.31	   down.0(g.0(f.0(f.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.0(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.1(b.)))) -> g_flat.0(down.0(f.0(g.1(b.))))
395.09/246.31	   down.0(g.0(f.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.0(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.1(b.))) -> g_flat.0(down.0(f.1(b.)))
395.09/246.31	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.31	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
395.09/246.31	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(g.0(g.0(g.0(x)))) -> up.1(b.)
395.09/246.31	   down.0(g.0(g.0(g.1(x)))) -> up.1(b.)
395.09/246.31	   down.0(g.0(g.1(b.))) -> g_flat.0(down.0(g.1(b.)))
395.09/246.31	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.31	   down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.31	
395.09/246.31	
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(143)
395.09/246.31	Obligation:
395.09/246.31	Q DP problem:
395.09/246.31	The TRS P consists of the following rules:
395.09/246.31	
395.09/246.31	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.31	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.1(x0))))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.1(b.))))) -> TOP.0(f_flat.0(down.0(g.0(g.1(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(f.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.1(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.1(b.))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.1(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.1(b.))))) -> TOP.0(g_flat.0(down.0(f.0(g.1(b.)))))
395.09/246.31	
395.09/246.31	The TRS R consists of the following rules:
395.09/246.31	
395.09/246.31	   down.0(f.0(g.1(b.))) -> f_flat.0(down.0(g.1(b.)))
395.09/246.31	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.31	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
395.09/246.31	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.31	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
395.09/246.31	   down.0(f.0(g.0(g.0(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.0(fresh_constant.))))
395.09/246.31	   down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.31	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.1(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(f.1(x))))) -> up.1(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.1(b.)))) -> f_flat.0(down.0(g.0(g.1(b.))))
395.09/246.31	   down.0(g.0(g.1(b.))) -> g_flat.0(down.0(g.1(b.)))
395.09/246.31	   down.0(g.0(g.0(g.0(x)))) -> up.1(b.)
395.09/246.31	   down.0(g.0(g.0(g.1(x)))) -> up.1(b.)
395.09/246.31	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.31	   down.0(a.) -> up.0(f.0(a.))
395.09/246.31	   down.0(a.) -> up.0(g.0(a.))
395.09/246.31	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(f.0(f.1(b.))) -> f_flat.0(down.0(f.1(b.)))
395.09/246.31	   down.0(f.0(f.0(f.0(x)))) -> up.1(b.)
395.09/246.31	   down.0(f.0(f.0(f.1(x)))) -> up.1(b.)
395.09/246.31	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.31	   down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.31	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.0(f.0(g.1(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.1(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.1(b.))) -> g_flat.0(down.0(f.1(b.)))
395.09/246.31	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.1(b.)))) -> g_flat.0(down.0(f.0(f.1(b.))))
395.09/246.31	   down.0(g.0(f.0(f.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.0(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.1(b.)))) -> g_flat.0(down.0(f.0(g.1(b.))))
395.09/246.31	   down.0(g.0(f.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.0(fresh_constant.))))
395.09/246.31	   down.0(f.0(g.0(fresh_constant.))) -> f_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.31	   down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.31	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.31	
395.09/246.31	Q is empty.
395.09/246.31	We have to consider all minimal (P,Q,R)-chains.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(144) QDPOrderProof (EQUIVALENT)
395.09/246.31	We use the reduction pair processor [LPAR04,JAR06].
395.09/246.31	
395.09/246.31	
395.09/246.31	The following pairs can be oriented strictly and are deleted.
395.09/246.31	
395.09/246.31	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.31	The remaining pairs can at least be oriented weakly.
395.09/246.31	Used ordering:  Polynomial interpretation [POLO]:
395.09/246.31	
395.09/246.31	   POL(TOP.0(x_1)) = x_1
395.09/246.31	   POL(a.) = 0
395.09/246.31	   POL(b.) = 0
395.09/246.31	   POL(down.0(x_1)) = x_1
395.09/246.31	   POL(f.0(x_1)) = 1
395.09/246.31	   POL(f.1(x_1)) = 0
395.09/246.31	   POL(f_flat.0(x_1)) = 1
395.09/246.31	   POL(fresh_constant.) = 0
395.09/246.31	   POL(g.0(x_1)) = x_1
395.09/246.31	   POL(g.1(x_1)) = 0
395.09/246.31	   POL(g_flat.0(x_1)) = x_1
395.09/246.31	   POL(up.0(x_1)) = 1
395.09/246.31	   POL(up.1(x_1)) = 1
395.09/246.31	
395.09/246.31	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
395.09/246.31	
395.09/246.31	   down.0(f.0(g.0(g.0(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.0(fresh_constant.))))
395.09/246.31	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.31	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.1(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(f.1(x))))) -> up.1(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.1(b.)))) -> f_flat.0(down.0(g.0(g.1(b.))))
395.09/246.31	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(f.0(g.0(fresh_constant.))) -> f_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.31	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.31	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
395.09/246.31	   down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.31	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.0(f.0(g.1(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.1(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.1(b.)))) -> g_flat.0(down.0(f.0(f.1(b.))))
395.09/246.31	   down.0(g.0(f.0(f.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.0(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.1(b.)))) -> g_flat.0(down.0(f.0(g.1(b.))))
395.09/246.31	   down.0(g.0(f.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.0(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.1(b.))) -> g_flat.0(down.0(f.1(b.)))
395.09/246.31	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.31	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
395.09/246.31	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(f.0(f.0(f.0(x)))) -> up.1(b.)
395.09/246.31	   down.0(f.0(f.0(f.1(x)))) -> up.1(b.)
395.09/246.31	   down.0(f.0(f.1(b.))) -> f_flat.0(down.0(f.1(b.)))
395.09/246.31	   down.0(f.0(g.1(b.))) -> f_flat.0(down.0(g.1(b.)))
395.09/246.31	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.31	   down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.31	
395.09/246.31	
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(145)
395.09/246.31	Obligation:
395.09/246.31	Q DP problem:
395.09/246.31	The TRS P consists of the following rules:
395.09/246.31	
395.09/246.31	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.31	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.0(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.1(x0))))))
395.09/246.31	   TOP.0(up.0(f.0(g.0(g.1(b.))))) -> TOP.0(f_flat.0(down.0(g.0(g.1(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.0(f.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.1(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(f.1(b.))))) -> TOP.0(g_flat.0(down.0(f.0(f.1(b.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.0(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.1(x0))))))
395.09/246.31	   TOP.0(up.0(g.0(f.0(g.1(b.))))) -> TOP.0(g_flat.0(down.0(f.0(g.1(b.)))))
395.09/246.31	
395.09/246.31	The TRS R consists of the following rules:
395.09/246.31	
395.09/246.31	   down.0(f.0(g.1(b.))) -> f_flat.0(down.0(g.1(b.)))
395.09/246.31	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.31	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
395.09/246.31	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.31	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
395.09/246.31	   down.0(f.0(g.0(g.0(fresh_constant.)))) -> f_flat.0(down.0(g.0(g.0(fresh_constant.))))
395.09/246.31	   down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.31	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.1(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(f.1(x))))) -> up.1(b.)
395.09/246.31	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.31	   down.0(f.0(g.0(g.1(b.)))) -> f_flat.0(down.0(g.0(g.1(b.))))
395.09/246.31	   down.0(g.0(g.1(b.))) -> g_flat.0(down.0(g.1(b.)))
395.09/246.31	   down.0(g.0(g.0(g.0(x)))) -> up.1(b.)
395.09/246.31	   down.0(g.0(g.0(g.1(x)))) -> up.1(b.)
395.09/246.31	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.31	   down.0(a.) -> up.0(f.0(a.))
395.09/246.31	   down.0(a.) -> up.0(g.0(a.))
395.09/246.31	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.31	   down.0(f.0(f.1(b.))) -> f_flat.0(down.0(f.1(b.)))
395.09/246.31	   down.0(f.0(f.0(f.0(x)))) -> up.1(b.)
395.09/246.31	   down.0(f.0(f.0(f.1(x)))) -> up.1(b.)
395.09/246.31	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.31	   down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.31	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.31	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.0(f.0(g.1(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.0(g.0(f.1(x))))) -> up.1(b.)
395.09/246.31	   down.0(g.0(f.1(b.))) -> g_flat.0(down.0(f.1(b.)))
395.09/246.31	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.31	   down.0(g.0(f.0(f.1(b.)))) -> g_flat.0(down.0(f.0(f.1(b.))))
395.09/246.31	   down.0(g.0(f.0(f.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.0(fresh_constant.))))
395.09/246.31	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.31	   down.0(g.0(f.0(g.1(b.)))) -> g_flat.0(down.0(f.0(g.1(b.))))
395.09/246.31	   down.0(g.0(f.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.0(fresh_constant.))))
395.09/246.31	   down.0(f.0(g.0(fresh_constant.))) -> f_flat.0(down.0(g.0(fresh_constant.)))
395.09/246.31	   down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.)))
395.09/246.31	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.31	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.31	
395.09/246.31	Q is empty.
395.09/246.31	We have to consider all minimal (P,Q,R)-chains.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(146) PisEmptyProof (SOUND)
395.09/246.31	The TRS P is empty. Hence, there is no (P,Q,R) chain.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(147)
395.09/246.31	TRUE
395.09/246.31	
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(148)
395.09/246.31	Obligation:
395.09/246.31	Q DP problem:
395.09/246.31	The TRS P consists of the following rules:
395.09/246.31	
395.09/246.31	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.31	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.31	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.31	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.31	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.31	   TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b)))))
395.09/246.31	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.31	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.31	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.09/246.31	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.31	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.31	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.31	
395.09/246.31	The TRS R consists of the following rules:
395.09/246.31	
395.09/246.31	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.31	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.31	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.31	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.31	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.31	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.31	   down(f(g(g(f(x))))) -> up(b)
395.09/246.31	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.31	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.31	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.31	   down(g(g(g(x)))) -> up(b)
395.09/246.31	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.31	   down(g(a)) -> g_flat(down(a))
395.09/246.31	   down(a) -> up(f(a))
395.09/246.31	   down(a) -> up(g(a))
395.09/246.31	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.31	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.31	   down(f(f(f(x)))) -> up(b)
395.09/246.31	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.31	   down(f(a)) -> f_flat(down(a))
395.09/246.31	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.31	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.31	   down(g(f(f(g(x))))) -> up(b)
395.09/246.31	   down(g(f(g(f(x))))) -> up(b)
395.09/246.31	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.31	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.31	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.31	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.31	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.31	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.31	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.31	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.31	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.31	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.31	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.31	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.31	
395.09/246.31	Q is empty.
