168.85/87.93	MAYBE
168.85/87.95	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
168.85/87.95	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
168.85/87.95	
168.85/87.95	
168.85/87.95	Outermost Termination of the given OTRS could not be shown:
168.85/87.95	
168.85/87.95	(0) OTRS
168.85/87.95	(1) Thiemann-SpecialC-Transformation [EQUIVALENT, 0 ms]
168.85/87.95	(2) QTRS
168.85/87.95	    (3) DependencyPairsProof [EQUIVALENT, 6 ms]
168.85/87.95	    (4) QDP
168.85/87.95	    (5) DependencyGraphProof [EQUIVALENT, 0 ms]
168.85/87.95	    (6) AND
168.85/87.95	        (7) QDP
168.85/87.95	            (8) UsableRulesProof [EQUIVALENT, 0 ms]
168.85/87.95	            (9) QDP
168.85/87.95	            (10) QReductionProof [EQUIVALENT, 0 ms]
168.85/87.95	            (11) QDP
168.85/87.95	            (12) MRRProof [EQUIVALENT, 0 ms]
168.85/87.95	            (13) QDP
168.85/87.95	            (14) PisEmptyProof [EQUIVALENT, 0 ms]
168.85/87.95	            (15) YES
168.85/87.95	        (16) QDP
168.85/87.95	            (17) UsableRulesProof [EQUIVALENT, 0 ms]
168.85/87.95	            (18) QDP
168.85/87.95	            (19) QReductionProof [EQUIVALENT, 0 ms]
168.85/87.95	            (20) QDP
168.85/87.95	            (21) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
168.85/87.95	            (22) QDP
168.85/87.95	            (23) DependencyGraphProof [EQUIVALENT, 0 ms]
168.85/87.95	            (24) TRUE
168.85/87.95	        (25) QDP
168.85/87.95	            (26) UsableRulesProof [EQUIVALENT, 0 ms]
168.85/87.95	            (27) QDP
168.85/87.95	                (28) QReductionProof [EQUIVALENT, 0 ms]
168.85/87.95	                (29) QDP
168.85/87.95	                (30) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	                (31) QDP
168.85/87.95	                (32) QDPOrderProof [EQUIVALENT, 11 ms]
168.85/87.95	                (33) QDP
168.85/87.95	            (34) UsableRulesProof [EQUIVALENT, 0 ms]
168.85/87.95	            (35) QDP
168.85/87.95	                (36) QReductionProof [EQUIVALENT, 0 ms]
168.85/87.95	                (37) QDP
168.85/87.95	(38) Trivial-Transformation [SOUND, 0 ms]
168.85/87.95	(39) QTRS
168.85/87.95	    (40) DependencyPairsProof [EQUIVALENT, 1 ms]
168.85/87.95	    (41) QDP
168.85/87.95	    (42) DependencyGraphProof [EQUIVALENT, 0 ms]
168.85/87.95	    (43) AND
168.85/87.95	        (44) QDP
168.85/87.95	            (45) UsableRulesProof [EQUIVALENT, 0 ms]
168.85/87.95	            (46) QDP
168.85/87.95	            (47) QDPSizeChangeProof [EQUIVALENT, 0 ms]
168.85/87.95	            (48) YES
168.85/87.95	        (49) QDP
168.85/87.95	            (50) UsableRulesProof [EQUIVALENT, 0 ms]
168.85/87.95	            (51) QDP
168.85/87.95	            (52) NonTerminationLoopProof [COMPLETE, 1 ms]
168.85/87.95	            (53) NO
168.85/87.95	(54) Raffelsieper-Zantema-Transformation [SOUND, 0 ms]
168.85/87.95	(55) QTRS
168.85/87.95	    (56) AAECC Innermost [EQUIVALENT, 9 ms]
168.85/87.95	    (57) QTRS
168.85/87.95	    (58) DependencyPairsProof [EQUIVALENT, 28 ms]
168.85/87.95	    (59) QDP
168.85/87.95	    (60) DependencyGraphProof [EQUIVALENT, 0 ms]
168.85/87.95	    (61) AND
168.85/87.95	        (62) QDP
168.85/87.95	            (63) UsableRulesProof [EQUIVALENT, 0 ms]
168.85/87.95	            (64) QDP
168.85/87.95	            (65) QReductionProof [EQUIVALENT, 0 ms]
168.85/87.95	            (66) QDP
168.85/87.95	            (67) QDPSizeChangeProof [EQUIVALENT, 0 ms]
168.85/87.95	            (68) YES
168.85/87.95	        (69) QDP
168.85/87.95	            (70) UsableRulesProof [EQUIVALENT, 0 ms]
168.85/87.95	            (71) QDP
168.85/87.95	            (72) QReductionProof [EQUIVALENT, 0 ms]
168.85/87.95	            (73) QDP
168.85/87.95	            (74) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (75) QDP
168.85/87.95	            (76) DependencyGraphProof [EQUIVALENT, 0 ms]
168.85/87.95	            (77) QDP
168.85/87.95	            (78) UsableRulesProof [EQUIVALENT, 0 ms]
168.85/87.95	            (79) QDP
168.85/87.95	            (80) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (81) QDP