395.09/246.31	We have to consider all minimal (P,Q,R)-chains.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(149) TransformationProof (EQUIVALENT)
395.09/246.31	By narrowing [LPAR04] the rule TOP(up(f(g(g(b))))) -> TOP(f_flat(down(g(g(b))))) at position [0] we obtained the following new rules [LPAR04]:
395.09/246.31	
395.09/246.31	   (TOP(up(f(g(g(b))))) -> TOP(f_flat(g_flat(down(g(b))))),TOP(up(f(g(g(b))))) -> TOP(f_flat(g_flat(down(g(b))))))
395.09/246.31	
395.09/246.31	
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(150)
395.09/246.31	Obligation:
395.09/246.31	Q DP problem:
395.09/246.31	The TRS P consists of the following rules:
395.09/246.31	
395.09/246.31	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.31	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.31	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.31	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.31	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.31	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.31	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.31	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.09/246.31	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.31	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.31	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.31	   TOP(up(f(g(g(b))))) -> TOP(f_flat(g_flat(down(g(b)))))
395.09/246.31	
395.09/246.31	The TRS R consists of the following rules:
395.09/246.31	
395.09/246.31	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.31	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.31	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.31	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.31	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.31	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.31	   down(f(g(g(f(x))))) -> up(b)
395.09/246.31	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.31	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.31	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.31	   down(g(g(g(x)))) -> up(b)
395.09/246.31	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.31	   down(g(a)) -> g_flat(down(a))
395.09/246.31	   down(a) -> up(f(a))
395.09/246.31	   down(a) -> up(g(a))
395.09/246.31	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.31	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.31	   down(f(f(f(x)))) -> up(b)
395.09/246.31	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.31	   down(f(a)) -> f_flat(down(a))
395.09/246.31	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.31	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.31	   down(g(f(f(g(x))))) -> up(b)
395.09/246.31	   down(g(f(g(f(x))))) -> up(b)
395.09/246.31	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.31	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.31	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.31	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.31	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.31	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.31	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.31	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.31	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.31	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.31	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.31	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.31	
395.09/246.31	Q is empty.
395.09/246.31	We have to consider all minimal (P,Q,R)-chains.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(151) DependencyGraphProof (EQUIVALENT)
395.09/246.31	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(152)
395.09/246.31	Obligation:
395.09/246.31	Q DP problem:
395.09/246.31	The TRS P consists of the following rules:
395.09/246.31	
395.09/246.31	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.31	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.31	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.31	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.31	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.31	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.31	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.31	   TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b)))))
395.09/246.31	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.31	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.31	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.31	
395.09/246.31	The TRS R consists of the following rules:
395.09/246.31	
395.09/246.31	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.31	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.31	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.31	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.31	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.31	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.31	   down(f(g(g(f(x))))) -> up(b)
395.09/246.31	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.31	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.31	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.31	   down(g(g(g(x)))) -> up(b)
395.09/246.31	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.31	   down(g(a)) -> g_flat(down(a))
395.09/246.31	   down(a) -> up(f(a))
395.09/246.31	   down(a) -> up(g(a))
395.09/246.31	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.31	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.31	   down(f(f(f(x)))) -> up(b)
395.09/246.31	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.31	   down(f(a)) -> f_flat(down(a))
395.09/246.31	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.31	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.31	   down(g(f(f(g(x))))) -> up(b)
395.09/246.31	   down(g(f(g(f(x))))) -> up(b)
395.09/246.31	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.31	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.31	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.31	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.31	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.31	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.31	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.31	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.31	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.31	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.31	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.31	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.31	
395.09/246.31	Q is empty.
395.09/246.31	We have to consider all minimal (P,Q,R)-chains.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(153) TransformationProof (EQUIVALENT)
395.09/246.31	By narrowing [LPAR04] the rule TOP(up(g(f(f(b))))) -> TOP(g_flat(down(f(f(b))))) at position [0] we obtained the following new rules [LPAR04]:
395.09/246.31	
395.09/246.31	   (TOP(up(g(f(f(b))))) -> TOP(g_flat(f_flat(down(f(b))))),TOP(up(g(f(f(b))))) -> TOP(g_flat(f_flat(down(f(b))))))
395.09/246.31	
395.09/246.31	
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(154)
395.09/246.31	Obligation:
395.09/246.31	Q DP problem:
395.09/246.31	The TRS P consists of the following rules:
395.09/246.31	
395.09/246.31	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.31	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.31	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.31	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.31	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.31	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.31	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.31	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.31	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.31	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.31	   TOP(up(g(f(f(b))))) -> TOP(g_flat(f_flat(down(f(b)))))
395.09/246.31	
395.09/246.31	The TRS R consists of the following rules:
395.09/246.31	
395.09/246.31	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.31	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.31	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.31	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.31	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.31	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.31	   down(f(g(g(f(x))))) -> up(b)
395.09/246.31	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.31	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.31	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.31	   down(g(g(g(x)))) -> up(b)
395.09/246.31	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.31	   down(g(a)) -> g_flat(down(a))
395.09/246.31	   down(a) -> up(f(a))
395.09/246.31	   down(a) -> up(g(a))
395.09/246.31	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.31	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.31	   down(f(f(f(x)))) -> up(b)
395.09/246.31	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.31	   down(f(a)) -> f_flat(down(a))
395.09/246.31	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.31	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.31	   down(g(f(f(g(x))))) -> up(b)
395.09/246.31	   down(g(f(g(f(x))))) -> up(b)
395.09/246.31	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.31	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.31	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.31	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.31	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.31	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.31	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.31	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.31	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.31	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.31	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.31	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.31	
395.09/246.31	Q is empty.
395.09/246.31	We have to consider all minimal (P,Q,R)-chains.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(155) DependencyGraphProof (EQUIVALENT)
395.09/246.31	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(156)
395.09/246.31	Obligation:
395.09/246.31	Q DP problem:
395.09/246.31	The TRS P consists of the following rules:
395.09/246.31	
395.09/246.31	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.31	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.31	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.31	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.31	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.31	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.31	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.31	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.31	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.31	   TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b)))))
395.09/246.31	
395.09/246.31	The TRS R consists of the following rules:
395.09/246.31	
395.09/246.31	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.31	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.31	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.31	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.31	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.31	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.31	   down(f(g(g(f(x))))) -> up(b)
395.09/246.31	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.31	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.31	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.31	   down(g(g(g(x)))) -> up(b)
395.09/246.31	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.31	   down(g(a)) -> g_flat(down(a))
395.09/246.31	   down(a) -> up(f(a))
395.09/246.31	   down(a) -> up(g(a))
395.09/246.31	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.31	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.31	   down(f(f(f(x)))) -> up(b)
395.09/246.31	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.31	   down(f(a)) -> f_flat(down(a))
395.09/246.31	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.31	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.31	   down(g(f(f(g(x))))) -> up(b)
395.09/246.31	   down(g(f(g(f(x))))) -> up(b)
395.09/246.31	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.31	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.31	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.31	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.31	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.31	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.31	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.31	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.31	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.31	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.31	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.31	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.31	
395.09/246.31	Q is empty.
395.09/246.31	We have to consider all minimal (P,Q,R)-chains.
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(157) TransformationProof (EQUIVALENT)
395.09/246.31	By narrowing [LPAR04] the rule TOP(up(g(f(g(b))))) -> TOP(g_flat(down(f(g(b))))) at position [0] we obtained the following new rules [LPAR04]:
395.09/246.31	
395.09/246.31	   (TOP(up(g(f(g(b))))) -> TOP(g_flat(f_flat(down(g(b))))),TOP(up(g(f(g(b))))) -> TOP(g_flat(f_flat(down(g(b))))))
395.09/246.31	
395.09/246.31	
395.09/246.31	----------------------------------------
395.09/246.31	
395.09/246.31	(158)
395.09/246.31	Obligation:
395.09/246.31	Q DP problem:
395.09/246.31	The TRS P consists of the following rules:
395.09/246.31	
395.09/246.31	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.31	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.32	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.32	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.32	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.32	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.32	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.32	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.32	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.32	   TOP(up(g(f(g(b))))) -> TOP(g_flat(f_flat(down(g(b)))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.32	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.32	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.32	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.32	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.32	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.32	   down(f(g(g(f(x))))) -> up(b)
395.09/246.32	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.32	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.32	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.32	   down(g(g(g(x)))) -> up(b)
395.09/246.32	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.32	   down(g(a)) -> g_flat(down(a))
395.09/246.32	   down(a) -> up(f(a))
395.09/246.32	   down(a) -> up(g(a))
395.09/246.32	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.32	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.32	   down(f(f(f(x)))) -> up(b)
395.09/246.32	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.32	   down(f(a)) -> f_flat(down(a))
395.09/246.32	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.32	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.32	   down(g(f(f(g(x))))) -> up(b)
395.09/246.32	   down(g(f(g(f(x))))) -> up(b)
395.09/246.32	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.32	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.32	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.32	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.32	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.32	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.32	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.32	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.32	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.32	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.32	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.32	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(159) DependencyGraphProof (EQUIVALENT)
395.09/246.32	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(160)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.32	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.32	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.32	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.32	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.32	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.32	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.32	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.32	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.32	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.32	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.32	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.32	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.32	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.32	   down(f(g(g(f(x))))) -> up(b)
395.09/246.32	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.32	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.32	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.32	   down(g(g(g(x)))) -> up(b)
395.09/246.32	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.32	   down(g(a)) -> g_flat(down(a))
395.09/246.32	   down(a) -> up(f(a))
395.09/246.32	   down(a) -> up(g(a))
395.09/246.32	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.32	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.32	   down(f(f(f(x)))) -> up(b)
395.09/246.32	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.32	   down(f(a)) -> f_flat(down(a))
395.09/246.32	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.32	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.32	   down(g(f(f(g(x))))) -> up(b)
395.09/246.32	   down(g(f(g(f(x))))) -> up(b)
395.09/246.32	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.32	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.32	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.32	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.32	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.32	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.32	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.32	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.32	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.32	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.32	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.32	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(161) SplitQDPProof (EQUIVALENT)
395.09/246.32	We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem
395.09/246.32	
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(162)
395.09/246.32	Complex Obligation (AND)
395.09/246.32	
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(163)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.32	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.32	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.32	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.32	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.32	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.32	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.32	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.32	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.32	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.32	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.32	   down(f(g(g(fresh_constant)))) -> f_flat(down(g(g(fresh_constant))))
395.09/246.32	   down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant)))
395.09/246.32	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.32	   down(f(g(g(f(x))))) -> up(b)
395.09/246.32	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.32	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.32	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.32	   down(g(g(g(x)))) -> up(b)
395.09/246.32	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.32	   down(g(a)) -> g_flat(down(a))
395.09/246.32	   down(a) -> up(f(a))
395.09/246.32	   down(a) -> up(g(a))
395.09/246.32	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.32	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.32	   down(f(f(f(x)))) -> up(b)
395.09/246.32	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.32	   down(f(a)) -> f_flat(down(a))
395.09/246.32	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.32	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.32	   down(g(f(f(g(x))))) -> up(b)
395.09/246.32	   down(g(f(g(f(x))))) -> up(b)
395.09/246.32	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.32	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.32	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.32	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.32	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.32	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.32	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.32	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.32	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.32	   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
395.09/246.32	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.32	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(164) SemLabProof (SOUND)
395.09/246.32	We found the following model for the rules of the TRSs R and P.
395.09/246.32	Interpretation over the domain with elements from 0 to 1.
395.09/246.32	a: 0
395.09/246.32	b: 0
395.09/246.32	down: 0
395.09/246.32	f: 0
395.09/246.32	fresh_constant: 1
395.09/246.32	up: 0
395.09/246.32	f_flat: 0
395.09/246.32	TOP: 0
395.09/246.32	g_flat: 0
395.09/246.32	g: x<sub>0</sub>
395.09/246.32	By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(165)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(f.1(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.1(g.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(f.1(g.1(g.1(g.1(x0)))))) -> TOP.0(f_flat.0(down.1(g.1(g.1(g.1(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(f.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.1(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.1(g.1(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.1(g.1(g.1(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.32	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.32	   g_flat.0(up.1(x_1)) -> up.1(g.1(x_1))
395.09/246.32	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.32	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
395.09/246.32	   down.0(f.1(g.1(g.1(fresh_constant.)))) -> f_flat.0(down.1(g.1(g.1(fresh_constant.))))
395.09/246.32	   down.1(g.1(g.1(fresh_constant.))) -> g_flat.0(down.1(g.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.32	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.0(f.1(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.32	   down.0(f.1(g.1(g.1(g.1(y22))))) -> f_flat.0(down.1(g.1(g.1(g.1(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.32	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.32	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.32	   down.1(g.1(g.1(g.1(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.32	   down.0(a.) -> up.0(f.0(a.))
395.09/246.32	   down.0(a.) -> up.0(g.0(a.))
395.09/246.32	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(f.0(f.0(f.1(x)))) -> up.0(b.)
395.09/246.32	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.32	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.32	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(f.1(g.1(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.1(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.32	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.32	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.32	   down.0(g.0(f.1(g.1(g.1(y28))))) -> g_flat.0(down.0(f.1(g.1(g.1(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.32	   down.0(g.0(f.1(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.1(g.1(fresh_constant.))))
395.09/246.32	   down.0(f.1(g.1(fresh_constant.))) -> f_flat.0(down.1(g.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.32	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(166) MRRProof (EQUIVALENT)
395.09/246.32	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.
395.09/246.32	
395.09/246.32	
395.09/246.32	Strictly oriented rules of the TRS R:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(g.0(f.1(x))))) -> up.0(b.)
395.09/246.32	   down.1(g.1(g.1(g.1(x)))) -> up.0(b.)
395.09/246.32	   down.0(f.0(f.0(f.1(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(f.1(g.1(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.1(x))))) -> up.0(b.)
395.09/246.32	
395.09/246.32	Used ordering: Polynomial interpretation [POLO]:
395.09/246.32	
395.09/246.32	   POL(TOP.0(x_1)) = x_1
395.09/246.32	   POL(a.) = 0
395.09/246.32	   POL(b.) = 0
395.09/246.32	   POL(down.0(x_1)) = x_1
395.09/246.32	   POL(down.1(x_1)) = 1 + x_1
395.09/246.32	   POL(f.0(x_1)) = x_1
395.09/246.32	   POL(f.1(x_1)) = 1 + x_1
395.09/246.32	   POL(f_flat.0(x_1)) = x_1
395.09/246.32	   POL(fresh_constant.) = 0
395.09/246.32	   POL(g.0(x_1)) = x_1
395.09/246.32	   POL(g.1(x_1)) = x_1
395.09/246.32	   POL(g_flat.0(x_1)) = x_1
395.09/246.32	   POL(up.0(x_1)) = x_1
395.09/246.32	   POL(up.1(x_1)) = 1 + x_1
395.09/246.32	
395.09/246.32	
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(167)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(f.1(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.1(g.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(f.1(g.1(g.1(g.1(x0)))))) -> TOP.0(f_flat.0(down.1(g.1(g.1(g.1(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(f.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.1(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.1(g.1(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.1(g.1(g.1(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.32	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.32	   g_flat.0(up.1(x_1)) -> up.1(g.1(x_1))
395.09/246.32	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.32	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
395.09/246.32	   down.0(f.1(g.1(g.1(fresh_constant.)))) -> f_flat.0(down.1(g.1(g.1(fresh_constant.))))
395.09/246.32	   down.1(g.1(g.1(fresh_constant.))) -> g_flat.0(down.1(g.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.32	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.32	   down.0(f.1(g.1(g.1(g.1(y22))))) -> f_flat.0(down.1(g.1(g.1(g.1(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.32	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.32	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.32	   down.0(a.) -> up.0(f.0(a.))
395.09/246.32	   down.0(a.) -> up.0(g.0(a.))
395.09/246.32	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.32	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.32	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.32	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.32	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.32	   down.0(g.0(f.1(g.1(g.1(y28))))) -> g_flat.0(down.0(f.1(g.1(g.1(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.32	   down.0(g.0(f.1(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.1(g.1(fresh_constant.))))
395.09/246.32	   down.0(f.1(g.1(fresh_constant.))) -> f_flat.0(down.1(g.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.32	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(168) DependencyGraphProof (EQUIVALENT)
395.09/246.32	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(169)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(f.1(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.1(g.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.1(g.1(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.1(g.1(g.1(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.32	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.32	   g_flat.0(up.1(x_1)) -> up.1(g.1(x_1))
395.09/246.32	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.32	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
395.09/246.32	   down.0(f.1(g.1(g.1(fresh_constant.)))) -> f_flat.0(down.1(g.1(g.1(fresh_constant.))))
395.09/246.32	   down.1(g.1(g.1(fresh_constant.))) -> g_flat.0(down.1(g.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.32	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.32	   down.0(f.1(g.1(g.1(g.1(y22))))) -> f_flat.0(down.1(g.1(g.1(g.1(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.32	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.32	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.32	   down.0(a.) -> up.0(f.0(a.))
395.09/246.32	   down.0(a.) -> up.0(g.0(a.))
395.09/246.32	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.32	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.32	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.32	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.32	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.32	   down.0(g.0(f.1(g.1(g.1(y28))))) -> g_flat.0(down.0(f.1(g.1(g.1(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.32	   down.0(g.0(f.1(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.1(g.1(fresh_constant.))))
395.09/246.32	   down.0(f.1(g.1(fresh_constant.))) -> f_flat.0(down.1(g.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.32	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(170) MRRProof (EQUIVALENT)
395.09/246.32	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.
395.09/246.32	
395.09/246.32	
395.09/246.32	Strictly oriented rules of the TRS R:
395.09/246.32	
395.09/246.32	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
395.09/246.32	
395.09/246.32	Used ordering: Polynomial interpretation [POLO]:
395.09/246.32	
395.09/246.32	   POL(TOP.0(x_1)) = x_1
395.09/246.32	   POL(a.) = 0
395.09/246.32	   POL(b.) = 0
395.09/246.32	   POL(down.0(x_1)) = x_1
395.09/246.32	   POL(down.1(x_1)) = x_1
395.09/246.32	   POL(f.0(x_1)) = x_1
395.09/246.32	   POL(f.1(x_1)) = x_1
395.09/246.32	   POL(f_flat.0(x_1)) = x_1
395.09/246.32	   POL(fresh_constant.) = 0
395.09/246.32	   POL(g.0(x_1)) = x_1
395.09/246.32	   POL(g.1(x_1)) = x_1
395.09/246.32	   POL(g_flat.0(x_1)) = x_1
395.09/246.32	   POL(up.0(x_1)) = x_1
395.09/246.32	   POL(up.1(x_1)) = 1 + x_1
395.09/246.32	
395.09/246.32	
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(171)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(f.1(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.1(g.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.1(g.1(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.1(g.1(g.1(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.32	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.32	   g_flat.0(up.1(x_1)) -> up.1(g.1(x_1))
395.09/246.32	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.32	   down.0(f.1(g.1(g.1(fresh_constant.)))) -> f_flat.0(down.1(g.1(g.1(fresh_constant.))))
395.09/246.32	   down.1(g.1(g.1(fresh_constant.))) -> g_flat.0(down.1(g.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.32	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.32	   down.0(f.1(g.1(g.1(g.1(y22))))) -> f_flat.0(down.1(g.1(g.1(g.1(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.32	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.32	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.32	   down.0(a.) -> up.0(f.0(a.))
395.09/246.32	   down.0(a.) -> up.0(g.0(a.))
395.09/246.32	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.32	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.32	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.32	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.32	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.32	   down.0(g.0(f.1(g.1(g.1(y28))))) -> g_flat.0(down.0(f.1(g.1(g.1(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.32	   down.0(g.0(f.1(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.1(g.1(fresh_constant.))))
395.09/246.32	   down.0(f.1(g.1(fresh_constant.))) -> f_flat.0(down.1(g.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.32	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(172) MRRProof (EQUIVALENT)
395.09/246.32	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.
395.09/246.32	
395.09/246.32	
395.09/246.32	Strictly oriented rules of the TRS R:
395.09/246.32	
395.09/246.32	   down.0(f.1(g.1(g.1(fresh_constant.)))) -> f_flat.0(down.1(g.1(g.1(fresh_constant.))))
395.09/246.32	   down.0(f.1(g.1(g.1(g.1(y22))))) -> f_flat.0(down.1(g.1(g.1(g.1(y22)))))
395.09/246.32	   down.0(f.1(g.1(fresh_constant.))) -> f_flat.0(down.1(g.1(fresh_constant.)))
395.09/246.32	
395.09/246.32	Used ordering: Polynomial interpretation [POLO]:
395.09/246.32	
395.09/246.32	   POL(TOP.0(x_1)) = x_1
395.09/246.32	   POL(a.) = 0
395.09/246.32	   POL(b.) = 0
395.09/246.32	   POL(down.0(x_1)) = x_1
395.09/246.32	   POL(down.1(x_1)) = x_1
395.09/246.32	   POL(f.0(x_1)) = x_1
395.09/246.32	   POL(f.1(x_1)) = 1 + x_1
395.09/246.32	   POL(f_flat.0(x_1)) = x_1
395.09/246.32	   POL(fresh_constant.) = 0
395.09/246.32	   POL(g.0(x_1)) = x_1
395.09/246.32	   POL(g.1(x_1)) = x_1
395.09/246.32	   POL(g_flat.0(x_1)) = x_1
395.09/246.32	   POL(up.0(x_1)) = x_1
395.09/246.32	   POL(up.1(x_1)) = x_1
395.09/246.32	
395.09/246.32	
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(173)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(f.1(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.1(g.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.1(g.1(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.1(g.1(g.1(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.32	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.32	   g_flat.0(up.1(x_1)) -> up.1(g.1(x_1))
395.09/246.32	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.32	   down.1(g.1(g.1(fresh_constant.))) -> g_flat.0(down.1(g.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.32	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.32	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.32	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.32	   down.0(a.) -> up.0(f.0(a.))
395.09/246.32	   down.0(a.) -> up.0(g.0(a.))
395.09/246.32	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.32	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.32	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.32	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.32	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.32	   down.0(g.0(f.1(g.1(g.1(y28))))) -> g_flat.0(down.0(f.1(g.1(g.1(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.32	   down.0(g.0(f.1(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.1(g.1(fresh_constant.))))
395.09/246.32	   down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.32	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(174) DependencyGraphProof (EQUIVALENT)
395.09/246.32	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(175)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.32	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.32	   g_flat.0(up.1(x_1)) -> up.1(g.1(x_1))
395.09/246.32	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.32	   down.1(g.1(g.1(fresh_constant.))) -> g_flat.0(down.1(g.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.32	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.32	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.32	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.32	   down.0(a.) -> up.0(f.0(a.))