168.85/87.95	            (82) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (83) QDP
168.85/87.95	            (84) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (85) QDP
168.85/87.95	            (86) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (87) QDP
168.85/87.95	            (88) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (89) QDP
168.85/87.95	            (90) DependencyGraphProof [EQUIVALENT, 0 ms]
168.85/87.95	            (91) QDP
168.85/87.95	            (92) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (93) QDP
168.85/87.95	            (94) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (95) QDP
168.85/87.95	            (96) DependencyGraphProof [EQUIVALENT, 0 ms]
168.85/87.95	            (97) QDP
168.85/87.95	            (98) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (99) QDP
168.85/87.95	            (100) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (101) QDP
168.85/87.95	            (102) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (103) QDP
168.85/87.95	            (104) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (105) QDP
168.85/87.95	            (106) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (107) QDP
168.85/87.95	            (108) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (109) QDP
168.85/87.95	            (110) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (111) QDP
168.85/87.95	            (112) UsableRulesProof [EQUIVALENT, 0 ms]
168.85/87.95	            (113) QDP
168.85/87.95	            (114) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (115) QDP
168.85/87.95	            (116) UsableRulesProof [EQUIVALENT, 0 ms]
168.85/87.95	            (117) QDP
168.85/87.95	            (118) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (119) QDP
168.85/87.95	            (120) UsableRulesProof [EQUIVALENT, 0 ms]
168.85/87.95	            (121) QDP
168.85/87.95	            (122) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (123) QDP
168.85/87.95	            (124) UsableRulesProof [EQUIVALENT, 0 ms]
168.85/87.95	            (125) QDP
168.85/87.95	            (126) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (127) QDP
168.85/87.95	            (128) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (129) QDP
168.85/87.95	            (130) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (131) QDP
168.85/87.95	            (132) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (133) QDP
168.85/87.95	            (134) DependencyGraphProof [EQUIVALENT, 0 ms]
168.85/87.95	            (135) QDP
168.85/87.95	            (136) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (137) QDP
168.85/87.95	            (138) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (139) QDP
168.85/87.95	            (140) DependencyGraphProof [EQUIVALENT, 0 ms]
168.85/87.95	            (141) QDP
168.85/87.95	            (142) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (143) QDP
168.85/87.95	            (144) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (145) QDP
168.85/87.95	            (146) UsableRulesProof [EQUIVALENT, 0 ms]
168.85/87.95	            (147) QDP
168.85/87.95	            (148) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (149) QDP
168.85/87.95	            (150) UsableRulesProof [EQUIVALENT, 0 ms]
168.85/87.95	            (151) QDP
168.85/87.95	            (152) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (153) QDP
168.85/87.95	            (154) UsableRulesProof [EQUIVALENT, 0 ms]
168.85/87.95	            (155) QDP
168.85/87.95	            (156) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (157) QDP
168.85/87.95	            (158) UsableRulesProof [EQUIVALENT, 0 ms]
168.85/87.95	            (159) QDP
168.85/87.95	            (160) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (161) QDP
168.85/87.95	            (162) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (163) QDP
168.85/87.95	            (164) UsableRulesProof [EQUIVALENT, 0 ms]
168.85/87.95	            (165) QDP
168.85/87.95	            (166) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (167) QDP
168.85/87.95	            (168) UsableRulesProof [EQUIVALENT, 0 ms]