395.09/246.32	   down.0(a.) -> up.0(g.0(a.))
395.09/246.32	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.32	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.32	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.32	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.32	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.32	   down.0(g.0(f.1(g.1(g.1(y28))))) -> g_flat.0(down.0(f.1(g.1(g.1(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.32	   down.0(g.0(f.1(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.1(g.1(fresh_constant.))))
395.09/246.32	   down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.32	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(176) UsableRulesReductionPairsProof (EQUIVALENT)
395.09/246.32	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.
395.09/246.32	
395.09/246.32	No dependency pairs are removed.
395.09/246.32	
395.09/246.32	The following rules are removed from R:
395.09/246.32	
395.09/246.32	   down.1(g.1(g.1(fresh_constant.))) -> g_flat.0(down.1(g.1(fresh_constant.)))
395.09/246.32	Used ordering: POLO with Polynomial interpretation [POLO]:
395.09/246.32	
395.09/246.32	   POL(TOP.0(x_1)) = x_1
395.09/246.32	   POL(a.) = 0
395.09/246.32	   POL(b.) = 0
395.09/246.32	   POL(down.0(x_1)) = 1 + x_1
395.09/246.32	   POL(f.0(x_1)) = x_1
395.09/246.32	   POL(f.1(x_1)) = x_1
395.09/246.32	   POL(f_flat.0(x_1)) = x_1
395.09/246.32	   POL(fresh_constant.) = 0
395.09/246.32	   POL(g.0(x_1)) = x_1
395.09/246.32	   POL(g.1(x_1)) = x_1
395.09/246.32	   POL(g_flat.0(x_1)) = x_1
395.09/246.32	   POL(up.0(x_1)) = 1 + x_1
395.09/246.32	   POL(up.1(x_1)) = x_1
395.09/246.32	
395.09/246.32	
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(177)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.32	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.32	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.32	   g_flat.0(up.1(x_1)) -> up.1(g.1(x_1))
395.09/246.32	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.32	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.32	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.32	   down.0(a.) -> up.0(f.0(a.))
395.09/246.32	   down.0(a.) -> up.0(g.0(a.))
395.09/246.32	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.32	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.32	   down.0(g.0(f.1(g.1(g.1(y28))))) -> g_flat.0(down.0(f.1(g.1(g.1(y28)))))
395.09/246.32	   down.0(g.0(f.1(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.1(g.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.32	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.32	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.32	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.32	   down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.32	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(178) QDPOrderProof (EQUIVALENT)
395.09/246.32	We use the reduction pair processor [LPAR04,JAR06].
395.09/246.32	
395.09/246.32	
395.09/246.32	The following pairs can be oriented strictly and are deleted.
395.09/246.32	
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.32	The remaining pairs can at least be oriented weakly.
395.09/246.32	Used ordering:  Polynomial interpretation [POLO]:
395.09/246.32	
395.09/246.32	   POL(TOP.0(x_1)) = x_1
395.09/246.32	   POL(a.) = 0
395.09/246.32	   POL(b.) = 0
395.09/246.32	   POL(down.0(x_1)) = x_1
395.09/246.32	   POL(f.0(x_1)) = 1
395.09/246.32	   POL(f.1(x_1)) = 0
395.09/246.32	   POL(f_flat.0(x_1)) = 1
395.09/246.32	   POL(fresh_constant.) = 0
395.09/246.32	   POL(g.0(x_1)) = x_1
395.09/246.32	   POL(g.1(x_1)) = 0
395.09/246.32	   POL(g_flat.0(x_1)) = x_1
395.09/246.32	   POL(up.0(x_1)) = 1
395.09/246.32	   POL(up.1(x_1)) = 1 + x_1
395.09/246.32	
395.09/246.32	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.32	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.32	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.32	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.32	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.32	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.32	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.32	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.32	   down.0(g.0(f.1(g.1(g.1(y28))))) -> g_flat.0(down.0(f.1(g.1(g.1(y28)))))
395.09/246.32	   down.0(g.0(f.1(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.1(g.1(fresh_constant.))))
395.09/246.32	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.32	   g_flat.0(up.1(x_1)) -> up.1(g.1(x_1))
395.09/246.32	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.32	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.32	
395.09/246.32	
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(179)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.32	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.32	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.32	   g_flat.0(up.1(x_1)) -> up.1(g.1(x_1))
395.09/246.32	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.32	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.32	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.32	   down.0(a.) -> up.0(f.0(a.))
395.09/246.32	   down.0(a.) -> up.0(g.0(a.))
395.09/246.32	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.32	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.32	   down.0(g.0(f.1(g.1(g.1(y28))))) -> g_flat.0(down.0(f.1(g.1(g.1(y28)))))
395.09/246.32	   down.0(g.0(f.1(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.1(g.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.32	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.32	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.32	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.32	   down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.32	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(180) QDPOrderProof (EQUIVALENT)
395.09/246.32	We use the reduction pair processor [LPAR04,JAR06].
395.09/246.32	
395.09/246.32	
395.09/246.32	The following pairs can be oriented strictly and are deleted.
395.09/246.32	
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.1(x0)))))
395.09/246.32	The remaining pairs can at least be oriented weakly.
395.09/246.32	Used ordering:  Polynomial interpretation [POLO]:
395.09/246.32	
395.09/246.32	   POL(TOP.0(x_1)) = x_1
395.09/246.32	   POL(a.) = 0
395.09/246.32	   POL(b.) = 0
395.09/246.32	   POL(down.0(x_1)) = 0
395.09/246.32	   POL(f.0(x_1)) = x_1
395.09/246.32	   POL(f.1(x_1)) = 1
395.09/246.32	   POL(f_flat.0(x_1)) = x_1
395.09/246.32	   POL(fresh_constant.) = 0
395.09/246.32	   POL(g.0(x_1)) = x_1
395.09/246.32	   POL(g.1(x_1)) = 0
395.09/246.32	   POL(g_flat.0(x_1)) = x_1
395.09/246.32	   POL(up.0(x_1)) = x_1
395.09/246.32	   POL(up.1(x_1)) = 1 + x_1
395.09/246.32	
395.09/246.32	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.32	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.32	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.32	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.32	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.32	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.32	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.32	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.32	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.32	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.32	   down.0(g.0(f.1(g.1(g.1(y28))))) -> g_flat.0(down.0(f.1(g.1(g.1(y28)))))
395.09/246.32	   down.0(g.0(f.1(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.1(g.1(fresh_constant.))))
395.09/246.32	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.32	   g_flat.0(up.1(x_1)) -> up.1(g.1(x_1))
395.09/246.32	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.32	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.32	   down.0(a.) -> up.0(f.0(a.))
395.09/246.32	   down.0(a.) -> up.0(g.0(a.))
395.09/246.32	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.32	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.32	
395.09/246.32	
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(181)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.32	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.32	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.32	   g_flat.0(up.1(x_1)) -> up.1(g.1(x_1))
395.09/246.32	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.32	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.32	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.32	   down.0(a.) -> up.0(f.0(a.))
395.09/246.32	   down.0(a.) -> up.0(g.0(a.))
395.09/246.32	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.32	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.32	   down.0(g.0(f.1(g.1(g.1(y28))))) -> g_flat.0(down.0(f.1(g.1(g.1(y28)))))
395.09/246.32	   down.0(g.0(f.1(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.1(g.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.1(y24))))) -> g_flat.0(down.0(f.0(f.0(f.1(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.32	   down.0(g.0(f.0(f.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(f.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.32	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.32	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.32	   down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.32	   down.0(f.0(g.0(f.1(y12)))) -> f_flat.0(down.0(g.0(f.1(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(182) PisEmptyProof (SOUND)
395.09/246.32	The TRS P is empty. Hence, there is no (P,Q,R) chain.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(183)
395.09/246.32	TRUE
395.09/246.32	
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(184)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.32	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.32	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.32	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.32	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.32	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.32	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.32	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.32	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.32	   down(f(g(g(f(x))))) -> up(b)
395.09/246.32	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.32	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.32	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.32	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.32	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.32	   down(g(g(g(x)))) -> up(b)
395.09/246.32	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.32	   down(g(a)) -> g_flat(down(a))
395.09/246.32	   down(a) -> up(f(a))
395.09/246.32	   down(a) -> up(g(a))
395.09/246.32	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.32	   down(f(f(f(x)))) -> up(b)
395.09/246.32	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.32	   down(f(a)) -> f_flat(down(a))
395.09/246.32	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.32	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.32	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.32	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.32	   down(g(f(f(g(x))))) -> up(b)
395.09/246.32	   down(g(f(g(f(x))))) -> up(b)
395.09/246.32	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.32	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.32	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.32	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.32	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.32	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.32	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.32	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.32	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.32	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.32	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(185) SplitQDPProof (EQUIVALENT)
395.09/246.32	We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem
395.09/246.32	
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(186)
395.09/246.32	Complex Obligation (AND)
395.09/246.32	
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(187)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.32	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.32	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.32	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.32	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.32	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.32	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.32	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.32	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.32	   down(f(g(g(f(x))))) -> up(b)
395.09/246.32	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.32	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.32	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.32	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.32	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.32	   down(g(g(g(x)))) -> up(b)
395.09/246.32	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.32	   down(g(a)) -> g_flat(down(a))
395.09/246.32	   down(a) -> up(f(a))
395.09/246.32	   down(a) -> up(g(a))
395.09/246.32	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.32	   down(f(f(f(x)))) -> up(b)
395.09/246.32	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.32	   down(f(a)) -> f_flat(down(a))
395.09/246.32	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
395.09/246.32	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.32	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.32	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.32	   down(g(f(f(g(x))))) -> up(b)
395.09/246.32	   down(g(f(g(f(x))))) -> up(b)
395.09/246.32	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.32	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.32	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.32	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.32	   down(g(f(f(fresh_constant)))) -> g_flat(down(f(f(fresh_constant))))
395.09/246.32	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.32	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.32	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.32	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
395.09/246.32	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.32	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(188) SemLabProof (SOUND)
395.09/246.32	We found the following model for the rules of the TRSs R and P.
395.09/246.32	Interpretation over the domain with elements from 0 to 1.
395.09/246.32	a: 0
395.09/246.32	b: 0
395.09/246.32	down: 0
395.09/246.32	f: x<sub>0</sub>
395.09/246.32	fresh_constant: 1
395.09/246.32	up: 0
395.09/246.32	f_flat: 0
395.09/246.32	TOP: 0
395.09/246.32	g_flat: 0
395.09/246.32	g: 0
395.09/246.32	By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(189)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.1(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.1(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.1(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.1(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.1(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.1(f.1(f.1(f.1(x0)))))) -> TOP.0(g_flat.0(down.1(f.1(f.1(f.1(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.1(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.32	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.1(f.1(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.32	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.32	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
395.09/246.32	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.32	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.32	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
395.09/246.32	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(g.0(g.1(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.32	   down.0(a.) -> up.0(f.0(a.))
395.09/246.32	   down.0(a.) -> up.0(g.0(a.))
395.09/246.32	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.32	   down.1(f.1(f.1(f.1(x)))) -> up.0(b.)