168.85/87.95	            (169) QDP
168.85/87.95	            (170) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (171) QDP
168.85/87.95	            (172) DependencyGraphProof [EQUIVALENT, 0 ms]
168.85/87.95	            (173) QDP
168.85/87.95	            (174) UsableRulesProof [EQUIVALENT, 0 ms]
168.85/87.95	            (175) QDP
168.85/87.95	            (176) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (177) QDP
168.85/87.95	            (178) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (179) QDP
168.85/87.95	            (180) DependencyGraphProof [EQUIVALENT, 0 ms]
168.85/87.95	            (181) QDP
168.85/87.95	            (182) UsableRulesProof [EQUIVALENT, 0 ms]
168.85/87.95	            (183) QDP
168.85/87.95	            (184) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (185) QDP
168.85/87.95	            (186) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (187) QDP
168.85/87.95	            (188) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (189) QDP
168.85/87.95	            (190) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (191) QDP
168.85/87.95	            (192) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (193) QDP
168.85/87.95	            (194) TransformationProof [EQUIVALENT, 0 ms]
168.85/87.95	            (195) QDP
168.85/87.95	            (196) DependencyGraphProof [EQUIVALENT, 0 ms]
168.85/87.95	            (197) QDP
168.85/87.95	            (198) QDPOrderProof [EQUIVALENT, 0 ms]
168.85/87.95	            (199) QDP
168.85/87.95	            (200) DependencyGraphProof [EQUIVALENT, 0 ms]
168.85/87.95	            (201) QDP
168.85/87.95	            (202) QDPOrderProof [EQUIVALENT, 10 ms]
168.85/87.95	            (203) QDP
168.85/87.95	            (204) QDPOrderProof [EQUIVALENT, 321 ms]
168.85/87.95	            (205) QDP
168.85/87.95	            (206) SplitQDPProof [EQUIVALENT, 0 ms]
168.85/87.95	            (207) AND
168.85/87.95	                (208) QDP
168.85/87.95	                    (209) SemLabProof [SOUND, 0 ms]
168.85/87.95	                    (210) QDP
168.85/87.95	                    (211) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
168.85/87.95	                    (212) QDP
168.85/87.95	                    (213) DependencyGraphProof [EQUIVALENT, 0 ms]
168.85/87.95	                    (214) QDP
168.85/87.95	                    (215) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
168.85/87.95	                    (216) QDP
168.85/87.95	                    (217) PisEmptyProof [SOUND, 0 ms]
168.85/87.95	                    (218) TRUE
168.85/87.95	                (219) QDP
168.85/87.95	                    (220) QReductionProof [EQUIVALENT, 0 ms]
168.85/87.95	                    (221) QDP
168.85/87.95	                    (222) SplitQDPProof [EQUIVALENT, 0 ms]
168.85/87.95	                    (223) AND
168.85/87.95	                        (224) QDP
168.85/87.95	                            (225) SemLabProof [SOUND, 0 ms]
168.85/87.95	                            (226) QDP
168.85/87.95	                            (227) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
168.85/87.95	                            (228) QDP
168.85/87.95	                            (229) DependencyGraphProof [EQUIVALENT, 0 ms]
168.85/87.95	                            (230) QDP
168.85/87.95	                            (231) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
168.85/87.95	                            (232) QDP
168.85/87.95	                            (233) PisEmptyProof [SOUND, 0 ms]
168.85/87.95	                            (234) TRUE
168.85/87.95	                        (235) QDP
168.85/87.95	                            (236) QReductionProof [EQUIVALENT, 0 ms]
168.85/87.95	                            (237) QDP
168.85/87.95	
168.85/87.95	
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(0)
168.85/87.95	Obligation:
168.85/87.95	Term rewrite system R:
168.85/87.95	The TRS R consists of the following rules:
168.85/87.95	
168.85/87.95	   a -> f(a)
168.85/87.95	   g(f(x)) -> f(g(x))
168.85/87.95	   f(f(f(f(f(x))))) -> b
168.85/87.95	
168.85/87.95	
168.85/87.95	
168.85/87.95	Outermost Strategy.
168.85/87.95	
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(1) Thiemann-SpecialC-Transformation (EQUIVALENT)
168.85/87.95	We applied the Thiemann-SpecialC transformation  to transform the outermost TRS to an innermost TRS.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(2)
168.85/87.95	Obligation:
168.85/87.95	Q restricted rewrite system:
168.85/87.95	The TRS R consists of the following rules:
168.85/87.95	
168.85/87.95	   top(go_up(x)) -> top(reduce(x))
168.85/87.95	   reduce(g(x_1)) -> check_g(redex_g(x_1))
168.85/87.95	   reduce(f(x_1)) -> check_f(redex_f(x_1))
168.85/87.95	   reduce(a) -> go_up(f(a))
168.85/87.95	   redex_g(f(x)) -> result_g(f(g(x)))
168.85/87.95	   redex_f(f(f(f(f(x))))) -> result_f(b)
168.85/87.95	   check_g(result_g(x)) -> go_up(x)
168.85/87.95	   check_f(result_f(x)) -> go_up(x)
168.85/87.95	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
168.85/87.95	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
168.85/87.95	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
168.85/87.95	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
168.85/87.95	
168.85/87.95	The set Q consists of the following terms:
168.85/87.95	
168.85/87.95	   top(go_up(x0))
168.85/87.95	   reduce(g(x0))
168.85/87.95	   reduce(f(x0))
168.85/87.95	   reduce(a)
168.85/87.95	   redex_g(f(x0))
168.85/87.95	   redex_f(f(f(f(f(x0)))))
168.85/87.95	   check_g(result_g(x0))
168.85/87.95	   check_f(result_f(x0))
168.85/87.95	   check_g(redex_g(x0))
168.85/87.95	   check_f(redex_f(x0))
168.85/87.95	   in_f_1(go_up(x0))
168.85/87.95	   in_g_1(go_up(x0))
168.85/87.95	
168.85/87.95	
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(3) DependencyPairsProof (EQUIVALENT)
168.85/87.95	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(4)
168.85/87.95	Obligation:
168.85/87.95	Q DP problem:
168.85/87.95	The TRS P consists of the following rules:
168.85/87.95	
168.85/87.95	   TOP(go_up(x)) -> TOP(reduce(x))
168.85/87.95	   TOP(go_up(x)) -> REDUCE(x)
168.85/87.95	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
168.85/87.95	   REDUCE(g(x_1)) -> REDEX_G(x_1)
168.85/87.95	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
168.85/87.95	   REDUCE(f(x_1)) -> REDEX_F(x_1)
168.85/87.95	   CHECK_G(redex_g(x_1)) -> IN_G_1(reduce(x_1))
168.85/87.95	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
168.85/87.95	   CHECK_F(redex_f(x_1)) -> IN_F_1(reduce(x_1))
168.85/87.95	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
168.85/87.95	
168.85/87.95	The TRS R consists of the following rules:
168.85/87.95	
168.85/87.95	   top(go_up(x)) -> top(reduce(x))
168.85/87.95	   reduce(g(x_1)) -> check_g(redex_g(x_1))
168.85/87.95	   reduce(f(x_1)) -> check_f(redex_f(x_1))
168.85/87.95	   reduce(a) -> go_up(f(a))
168.85/87.95	   redex_g(f(x)) -> result_g(f(g(x)))
168.85/87.95	   redex_f(f(f(f(f(x))))) -> result_f(b)
168.85/87.95	   check_g(result_g(x)) -> go_up(x)
168.85/87.95	   check_f(result_f(x)) -> go_up(x)
168.85/87.95	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
168.85/87.95	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
168.85/87.95	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
168.85/87.95	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
168.85/87.95	
168.85/87.95	The set Q consists of the following terms:
168.85/87.95	
168.85/87.95	   top(go_up(x0))
168.85/87.95	   reduce(g(x0))
168.85/87.95	   reduce(f(x0))
168.85/87.95	   reduce(a)
168.85/87.95	   redex_g(f(x0))
168.85/87.95	   redex_f(f(f(f(f(x0)))))
168.85/87.95	   check_g(result_g(x0))
168.85/87.95	   check_f(result_f(x0))
168.85/87.95	   check_g(redex_g(x0))
168.85/87.95	   check_f(redex_f(x0))
168.85/87.95	   in_f_1(go_up(x0))
168.85/87.95	   in_g_1(go_up(x0))
168.85/87.95	
168.85/87.95	We have to consider all minimal (P,Q,R)-chains.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(5) DependencyGraphProof (EQUIVALENT)
168.85/87.95	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 5 less nodes.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(6)
168.85/87.95	Complex Obligation (AND)
168.85/87.95	
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(7)
168.85/87.95	Obligation:
168.85/87.95	Q DP problem:
168.85/87.95	The TRS P consists of the following rules:
168.85/87.95	
168.85/87.95	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
168.85/87.95	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
168.85/87.95	
168.85/87.95	The TRS R consists of the following rules:
168.85/87.95	
168.85/87.95	   top(go_up(x)) -> top(reduce(x))
168.85/87.95	   reduce(g(x_1)) -> check_g(redex_g(x_1))