395.09/246.32	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.32	   down.0(g.1(f.1(fresh_constant.))) -> g_flat.0(down.1(f.1(fresh_constant.)))
395.09/246.32	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(f.0(g.1(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.1(f.1(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.32	   down.0(g.1(f.1(f.1(f.1(y24))))) -> g_flat.0(down.1(f.1(f.1(f.1(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.32	   down.0(g.1(f.1(f.1(fresh_constant.)))) -> g_flat.0(down.1(f.1(f.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.32	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.32	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.32	   down.1(f.1(f.1(fresh_constant.))) -> f_flat.0(down.1(f.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.32	   down.0(f.0(g.1(f.1(y12)))) -> f_flat.0(down.0(g.1(f.1(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(190) MRRProof (EQUIVALENT)
395.09/246.32	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.
395.09/246.32	
395.09/246.32	
395.09/246.32	Strictly oriented rules of the TRS R:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(g.1(f.1(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(g.0(g.1(x)))) -> up.0(b.)
395.09/246.32	   down.1(f.1(f.1(f.1(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(f.0(g.1(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.1(f.1(x))))) -> up.0(b.)
395.09/246.32	
395.09/246.32	Used ordering: Polynomial interpretation [POLO]:
395.09/246.32	
395.09/246.32	   POL(TOP.0(x_1)) = x_1
395.09/246.32	   POL(a.) = 0
395.09/246.32	   POL(b.) = 0
395.09/246.32	   POL(down.0(x_1)) = x_1
395.09/246.32	   POL(down.1(x_1)) = 1 + x_1
395.09/246.32	   POL(f.0(x_1)) = x_1
395.09/246.32	   POL(f.1(x_1)) = x_1
395.09/246.32	   POL(f_flat.0(x_1)) = x_1
395.09/246.32	   POL(fresh_constant.) = 0
395.09/246.32	   POL(g.0(x_1)) = x_1
395.09/246.32	   POL(g.1(x_1)) = 1 + x_1
395.09/246.32	   POL(g_flat.0(x_1)) = x_1
395.09/246.32	   POL(up.0(x_1)) = x_1
395.09/246.32	   POL(up.1(x_1)) = 1 + x_1
395.09/246.32	
395.09/246.32	
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(191)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.1(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.1(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.1(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.1(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(g.1(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.1(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.1(f.1(f.1(f.1(x0)))))) -> TOP.0(g_flat.0(down.1(f.1(f.1(f.1(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(g.1(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.1(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.32	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.32	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.32	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
395.09/246.32	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.32	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.32	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
395.09/246.32	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.32	   down.0(a.) -> up.0(f.0(a.))
395.09/246.32	   down.0(a.) -> up.0(g.0(a.))
395.09/246.32	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.32	   down.0(g.1(f.1(fresh_constant.))) -> g_flat.0(down.1(f.1(fresh_constant.)))
395.09/246.32	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.32	   down.0(g.1(f.1(f.1(f.1(y24))))) -> g_flat.0(down.1(f.1(f.1(f.1(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.32	   down.0(g.1(f.1(f.1(fresh_constant.)))) -> g_flat.0(down.1(f.1(f.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.32	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.32	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.32	   down.1(f.1(f.1(fresh_constant.))) -> f_flat.0(down.1(f.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.32	   down.0(f.0(g.1(f.1(y12)))) -> f_flat.0(down.0(g.1(f.1(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(192) DependencyGraphProof (EQUIVALENT)
395.09/246.32	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(193)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.1(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.1(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.1(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.1(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.32	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.32	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.32	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
395.09/246.32	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.32	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.32	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
395.09/246.32	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.32	   down.0(a.) -> up.0(f.0(a.))
395.09/246.32	   down.0(a.) -> up.0(g.0(a.))
395.09/246.32	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.32	   down.0(g.1(f.1(fresh_constant.))) -> g_flat.0(down.1(f.1(fresh_constant.)))
395.09/246.32	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.32	   down.0(g.1(f.1(f.1(f.1(y24))))) -> g_flat.0(down.1(f.1(f.1(f.1(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.32	   down.0(g.1(f.1(f.1(fresh_constant.)))) -> g_flat.0(down.1(f.1(f.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.32	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.32	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.32	   down.1(f.1(f.1(fresh_constant.))) -> f_flat.0(down.1(f.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.32	   down.0(f.0(g.1(f.1(y12)))) -> f_flat.0(down.0(g.1(f.1(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(194) MRRProof (EQUIVALENT)
395.09/246.32	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.
395.09/246.32	
395.09/246.32	
395.09/246.32	Strictly oriented rules of the TRS R:
395.09/246.32	
395.09/246.32	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
395.09/246.32	
395.09/246.32	Used ordering: Polynomial interpretation [POLO]:
395.09/246.32	
395.09/246.32	   POL(TOP.0(x_1)) = x_1
395.09/246.32	   POL(a.) = 0
395.09/246.32	   POL(b.) = 0
395.09/246.32	   POL(down.0(x_1)) = x_1
395.09/246.32	   POL(down.1(x_1)) = x_1
395.09/246.32	   POL(f.0(x_1)) = x_1
395.09/246.32	   POL(f.1(x_1)) = x_1
395.09/246.32	   POL(f_flat.0(x_1)) = x_1
395.09/246.32	   POL(fresh_constant.) = 0
395.09/246.32	   POL(g.0(x_1)) = x_1
395.09/246.32	   POL(g.1(x_1)) = x_1
395.09/246.32	   POL(g_flat.0(x_1)) = x_1
395.09/246.32	   POL(up.0(x_1)) = x_1
395.09/246.32	   POL(up.1(x_1)) = 1 + x_1
395.09/246.32	
395.09/246.32	
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(195)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.1(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.1(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.1(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.1(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.32	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.32	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.32	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.32	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.32	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
395.09/246.32	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.32	   down.0(a.) -> up.0(f.0(a.))
395.09/246.32	   down.0(a.) -> up.0(g.0(a.))
395.09/246.32	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.32	   down.0(g.1(f.1(fresh_constant.))) -> g_flat.0(down.1(f.1(fresh_constant.)))
395.09/246.32	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.32	   down.0(g.1(f.1(f.1(f.1(y24))))) -> g_flat.0(down.1(f.1(f.1(f.1(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.32	   down.0(g.1(f.1(f.1(fresh_constant.)))) -> g_flat.0(down.1(f.1(f.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.32	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.32	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.32	   down.1(f.1(f.1(fresh_constant.))) -> f_flat.0(down.1(f.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.32	   down.0(f.0(g.1(f.1(y12)))) -> f_flat.0(down.0(g.1(f.1(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(196) MRRProof (EQUIVALENT)
395.09/246.32	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.
395.09/246.32	
395.09/246.32	
395.09/246.32	Strictly oriented rules of the TRS R:
395.09/246.32	
395.09/246.32	   down.0(g.1(f.1(fresh_constant.))) -> g_flat.0(down.1(f.1(fresh_constant.)))
395.09/246.32	   down.0(g.1(f.1(f.1(f.1(y24))))) -> g_flat.0(down.1(f.1(f.1(f.1(y24)))))
395.09/246.32	   down.0(g.1(f.1(f.1(fresh_constant.)))) -> g_flat.0(down.1(f.1(f.1(fresh_constant.))))
395.09/246.32	
395.09/246.32	Used ordering: Polynomial interpretation [POLO]:
395.09/246.32	
395.09/246.32	   POL(TOP.0(x_1)) = x_1
395.09/246.32	   POL(a.) = 0
395.09/246.32	   POL(b.) = 0
395.09/246.32	   POL(down.0(x_1)) = x_1
395.09/246.32	   POL(down.1(x_1)) = x_1
395.09/246.32	   POL(f.0(x_1)) = x_1
395.09/246.32	   POL(f.1(x_1)) = x_1
395.09/246.32	   POL(f_flat.0(x_1)) = x_1
395.09/246.32	   POL(fresh_constant.) = 0
395.09/246.32	   POL(g.0(x_1)) = x_1
395.09/246.32	   POL(g.1(x_1)) = 1 + x_1
395.09/246.32	   POL(g_flat.0(x_1)) = x_1
395.09/246.32	   POL(up.0(x_1)) = x_1
395.09/246.32	   POL(up.1(x_1)) = x_1
395.09/246.32	
395.09/246.32	
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(197)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.1(f.1(x0))))) -> TOP.0(f_flat.0(down.0(g.1(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.1(f.1(x0))))) -> TOP.0(g_flat.0(down.0(g.1(f.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.32	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.32	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.32	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.32	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.32	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
395.09/246.32	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.32	   down.0(a.) -> up.0(f.0(a.))
395.09/246.32	   down.0(a.) -> up.0(g.0(a.))
395.09/246.32	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.32	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.32	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.32	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.32	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.32	   down.1(f.1(f.1(fresh_constant.))) -> f_flat.0(down.1(f.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.32	   down.0(f.0(g.1(f.1(y12)))) -> f_flat.0(down.0(g.1(f.1(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(198) DependencyGraphProof (EQUIVALENT)
395.09/246.32	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(199)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.32	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.32	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.32	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.32	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.32	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
395.09/246.32	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.32	   down.0(a.) -> up.0(f.0(a.))
395.09/246.32	   down.0(a.) -> up.0(g.0(a.))
395.09/246.32	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.32	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.32	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.32	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.32	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.32	   down.1(f.1(f.1(fresh_constant.))) -> f_flat.0(down.1(f.1(fresh_constant.)))
395.09/246.32	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.32	   down.0(f.0(g.1(f.1(y12)))) -> f_flat.0(down.0(g.1(f.1(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(200) UsableRulesReductionPairsProof (EQUIVALENT)
395.09/246.32	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.
395.09/246.32	
395.09/246.32	No dependency pairs are removed.
395.09/246.32	
395.09/246.32	The following rules are removed from R:
395.09/246.32	
395.09/246.32	   down.1(f.1(f.1(fresh_constant.))) -> f_flat.0(down.1(f.1(fresh_constant.)))
395.09/246.32	Used ordering: POLO with Polynomial interpretation [POLO]:
395.09/246.32	
395.09/246.32	   POL(TOP.0(x_1)) = x_1
395.09/246.32	   POL(a.) = 0
395.09/246.32	   POL(b.) = 0
395.09/246.32	   POL(down.0(x_1)) = 1 + x_1
395.09/246.32	   POL(f.0(x_1)) = x_1
395.09/246.32	   POL(f.1(x_1)) = x_1
395.09/246.32	   POL(f_flat.0(x_1)) = x_1
395.09/246.32	   POL(fresh_constant.) = 0
395.09/246.32	   POL(g.0(x_1)) = x_1
395.09/246.32	   POL(g.1(x_1)) = x_1
395.09/246.32	   POL(g_flat.0(x_1)) = x_1
395.09/246.32	   POL(up.0(x_1)) = 1 + x_1
395.09/246.32	   POL(up.1(x_1)) = x_1
395.09/246.32	
395.09/246.32	
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(201)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.32	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.32	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.32	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.32	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.32	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
395.09/246.32	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.32	   down.0(a.) -> up.0(f.0(a.))
395.09/246.32	   down.0(a.) -> up.0(g.0(a.))
395.09/246.32	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.32	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.32	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.32	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.32	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.32	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(f.0(g.1(f.1(y12)))) -> f_flat.0(down.0(g.1(f.1(y12))))
395.09/246.32	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(202) QDPOrderProof (EQUIVALENT)
395.09/246.32	We use the reduction pair processor [LPAR04,JAR06].
395.09/246.32	
395.09/246.32	
395.09/246.32	The following pairs can be oriented strictly and are deleted.
395.09/246.32	
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.1(x0)))))
395.09/246.32	The remaining pairs can at least be oriented weakly.
395.09/246.32	Used ordering:  Polynomial interpretation [POLO]:
395.09/246.32	
395.09/246.32	   POL(TOP.0(x_1)) = x_1
395.09/246.32	   POL(a.) = 0
395.09/246.32	   POL(b.) = 0
395.09/246.32	   POL(down.0(x_1)) = x_1
395.09/246.32	   POL(f.0(x_1)) = x_1
395.09/246.32	   POL(f.1(x_1)) = x_1
395.09/246.32	   POL(f_flat.0(x_1)) = x_1
395.09/246.32	   POL(fresh_constant.) = 0
395.09/246.32	   POL(g.0(x_1)) = 1
395.09/246.32	   POL(g.1(x_1)) = 0
395.09/246.32	   POL(g_flat.0(x_1)) = 1
395.09/246.32	   POL(up.0(x_1)) = 1
395.09/246.32	   POL(up.1(x_1)) = 1 + x_1
395.09/246.32	
395.09/246.32	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.32	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.32	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.32	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.32	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.32	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
395.09/246.32	   down.0(f.0(g.1(f.1(y12)))) -> f_flat.0(down.0(g.1(f.1(y12))))
395.09/246.32	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.32	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.32	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.32	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.32	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.32	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.32	
395.09/246.32	
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(203)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.32	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.32	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.32	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.32	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.32	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
395.09/246.32	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.32	   down.0(a.) -> up.0(f.0(a.))
395.09/246.32	   down.0(a.) -> up.0(g.0(a.))
395.09/246.32	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.32	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.32	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.32	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.32	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.32	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(f.0(g.1(f.1(y12)))) -> f_flat.0(down.0(g.1(f.1(y12))))
395.09/246.32	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(204) UsableRulesReductionPairsProof (EQUIVALENT)
395.09/246.32	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.
395.09/246.32	
395.09/246.32	No dependency pairs are removed.
395.09/246.32	
395.09/246.32	The following rules are removed from R:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.1(f.1(y12)))) -> f_flat.0(down.0(g.1(f.1(y12))))
395.09/246.32	Used ordering: POLO with Polynomial interpretation [POLO]:
395.09/246.32	
395.09/246.32	   POL(TOP.0(x_1)) = x_1
395.09/246.32	   POL(a.) = 0
395.09/246.32	   POL(b.) = 0
395.09/246.32	   POL(down.0(x_1)) = 1 + x_1
395.09/246.32	   POL(f.0(x_1)) = x_1
395.09/246.32	   POL(f.1(x_1)) = x_1
395.09/246.32	   POL(f_flat.0(x_1)) = x_1
395.09/246.32	   POL(fresh_constant.) = 0
395.09/246.32	   POL(g.0(x_1)) = x_1
395.09/246.32	   POL(g.1(x_1)) = x_1
395.09/246.32	   POL(g_flat.0(x_1)) = x_1
395.09/246.32	   POL(up.0(x_1)) = 1 + x_1
395.09/246.32	   POL(up.1(x_1)) = x_1
395.09/246.32	
395.09/246.32	
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(205)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.32	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.32	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(g.1(y22))))) -> f_flat.0(down.0(g.0(g.0(g.1(y22)))))
395.09/246.32	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.32	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.32	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.32	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.32	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
395.09/246.32	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.32	   down.0(a.) -> up.0(f.0(a.))
395.09/246.32	   down.0(a.) -> up.0(g.0(a.))
395.09/246.32	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.32	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.32	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.32	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.0(g.1(y28))))) -> g_flat.0(down.0(f.0(g.0(g.1(y28)))))
395.09/246.32	   down.0(g.0(f.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(f.0(g.1(fresh_constant.))))
395.09/246.32	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.32	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.32	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.32	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.32	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.32	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.32	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.32	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.32	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.32	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(206) PisEmptyProof (SOUND)
395.09/246.32	The TRS P is empty. Hence, there is no (P,Q,R) chain.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(207)
395.09/246.32	TRUE
395.09/246.32	
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(208)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.32	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.32	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.32	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.32	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.32	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.32	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.32	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.32	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.32	   down(f(g(g(f(x))))) -> up(b)
395.09/246.32	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.32	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.32	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.32	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.32	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.32	   down(g(g(g(x)))) -> up(b)
395.09/246.32	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.32	   down(g(a)) -> g_flat(down(a))
395.09/246.32	   down(a) -> up(f(a))
395.09/246.32	   down(a) -> up(g(a))
395.09/246.32	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.32	   down(f(f(f(x)))) -> up(b)
395.09/246.32	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.32	   down(f(a)) -> f_flat(down(a))
395.09/246.32	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.32	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.32	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.32	   down(g(f(f(g(x))))) -> up(b)
395.09/246.32	   down(g(f(g(f(x))))) -> up(b)
395.09/246.32	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.32	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.32	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.32	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.32	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.32	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.32	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.32	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.32	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.32	
395.09/246.32	Q is empty.
395.09/246.32	We have to consider all minimal (P,Q,R)-chains.
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(209) SplitQDPProof (EQUIVALENT)
395.09/246.32	We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem
395.09/246.32	
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(210)
395.09/246.32	Complex Obligation (AND)
395.09/246.32	
395.09/246.32	----------------------------------------
395.09/246.32	
395.09/246.32	(211)
395.09/246.32	Obligation:
395.09/246.32	Q DP problem:
395.09/246.32	The TRS P consists of the following rules:
395.09/246.32	
395.09/246.32	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.32	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.32	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.32	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.32	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.32	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.32	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.32	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.32	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.32	
395.09/246.32	The TRS R consists of the following rules:
395.09/246.32	
395.09/246.32	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.32	   down(f(g(g(f(x))))) -> up(b)
395.09/246.32	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.32	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.32	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.32	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.32	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.32	   down(g(g(g(x)))) -> up(b)
395.09/246.32	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.33	   down(g(a)) -> g_flat(down(a))
395.09/246.33	   down(a) -> up(f(a))
395.09/246.33	   down(a) -> up(g(a))
395.09/246.33	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.33	   down(f(f(f(x)))) -> up(b)
395.09/246.33	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.33	   down(f(a)) -> f_flat(down(a))
395.09/246.33	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.33	   down(g(f(g(fresh_constant)))) -> g_flat(down(f(g(fresh_constant))))
395.09/246.33	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.33	   down(g(f(f(g(x))))) -> up(b)
395.09/246.33	   down(g(f(g(f(x))))) -> up(b)
395.09/246.33	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.33	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.33	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.33	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.33	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.33	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.33	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.33	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.33	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.33	
395.09/246.33	Q is empty.
395.09/246.33	We have to consider all minimal (P,Q,R)-chains.
395.09/246.33	----------------------------------------
395.09/246.33	
395.09/246.33	(212) SemLabProof (SOUND)
395.09/246.33	We found the following model for the rules of the TRSs R and P.
395.09/246.33	Interpretation over the domain with elements from 0 to 1.
395.09/246.33	a: 0
395.09/246.33	b: 0
395.09/246.33	down: 0
395.09/246.33	f: x<sub>0</sub>
395.09/246.33	fresh_constant: 1
395.09/246.33	up: 0
395.09/246.33	f_flat: 0
395.09/246.33	TOP: 0
395.09/246.33	g_flat: 0
395.09/246.33	g: x<sub>0</sub>
395.09/246.33	By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
395.09/246.33	----------------------------------------
395.09/246.33	
395.09/246.33	(213)
395.09/246.33	Obligation:
395.09/246.33	Q DP problem:
395.09/246.33	The TRS P consists of the following rules:
395.09/246.33	
395.09/246.33	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.33	   TOP.0(up.1(f.1(f.1(g.1(x0))))) -> TOP.0(f_flat.0(down.1(f.1(g.1(x0)))))
395.09/246.33	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.33	   TOP.0(up.1(f.1(g.1(f.1(x0))))) -> TOP.0(f_flat.0(down.1(g.1(f.1(x0)))))
395.09/246.33	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.33	   TOP.0(up.1(g.1(g.1(f.1(x0))))) -> TOP.0(g_flat.0(down.1(g.1(f.1(x0)))))
395.09/246.33	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.33	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.33	   TOP.0(up.1(f.1(g.1(g.1(g.1(x0)))))) -> TOP.0(f_flat.0(down.1(g.1(g.1(g.1(x0))))))
395.09/246.33	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.33	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.33	   TOP.0(up.1(g.1(f.1(f.1(f.1(x0)))))) -> TOP.0(g_flat.0(down.1(f.1(f.1(f.1(x0))))))
395.09/246.33	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.33	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.33	   TOP.0(up.1(g.1(f.1(g.1(g.1(x0)))))) -> TOP.0(g_flat.0(down.1(f.1(g.1(g.1(x0))))))
395.09/246.33	
395.09/246.33	The TRS R consists of the following rules:
395.09/246.33	
395.09/246.33	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.33	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.33	   down.1(f.1(g.1(g.1(f.1(x))))) -> up.0(b.)
395.09/246.33	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.33	   down.1(f.1(g.1(g.1(g.1(y22))))) -> f_flat.0(down.1(g.1(g.1(g.1(y22)))))
395.09/246.33	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.33	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.33	   g_flat.0(up.1(x_1)) -> up.1(g.1(x_1))
395.09/246.33	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.33	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.33	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
395.09/246.33	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.33	   down.1(g.1(g.1(g.1(x)))) -> up.0(b.)
395.09/246.33	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.33	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.33	   down.0(a.) -> up.0(f.0(a.))
395.09/246.33	   down.0(a.) -> up.0(g.0(a.))
395.09/246.33	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.33	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.33	   down.1(f.1(f.1(f.1(x)))) -> up.0(b.)
395.09/246.33	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.33	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.33	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.33	   down.1(g.1(f.1(g.1(g.1(y28))))) -> g_flat.0(down.1(f.1(g.1(g.1(y28)))))
395.09/246.33	   down.1(g.1(f.1(g.1(fresh_constant.)))) -> g_flat.0(down.1(f.1(g.1(fresh_constant.))))
395.09/246.33	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.33	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.33	   down.1(g.1(f.1(f.1(g.1(x))))) -> up.0(b.)
395.09/246.33	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.33	   down.1(g.1(f.1(g.1(f.1(x))))) -> up.0(b.)
395.09/246.33	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.33	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.33	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.33	   down.1(g.1(f.1(f.1(f.1(y24))))) -> g_flat.0(down.1(f.1(f.1(f.1(y24)))))
395.09/246.33	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.33	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.33	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.33	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.33	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.33	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.33	   down.1(f.1(g.1(f.1(y12)))) -> f_flat.0(down.1(g.1(f.1(y12))))
395.09/246.33	
395.09/246.33	Q is empty.