168.85/87.95	   reduce(f(x_1)) -> check_f(redex_f(x_1))
168.85/87.95	   reduce(a) -> go_up(f(a))
168.85/87.95	   redex_g(f(x)) -> result_g(f(g(x)))
168.85/87.95	   redex_f(f(f(f(f(x))))) -> result_f(b)
168.85/87.95	   check_g(result_g(x)) -> go_up(x)
168.85/87.95	   check_f(result_f(x)) -> go_up(x)
168.85/87.95	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
168.85/87.95	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
168.85/87.95	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
168.85/87.95	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
168.85/87.95	
168.85/87.95	The set Q consists of the following terms:
168.85/87.95	
168.85/87.95	   top(go_up(x0))
168.85/87.95	   reduce(g(x0))
168.85/87.95	   reduce(f(x0))
168.85/87.95	   reduce(a)
168.85/87.95	   redex_g(f(x0))
168.85/87.95	   redex_f(f(f(f(f(x0)))))
168.85/87.95	   check_g(result_g(x0))
168.85/87.95	   check_f(result_f(x0))
168.85/87.95	   check_g(redex_g(x0))
168.85/87.95	   check_f(redex_f(x0))
168.85/87.95	   in_f_1(go_up(x0))
168.85/87.95	   in_g_1(go_up(x0))
168.85/87.95	
168.85/87.95	We have to consider all minimal (P,Q,R)-chains.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(8) UsableRulesProof (EQUIVALENT)
168.85/87.95	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.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(9)
168.85/87.95	Obligation:
168.85/87.95	Q DP problem:
168.85/87.95	The TRS P consists of the following rules:
168.85/87.95	
168.85/87.95	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
168.85/87.95	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
168.85/87.95	
168.85/87.95	The TRS R consists of the following rules:
168.85/87.95	
168.85/87.95	   redex_g(f(x)) -> result_g(f(g(x)))
168.85/87.95	
168.85/87.95	The set Q consists of the following terms:
168.85/87.95	
168.85/87.95	   top(go_up(x0))
168.85/87.95	   reduce(g(x0))
168.85/87.95	   reduce(f(x0))
168.85/87.95	   reduce(a)
168.85/87.95	   redex_g(f(x0))
168.85/87.95	   redex_f(f(f(f(f(x0)))))
168.85/87.95	   check_g(result_g(x0))
168.85/87.95	   check_f(result_f(x0))
168.85/87.95	   check_g(redex_g(x0))
168.85/87.95	   check_f(redex_f(x0))
168.85/87.95	   in_f_1(go_up(x0))
168.85/87.95	   in_g_1(go_up(x0))
168.85/87.95	
168.85/87.95	We have to consider all minimal (P,Q,R)-chains.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(10) QReductionProof (EQUIVALENT)
168.85/87.95	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
168.85/87.95	
168.85/87.95	   top(go_up(x0))
168.85/87.95	   reduce(g(x0))
168.85/87.95	   reduce(f(x0))
168.85/87.95	   reduce(a)
168.85/87.95	   redex_f(f(f(f(f(x0)))))
168.85/87.95	   check_g(result_g(x0))
168.85/87.95	   check_f(result_f(x0))
168.85/87.95	   check_g(redex_g(x0))
168.85/87.95	   check_f(redex_f(x0))
168.85/87.95	   in_f_1(go_up(x0))
168.85/87.95	   in_g_1(go_up(x0))
168.85/87.95	
168.85/87.95	
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(11)
168.85/87.95	Obligation:
168.85/87.95	Q DP problem:
168.85/87.95	The TRS P consists of the following rules:
168.85/87.95	
168.85/87.95	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
168.85/87.95	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
168.85/87.95	
168.85/87.95	The TRS R consists of the following rules:
168.85/87.95	
168.85/87.95	   redex_g(f(x)) -> result_g(f(g(x)))
168.85/87.95	
168.85/87.95	The set Q consists of the following terms:
168.85/87.95	
168.85/87.95	   redex_g(f(x0))
168.85/87.95	
168.85/87.95	We have to consider all minimal (P,Q,R)-chains.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(12) MRRProof (EQUIVALENT)
168.85/87.95	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.
168.85/87.95	
168.85/87.95	Strictly oriented dependency pairs:
168.85/87.95	
168.85/87.95	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
168.85/87.95	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
168.85/87.95	
168.85/87.95	Strictly oriented rules of the TRS R:
168.85/87.95	
168.85/87.95	   redex_g(f(x)) -> result_g(f(g(x)))
168.85/87.95	
168.85/87.95	Used ordering: Polynomial interpretation [POLO]:
168.85/87.95	
168.85/87.95	   POL(CHECK_G(x_1)) = x_1
168.85/87.95	   POL(REDUCE(x_1)) = 2*x_1
168.85/87.95	   POL(f(x_1)) = 1 + x_1
168.85/87.95	   POL(g(x_1)) = 2 + 2*x_1
168.85/87.95	   POL(redex_g(x_1)) = 2 + 2*x_1
168.85/87.95	   POL(result_g(x_1)) = x_1
168.85/87.95	
168.85/87.95	
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(13)
168.85/87.95	Obligation:
168.85/87.95	Q DP problem:
168.85/87.95	P is empty.
168.85/87.95	R is empty.