395.09/246.33	We have to consider all minimal (P,Q,R)-chains.
395.09/246.33	----------------------------------------
395.09/246.33	
395.09/246.33	(214) MRRProof (EQUIVALENT)
395.09/246.33	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.
395.09/246.33	
395.09/246.33	
395.09/246.33	Strictly oriented rules of the TRS R:
395.09/246.33	
395.09/246.33	   down.1(f.1(g.1(g.1(f.1(x))))) -> up.0(b.)
395.09/246.33	   down.1(g.1(g.1(g.1(x)))) -> up.0(b.)
395.09/246.33	   down.1(f.1(f.1(f.1(x)))) -> up.0(b.)
395.09/246.33	   down.1(g.1(f.1(f.1(g.1(x))))) -> up.0(b.)
395.09/246.33	   down.1(g.1(f.1(g.1(f.1(x))))) -> up.0(b.)
395.09/246.33	
395.09/246.33	Used ordering: Polynomial interpretation [POLO]:
395.09/246.33	
395.09/246.33	   POL(TOP.0(x_1)) = x_1
395.09/246.33	   POL(a.) = 0
395.09/246.33	   POL(b.) = 0
395.09/246.33	   POL(down.0(x_1)) = x_1
395.09/246.33	   POL(down.1(x_1)) = 1 + x_1
395.09/246.33	   POL(f.0(x_1)) = x_1
395.09/246.33	   POL(f.1(x_1)) = x_1
395.09/246.33	   POL(f_flat.0(x_1)) = x_1
395.09/246.33	   POL(fresh_constant.) = 0
395.09/246.33	   POL(g.0(x_1)) = x_1
395.09/246.33	   POL(g.1(x_1)) = x_1
395.09/246.33	   POL(g_flat.0(x_1)) = x_1
395.09/246.33	   POL(up.0(x_1)) = x_1
395.09/246.33	   POL(up.1(x_1)) = 1 + x_1
395.09/246.33	
395.09/246.33	
395.09/246.33	----------------------------------------
395.09/246.33	
395.09/246.33	(215)
395.09/246.33	Obligation:
395.09/246.33	Q DP problem:
395.09/246.33	The TRS P consists of the following rules:
395.09/246.33	
395.09/246.33	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.33	   TOP.0(up.1(f.1(f.1(g.1(x0))))) -> TOP.0(f_flat.0(down.1(f.1(g.1(x0)))))
395.09/246.33	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.33	   TOP.0(up.1(f.1(g.1(f.1(x0))))) -> TOP.0(f_flat.0(down.1(g.1(f.1(x0)))))
395.09/246.33	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.33	   TOP.0(up.1(g.1(g.1(f.1(x0))))) -> TOP.0(g_flat.0(down.1(g.1(f.1(x0)))))
395.09/246.33	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.33	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.33	   TOP.0(up.1(f.1(g.1(g.1(g.1(x0)))))) -> TOP.0(f_flat.0(down.1(g.1(g.1(g.1(x0))))))
395.09/246.33	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.33	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.33	   TOP.0(up.1(g.1(f.1(f.1(f.1(x0)))))) -> TOP.0(g_flat.0(down.1(f.1(f.1(f.1(x0))))))
395.09/246.33	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.33	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.33	   TOP.0(up.1(g.1(f.1(g.1(g.1(x0)))))) -> TOP.0(g_flat.0(down.1(f.1(g.1(g.1(x0))))))
395.09/246.33	
395.09/246.33	The TRS R consists of the following rules:
395.09/246.33	
395.09/246.33	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.33	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.33	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.33	   down.1(f.1(g.1(g.1(g.1(y22))))) -> f_flat.0(down.1(g.1(g.1(g.1(y22)))))
395.09/246.33	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.33	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.33	   g_flat.0(up.1(x_1)) -> up.1(g.1(x_1))
395.09/246.33	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.33	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.33	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
395.09/246.33	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.33	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.33	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.33	   down.0(a.) -> up.0(f.0(a.))
395.09/246.33	   down.0(a.) -> up.0(g.0(a.))
395.09/246.33	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.33	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.33	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.33	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.33	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.33	   down.1(g.1(f.1(g.1(g.1(y28))))) -> g_flat.0(down.1(f.1(g.1(g.1(y28)))))
395.09/246.33	   down.1(g.1(f.1(g.1(fresh_constant.)))) -> g_flat.0(down.1(f.1(g.1(fresh_constant.))))
395.09/246.33	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.33	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.33	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.33	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.33	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.33	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.33	   down.1(g.1(f.1(f.1(f.1(y24))))) -> g_flat.0(down.1(f.1(f.1(f.1(y24)))))
395.09/246.33	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.33	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.33	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.33	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.33	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.33	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.33	   down.1(f.1(g.1(f.1(y12)))) -> f_flat.0(down.1(g.1(f.1(y12))))
395.09/246.33	
395.09/246.33	Q is empty.
395.09/246.33	We have to consider all minimal (P,Q,R)-chains.
395.09/246.33	----------------------------------------
395.09/246.33	
395.09/246.33	(216) DependencyGraphProof (EQUIVALENT)
395.09/246.33	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
395.09/246.33	----------------------------------------
395.09/246.33	
395.09/246.33	(217)
395.09/246.33	Obligation:
395.09/246.33	Q DP problem:
395.09/246.33	The TRS P consists of the following rules:
395.09/246.33	
395.09/246.33	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.33	   TOP.0(up.1(f.1(f.1(g.1(x0))))) -> TOP.0(f_flat.0(down.1(f.1(g.1(x0)))))
395.09/246.33	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.33	   TOP.0(up.1(f.1(g.1(f.1(x0))))) -> TOP.0(f_flat.0(down.1(g.1(f.1(x0)))))
395.09/246.33	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.33	   TOP.0(up.1(g.1(g.1(f.1(x0))))) -> TOP.0(g_flat.0(down.1(g.1(f.1(x0)))))
395.09/246.33	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.33	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.33	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.33	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.33	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.33	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.33	   TOP.0(up.1(g.1(f.1(g.1(g.1(x0)))))) -> TOP.0(g_flat.0(down.1(f.1(g.1(g.1(x0))))))
395.09/246.33	
395.09/246.33	The TRS R consists of the following rules:
395.09/246.33	
395.09/246.33	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.33	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.33	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.33	   down.1(f.1(g.1(g.1(g.1(y22))))) -> f_flat.0(down.1(g.1(g.1(g.1(y22)))))
395.09/246.33	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.33	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.33	   g_flat.0(up.1(x_1)) -> up.1(g.1(x_1))
395.09/246.33	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.33	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.33	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
395.09/246.33	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.33	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.33	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.33	   down.0(a.) -> up.0(f.0(a.))
395.09/246.33	   down.0(a.) -> up.0(g.0(a.))
395.09/246.33	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.33	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.33	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.33	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.33	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.33	   down.1(g.1(f.1(g.1(g.1(y28))))) -> g_flat.0(down.1(f.1(g.1(g.1(y28)))))
395.09/246.33	   down.1(g.1(f.1(g.1(fresh_constant.)))) -> g_flat.0(down.1(f.1(g.1(fresh_constant.))))
395.09/246.33	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.33	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.33	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.33	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.33	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.33	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.33	   down.1(g.1(f.1(f.1(f.1(y24))))) -> g_flat.0(down.1(f.1(f.1(f.1(y24)))))
395.09/246.33	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.33	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.33	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.33	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.33	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.33	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.33	   down.1(f.1(g.1(f.1(y12)))) -> f_flat.0(down.1(g.1(f.1(y12))))
395.09/246.33	
395.09/246.33	Q is empty.
395.09/246.33	We have to consider all minimal (P,Q,R)-chains.
395.09/246.33	----------------------------------------
395.09/246.33	
395.09/246.33	(218) MRRProof (EQUIVALENT)
395.09/246.33	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.
395.09/246.33	
395.09/246.33	Strictly oriented dependency pairs:
395.09/246.33	
395.09/246.33	   TOP.0(up.1(f.1(f.1(g.1(x0))))) -> TOP.0(f_flat.0(down.1(f.1(g.1(x0)))))
395.09/246.33	   TOP.0(up.1(f.1(g.1(f.1(x0))))) -> TOP.0(f_flat.0(down.1(g.1(f.1(x0)))))
395.09/246.33	   TOP.0(up.1(g.1(g.1(f.1(x0))))) -> TOP.0(g_flat.0(down.1(g.1(f.1(x0)))))
395.09/246.33	   TOP.0(up.1(g.1(f.1(g.1(g.1(x0)))))) -> TOP.0(g_flat.0(down.1(f.1(g.1(g.1(x0))))))
395.09/246.33	
395.09/246.33	
395.09/246.33	Used ordering: Polynomial interpretation [POLO]:
395.09/246.33	
395.09/246.33	   POL(TOP.0(x_1)) = x_1
395.09/246.33	   POL(a.) = 0
395.09/246.33	   POL(b.) = 0
395.09/246.33	   POL(down.0(x_1)) = 1 + x_1
395.09/246.33	   POL(down.1(x_1)) = x_1
395.09/246.33	   POL(f.0(x_1)) = x_1
395.09/246.33	   POL(f.1(x_1)) = x_1
395.09/246.33	   POL(f_flat.0(x_1)) = x_1
395.09/246.33	   POL(fresh_constant.) = 0
395.09/246.33	   POL(g.0(x_1)) = x_1
395.09/246.33	   POL(g.1(x_1)) = x_1
395.09/246.33	   POL(g_flat.0(x_1)) = x_1
395.09/246.33	   POL(up.0(x_1)) = 1 + x_1
395.09/246.33	   POL(up.1(x_1)) = 1 + x_1
395.09/246.33	
395.09/246.33	
395.09/246.33	----------------------------------------
395.09/246.33	
395.09/246.33	(219)
395.09/246.33	Obligation:
395.09/246.33	Q DP problem:
395.09/246.33	The TRS P consists of the following rules:
395.09/246.33	
395.09/246.33	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.33	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.33	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.33	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.33	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.33	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.33	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.33	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.33	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.33	
395.09/246.33	The TRS R consists of the following rules:
395.09/246.33	
395.09/246.33	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.33	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.33	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.33	   down.1(f.1(g.1(g.1(g.1(y22))))) -> f_flat.0(down.1(g.1(g.1(g.1(y22)))))
395.09/246.33	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.33	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.33	   g_flat.0(up.1(x_1)) -> up.1(g.1(x_1))
395.09/246.33	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.33	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.33	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
395.09/246.33	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.33	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.33	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.33	   down.0(a.) -> up.0(f.0(a.))
395.09/246.33	   down.0(a.) -> up.0(g.0(a.))
395.09/246.33	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.33	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.33	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.33	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.33	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.33	   down.1(g.1(f.1(g.1(g.1(y28))))) -> g_flat.0(down.1(f.1(g.1(g.1(y28)))))
395.09/246.33	   down.1(g.1(f.1(g.1(fresh_constant.)))) -> g_flat.0(down.1(f.1(g.1(fresh_constant.))))