168.85/87.95	The set Q consists of the following terms:
168.85/87.95	
168.85/87.95	   redex_g(f(x0))
168.85/87.95	
168.85/87.95	We have to consider all minimal (P,Q,R)-chains.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(14) PisEmptyProof (EQUIVALENT)
168.85/87.95	The TRS P is empty. Hence, there is no (P,Q,R) chain.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(15)
168.85/87.95	YES
168.85/87.95	
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(16)
168.85/87.95	Obligation:
168.85/87.95	Q DP problem:
168.85/87.95	The TRS P consists of the following rules:
168.85/87.95	
168.85/87.95	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
168.85/87.95	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
168.85/87.95	
168.85/87.95	The TRS R consists of the following rules:
168.85/87.95	
168.85/87.95	   top(go_up(x)) -> top(reduce(x))
168.85/87.95	   reduce(g(x_1)) -> check_g(redex_g(x_1))
168.85/87.95	   reduce(f(x_1)) -> check_f(redex_f(x_1))
168.85/87.95	   reduce(a) -> go_up(f(a))
168.85/87.95	   redex_g(f(x)) -> result_g(f(g(x)))
168.85/87.95	   redex_f(f(f(f(f(x))))) -> result_f(b)
168.85/87.95	   check_g(result_g(x)) -> go_up(x)
168.85/87.95	   check_f(result_f(x)) -> go_up(x)
168.85/87.95	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
168.85/87.95	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
168.85/87.95	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
168.85/87.95	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
168.85/87.95	
168.85/87.95	The set Q consists of the following terms:
168.85/87.95	
168.85/87.95	   top(go_up(x0))
168.85/87.95	   reduce(g(x0))
168.85/87.95	   reduce(f(x0))
168.85/87.95	   reduce(a)
168.85/87.95	   redex_g(f(x0))
168.85/87.95	   redex_f(f(f(f(f(x0)))))
168.85/87.95	   check_g(result_g(x0))
168.85/87.95	   check_f(result_f(x0))
168.85/87.95	   check_g(redex_g(x0))
168.85/87.95	   check_f(redex_f(x0))
168.85/87.95	   in_f_1(go_up(x0))
168.85/87.95	   in_g_1(go_up(x0))
168.85/87.95	
168.85/87.95	We have to consider all minimal (P,Q,R)-chains.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(17) UsableRulesProof (EQUIVALENT)
168.85/87.95	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.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(18)
168.85/87.95	Obligation:
168.85/87.95	Q DP problem:
168.85/87.95	The TRS P consists of the following rules:
168.85/87.95	
168.85/87.95	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
168.85/87.95	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
168.85/87.95	
168.85/87.95	The TRS R consists of the following rules:
168.85/87.95	
168.85/87.95	   redex_f(f(f(f(f(x))))) -> result_f(b)
168.85/87.95	
168.85/87.95	The set Q consists of the following terms:
168.85/87.95	
168.85/87.95	   top(go_up(x0))
168.85/87.95	   reduce(g(x0))
168.85/87.95	   reduce(f(x0))
168.85/87.95	   reduce(a)
168.85/87.95	   redex_g(f(x0))
168.85/87.95	   redex_f(f(f(f(f(x0)))))
168.85/87.95	   check_g(result_g(x0))
168.85/87.95	   check_f(result_f(x0))
168.85/87.95	   check_g(redex_g(x0))
168.85/87.95	   check_f(redex_f(x0))
168.85/87.95	   in_f_1(go_up(x0))
168.85/87.95	   in_g_1(go_up(x0))
168.85/87.95	
168.85/87.95	We have to consider all minimal (P,Q,R)-chains.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(19) QReductionProof (EQUIVALENT)
168.85/87.95	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
168.85/87.95	
168.85/87.95	   top(go_up(x0))
168.85/87.95	   reduce(g(x0))
168.85/87.95	   reduce(f(x0))
168.85/87.95	   reduce(a)
168.85/87.95	   redex_g(f(x0))
168.85/87.95	   check_g(result_g(x0))
168.85/87.95	   check_f(result_f(x0))
168.85/87.95	   check_g(redex_g(x0))
168.85/87.95	   check_f(redex_f(x0))
168.85/87.95	   in_f_1(go_up(x0))
168.85/87.95	   in_g_1(go_up(x0))
168.85/87.95	
168.85/87.95	
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(20)
168.85/87.95	Obligation:
168.85/87.95	Q DP problem:
168.85/87.95	The TRS P consists of the following rules:
168.85/87.95	
168.85/87.95	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
168.85/87.95	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
168.85/87.95	
168.85/87.95	The TRS R consists of the following rules:
168.85/87.95	
168.85/87.95	   redex_f(f(f(f(f(x))))) -> result_f(b)
168.85/87.95	
168.85/87.95	The set Q consists of the following terms:
168.85/87.95	
168.85/87.95	   redex_f(f(f(f(f(x0)))))
168.85/87.95	
168.85/87.95	We have to consider all minimal (P,Q,R)-chains.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(21) UsableRulesReductionPairsProof (EQUIVALENT)
168.85/87.95	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.
168.85/87.95	
168.85/87.95	The following dependency pairs can be deleted:
168.85/87.95	
168.85/87.95	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
168.85/87.95	The following rules are removed from R:
168.85/87.95	
168.85/87.95	   redex_f(f(f(f(f(x))))) -> result_f(b)
168.85/87.95	Used ordering: POLO with Polynomial interpretation [POLO]:
168.85/87.95	
168.85/87.95	   POL(CHECK_F(x_1)) = x_1
168.85/87.95	   POL(REDUCE(x_1)) = 2*x_1
168.85/87.95	   POL(b) = 0
168.85/87.95	   POL(f(x_1)) = 2*x_1
168.85/87.95	   POL(redex_f(x_1)) = 2*x_1
168.85/87.95	   POL(result_f(x_1)) = x_1
168.85/87.95	
168.85/87.95	
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(22)
168.85/87.95	Obligation:
168.85/87.95	Q DP problem:
168.85/87.95	The TRS P consists of the following rules:
168.85/87.95	
168.85/87.95	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
168.85/87.95	
168.85/87.95	R is empty.
168.85/87.95	The set Q consists of the following terms:
168.85/87.95	
168.85/87.95	   redex_f(f(f(f(f(x0)))))
168.85/87.95	
168.85/87.95	We have to consider all minimal (P,Q,R)-chains.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(23) DependencyGraphProof (EQUIVALENT)
168.85/87.95	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(24)
168.85/87.95	TRUE
168.85/87.95	
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(25)
168.85/87.95	Obligation:
168.85/87.95	Q DP problem:
168.85/87.95	The TRS P consists of the following rules:
168.85/87.95	
168.85/87.95	   TOP(go_up(x)) -> TOP(reduce(x))
168.85/87.95	
168.85/87.95	The TRS R consists of the following rules:
168.85/87.95	
168.85/87.95	   top(go_up(x)) -> top(reduce(x))
168.85/87.95	   reduce(g(x_1)) -> check_g(redex_g(x_1))
168.85/87.95	   reduce(f(x_1)) -> check_f(redex_f(x_1))
168.85/87.95	   reduce(a) -> go_up(f(a))
168.85/87.95	   redex_g(f(x)) -> result_g(f(g(x)))
168.85/87.95	   redex_f(f(f(f(f(x))))) -> result_f(b)
168.85/87.95	   check_g(result_g(x)) -> go_up(x)
168.85/87.95	   check_f(result_f(x)) -> go_up(x)
168.85/87.95	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
168.85/87.95	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
168.85/87.95	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
168.85/87.95	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
168.85/87.95	
168.85/87.95	The set Q consists of the following terms:
168.85/87.95	
168.85/87.95	   top(go_up(x0))
168.85/87.95	   reduce(g(x0))
168.85/87.95	   reduce(f(x0))
168.85/87.95	   reduce(a)
168.85/87.95	   redex_g(f(x0))
168.85/87.95	   redex_f(f(f(f(f(x0)))))
168.85/87.95	   check_g(result_g(x0))
168.85/87.95	   check_f(result_f(x0))
168.85/87.95	   check_g(redex_g(x0))
168.85/87.95	   check_f(redex_f(x0))
168.85/87.95	   in_f_1(go_up(x0))
168.85/87.95	   in_g_1(go_up(x0))
168.85/87.95	
168.85/87.95	We have to consider all minimal (P,Q,R)-chains.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(26) UsableRulesProof (EQUIVALENT)
168.85/87.95	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.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(27)
168.85/87.95	Obligation:
168.85/87.95	Q DP problem:
168.85/87.95	The TRS P consists of the following rules:
168.85/87.95	
168.85/87.95	   TOP(go_up(x)) -> TOP(reduce(x))
168.85/87.95	
168.85/87.95	The TRS R consists of the following rules:
168.85/87.95	
168.85/87.95	   reduce(g(x_1)) -> check_g(redex_g(x_1))
168.85/87.95	   reduce(f(x_1)) -> check_f(redex_f(x_1))
168.85/87.95	   reduce(a) -> go_up(f(a))
168.85/87.95	   redex_f(f(f(f(f(x))))) -> result_f(b)
168.85/87.95	   check_f(result_f(x)) -> go_up(x)
168.85/87.95	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
168.85/87.95	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
168.85/87.95	   redex_g(f(x)) -> result_g(f(g(x)))
168.85/87.95	   check_g(result_g(x)) -> go_up(x)
168.85/87.95	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
168.85/87.95	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
168.85/87.95	
168.85/87.95	The set Q consists of the following terms:
168.85/87.95	
168.85/87.95	   top(go_up(x0))