395.09/246.33	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.33	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.33	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.33	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.33	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.33	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.33	   down.1(g.1(f.1(f.1(f.1(y24))))) -> g_flat.0(down.1(f.1(f.1(f.1(y24)))))
395.09/246.33	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.33	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.33	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.33	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.33	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.33	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.33	   down.1(f.1(g.1(f.1(y12)))) -> f_flat.0(down.1(g.1(f.1(y12))))
395.09/246.33	
395.09/246.33	Q is empty.
395.09/246.33	We have to consider all minimal (P,Q,R)-chains.
395.09/246.33	----------------------------------------
395.09/246.33	
395.09/246.33	(220) UsableRulesReductionPairsProof (EQUIVALENT)
395.09/246.33	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.
395.09/246.33	
395.09/246.33	No dependency pairs are removed.
395.09/246.33	
395.09/246.33	The following rules are removed from R:
395.09/246.33	
395.09/246.33	   down.1(f.1(g.1(g.1(g.1(y22))))) -> f_flat.0(down.1(g.1(g.1(g.1(y22)))))
395.09/246.33	   down.1(g.1(f.1(g.1(g.1(y28))))) -> g_flat.0(down.1(f.1(g.1(g.1(y28)))))
395.09/246.33	   down.1(g.1(f.1(g.1(fresh_constant.)))) -> g_flat.0(down.1(f.1(g.1(fresh_constant.))))
395.09/246.33	   down.1(g.1(f.1(f.1(f.1(y24))))) -> g_flat.0(down.1(f.1(f.1(f.1(y24)))))
395.09/246.33	   down.1(f.1(g.1(f.1(y12)))) -> f_flat.0(down.1(g.1(f.1(y12))))
395.09/246.33	Used ordering: POLO with Polynomial interpretation [POLO]:
395.09/246.33	
395.09/246.33	   POL(TOP.0(x_1)) = x_1
395.09/246.33	   POL(a.) = 0
395.09/246.33	   POL(b.) = 0
395.09/246.33	   POL(down.0(x_1)) = 1 + x_1
395.09/246.33	   POL(f.0(x_1)) = x_1
395.09/246.33	   POL(f.1(x_1)) = x_1
395.09/246.33	   POL(f_flat.0(x_1)) = x_1
395.09/246.33	   POL(g.0(x_1)) = x_1
395.09/246.33	   POL(g.1(x_1)) = x_1
395.09/246.33	   POL(g_flat.0(x_1)) = x_1
395.09/246.33	   POL(up.0(x_1)) = 1 + x_1
395.09/246.33	   POL(up.1(x_1)) = x_1
395.09/246.33	
395.09/246.33	
395.09/246.33	----------------------------------------
395.09/246.33	
395.09/246.33	(221)
395.09/246.33	Obligation:
395.09/246.33	Q DP problem:
395.09/246.33	The TRS P consists of the following rules:
395.09/246.33	
395.09/246.33	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(down.0(f.0(g.0(x0)))))
395.09/246.33	   TOP.0(up.0(f.0(g.0(f.0(x0))))) -> TOP.0(f_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.33	   TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(down.0(g.0(f.0(x0)))))
395.09/246.33	   TOP.0(up.0(f.0(g.0(g.0(a.))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(a.)))))
395.09/246.33	   TOP.0(up.0(f.0(g.0(g.0(g.0(x0)))))) -> TOP.0(f_flat.0(down.0(g.0(g.0(g.0(x0))))))
395.09/246.33	   TOP.0(up.0(g.0(f.0(f.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(a.)))))
395.09/246.33	   TOP.0(up.0(g.0(f.0(f.0(f.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(f.0(f.0(x0))))))
395.09/246.33	   TOP.0(up.0(g.0(f.0(g.0(a.))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(a.)))))
395.09/246.33	   TOP.0(up.0(g.0(f.0(g.0(g.0(x0)))))) -> TOP.0(g_flat.0(down.0(f.0(g.0(g.0(x0))))))
395.09/246.33	
395.09/246.33	The TRS R consists of the following rules:
395.09/246.33	
395.09/246.33	   down.0(f.0(g.0(g.0(a.)))) -> f_flat.0(down.0(g.0(g.0(a.))))
395.09/246.33	   down.0(f.0(g.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.33	   down.0(f.0(g.0(g.0(g.0(y22))))) -> f_flat.0(down.0(g.0(g.0(g.0(y22)))))
395.09/246.33	   down.0(f.0(g.0(g.0(b.)))) -> f_flat.0(down.0(g.0(g.0(b.))))
395.09/246.33	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
395.09/246.33	   g_flat.0(up.1(x_1)) -> up.1(g.1(x_1))
395.09/246.33	   down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.)))
395.09/246.33	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
395.09/246.33	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
395.09/246.33	   down.0(g.0(g.0(g.0(x)))) -> up.0(b.)
395.09/246.33	   down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.)))
395.09/246.33	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
395.09/246.33	   down.0(a.) -> up.0(f.0(a.))
395.09/246.33	   down.0(a.) -> up.0(g.0(a.))
395.09/246.33	   down.0(f.0(g.0(a.))) -> f_flat.0(down.0(g.0(a.)))
395.09/246.33	   down.0(f.0(f.0(f.0(x)))) -> up.0(b.)
395.09/246.33	   down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.)))
395.09/246.33	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
395.09/246.33	   down.0(g.0(f.0(g.0(g.0(y28))))) -> g_flat.0(down.0(f.0(g.0(g.0(y28)))))
395.09/246.33	   down.0(g.0(f.0(a.))) -> g_flat.0(down.0(f.0(a.)))
395.09/246.33	   down.0(g.0(f.0(f.0(g.0(x))))) -> up.0(b.)
395.09/246.33	   down.0(g.0(f.0(g.0(f.0(x))))) -> up.0(b.)
395.09/246.33	   down.0(g.0(f.0(b.))) -> g_flat.0(down.0(f.0(b.)))
395.09/246.33	   down.0(g.0(f.0(f.0(a.)))) -> g_flat.0(down.0(f.0(f.0(a.))))
395.09/246.33	   down.0(g.0(f.0(f.0(f.0(y24))))) -> g_flat.0(down.0(f.0(f.0(f.0(y24)))))
395.09/246.33	   down.0(g.0(f.0(f.0(b.)))) -> g_flat.0(down.0(f.0(f.0(b.))))
395.09/246.33	   down.0(g.0(f.0(g.0(a.)))) -> g_flat.0(down.0(f.0(g.0(a.))))
395.09/246.33	   down.0(g.0(f.0(g.0(b.)))) -> g_flat.0(down.0(f.0(g.0(b.))))
395.09/246.33	   down.0(f.0(g.0(b.))) -> f_flat.0(down.0(g.0(b.)))
395.09/246.33	   down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.)))
395.09/246.33	   down.0(f.0(g.0(f.0(y12)))) -> f_flat.0(down.0(g.0(f.0(y12))))
395.09/246.33	
395.09/246.33	Q is empty.
395.09/246.33	We have to consider all minimal (P,Q,R)-chains.
395.09/246.33	----------------------------------------
395.09/246.33	
395.09/246.33	(222) PisEmptyProof (SOUND)
395.09/246.33	The TRS P is empty. Hence, there is no (P,Q,R) chain.
395.09/246.33	----------------------------------------
395.09/246.33	
395.09/246.33	(223)
395.09/246.33	TRUE
395.09/246.33	
395.09/246.33	----------------------------------------
395.09/246.33	
395.09/246.33	(224)
395.09/246.33	Obligation:
395.09/246.33	Q DP problem:
395.09/246.33	The TRS P consists of the following rules:
395.09/246.33	
395.09/246.33	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
395.09/246.33	   TOP(up(f(g(f(x0))))) -> TOP(f_flat(down(g(f(x0)))))
395.09/246.33	   TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))
395.09/246.33	   TOP(up(f(g(g(a))))) -> TOP(f_flat(down(g(g(a)))))
395.09/246.33	   TOP(up(f(g(g(g(x0)))))) -> TOP(f_flat(down(g(g(g(x0))))))
395.09/246.33	   TOP(up(g(f(f(a))))) -> TOP(g_flat(down(f(f(a)))))
395.09/246.33	   TOP(up(g(f(f(f(x0)))))) -> TOP(g_flat(down(f(f(f(x0))))))
395.09/246.33	   TOP(up(g(f(g(a))))) -> TOP(g_flat(down(f(g(a)))))
395.09/246.33	   TOP(up(g(f(g(g(x0)))))) -> TOP(g_flat(down(f(g(g(x0))))))
395.09/246.33	
395.09/246.33	The TRS R consists of the following rules:
395.09/246.33	
395.09/246.33	   down(f(g(g(a)))) -> f_flat(down(g(g(a))))
395.09/246.33	   down(f(g(g(f(x))))) -> up(b)
395.09/246.33	   down(f(g(g(g(y22))))) -> f_flat(down(g(g(g(y22)))))
395.09/246.33	   down(f(g(g(b)))) -> f_flat(down(g(g(b))))
395.09/246.33	   g_flat(up(x_1)) -> up(g(x_1))
395.09/246.33	   down(g(g(b))) -> g_flat(down(g(b)))
395.09/246.33	   f_flat(up(x_1)) -> up(f(x_1))
395.09/246.33	   down(g(g(g(x)))) -> up(b)
395.09/246.33	   down(g(g(a))) -> g_flat(down(g(a)))
395.09/246.33	   down(g(a)) -> g_flat(down(a))
395.09/246.33	   down(a) -> up(f(a))
395.09/246.33	   down(a) -> up(g(a))
395.09/246.33	   down(f(g(a))) -> f_flat(down(g(a)))
395.09/246.33	   down(f(f(f(x)))) -> up(b)
395.09/246.33	   down(f(f(a))) -> f_flat(down(f(a)))
395.09/246.33	   down(f(a)) -> f_flat(down(a))
395.09/246.33	   down(g(f(g(g(y28))))) -> g_flat(down(f(g(g(y28)))))
395.09/246.33	   down(g(f(a))) -> g_flat(down(f(a)))
395.09/246.33	   down(g(f(f(g(x))))) -> up(b)
395.09/246.33	   down(g(f(g(f(x))))) -> up(b)
395.09/246.33	   down(g(f(b))) -> g_flat(down(f(b)))
395.09/246.33	   down(g(f(f(a)))) -> g_flat(down(f(f(a))))
395.09/246.33	   down(g(f(f(f(y24))))) -> g_flat(down(f(f(f(y24)))))
395.09/246.33	   down(g(f(f(b)))) -> g_flat(down(f(f(b))))
395.09/246.33	   down(g(f(g(a)))) -> g_flat(down(f(g(a))))
395.09/246.33	   down(g(f(g(b)))) -> g_flat(down(f(g(b))))
395.09/246.33	   down(f(g(b))) -> f_flat(down(g(b)))
395.09/246.33	   down(f(f(b))) -> f_flat(down(f(b)))
395.09/246.33	   down(f(g(f(y12)))) -> f_flat(down(g(f(y12))))
395.09/246.33	
395.09/246.33	Q is empty.
395.09/246.33	We have to consider all minimal (P,Q,R)-chains.
395.26/246.39	EOF