168.85/87.95	   reduce(g(x0))
168.85/87.95	   reduce(f(x0))
168.85/87.95	   reduce(a)
168.85/87.95	   redex_g(f(x0))
168.85/87.95	   redex_f(f(f(f(f(x0)))))
168.85/87.95	   check_g(result_g(x0))
168.85/87.95	   check_f(result_f(x0))
168.85/87.95	   check_g(redex_g(x0))
168.85/87.95	   check_f(redex_f(x0))
168.85/87.95	   in_f_1(go_up(x0))
168.85/87.95	   in_g_1(go_up(x0))
168.85/87.95	
168.85/87.95	We have to consider all minimal (P,Q,R)-chains.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(28) QReductionProof (EQUIVALENT)
168.85/87.95	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
168.85/87.95	
168.85/87.95	   top(go_up(x0))
168.85/87.95	
168.85/87.95	
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(29)
168.85/87.95	Obligation:
168.85/87.95	Q DP problem:
168.85/87.95	The TRS P consists of the following rules:
168.85/87.95	
168.85/87.95	   TOP(go_up(x)) -> TOP(reduce(x))
168.85/87.95	
168.85/87.95	The TRS R consists of the following rules:
168.85/87.95	
168.85/87.95	   reduce(g(x_1)) -> check_g(redex_g(x_1))
168.85/87.95	   reduce(f(x_1)) -> check_f(redex_f(x_1))
168.85/87.95	   reduce(a) -> go_up(f(a))
168.85/87.95	   redex_f(f(f(f(f(x))))) -> result_f(b)
168.85/87.95	   check_f(result_f(x)) -> go_up(x)
168.85/87.95	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
168.85/87.95	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
168.85/87.95	   redex_g(f(x)) -> result_g(f(g(x)))
168.85/87.95	   check_g(result_g(x)) -> go_up(x)
168.85/87.95	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
168.85/87.95	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
168.85/87.95	
168.85/87.95	The set Q consists of the following terms:
168.85/87.95	
168.85/87.95	   reduce(g(x0))
168.85/87.95	   reduce(f(x0))
168.85/87.95	   reduce(a)
168.85/87.95	   redex_g(f(x0))
168.85/87.95	   redex_f(f(f(f(f(x0)))))
168.85/87.95	   check_g(result_g(x0))
168.85/87.95	   check_f(result_f(x0))
168.85/87.95	   check_g(redex_g(x0))
168.85/87.95	   check_f(redex_f(x0))
168.85/87.95	   in_f_1(go_up(x0))
168.85/87.95	   in_g_1(go_up(x0))
168.85/87.95	
168.85/87.95	We have to consider all minimal (P,Q,R)-chains.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(30) TransformationProof (EQUIVALENT)
168.85/87.95	By narrowing [LPAR04] the rule TOP(go_up(x)) -> TOP(reduce(x)) at position [0] we obtained the following new rules [LPAR04]:
168.85/87.95	
168.85/87.95	   (TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))),TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))))
168.85/87.95	   (TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))),TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))))
168.85/87.95	   (TOP(go_up(a)) -> TOP(go_up(f(a))),TOP(go_up(a)) -> TOP(go_up(f(a))))
168.85/87.95	
168.85/87.95	
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(31)
168.85/87.95	Obligation:
168.85/87.95	Q DP problem:
168.85/87.95	The TRS P consists of the following rules:
168.85/87.95	
168.85/87.95	   TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0)))
168.85/87.95	   TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0)))
168.85/87.95	   TOP(go_up(a)) -> TOP(go_up(f(a)))
168.85/87.95	
168.85/87.95	The TRS R consists of the following rules:
168.85/87.95	
168.85/87.95	   reduce(g(x_1)) -> check_g(redex_g(x_1))
168.85/87.95	   reduce(f(x_1)) -> check_f(redex_f(x_1))
168.85/87.95	   reduce(a) -> go_up(f(a))
168.85/87.95	   redex_f(f(f(f(f(x))))) -> result_f(b)
168.85/87.95	   check_f(result_f(x)) -> go_up(x)
168.85/87.95	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
168.85/87.95	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
168.85/87.95	   redex_g(f(x)) -> result_g(f(g(x)))
168.85/87.95	   check_g(result_g(x)) -> go_up(x)
168.85/87.95	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
168.85/87.95	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
168.85/87.95	
168.85/87.95	The set Q consists of the following terms:
168.85/87.95	
168.85/87.95	   reduce(g(x0))
168.85/87.95	   reduce(f(x0))
168.85/87.95	   reduce(a)
168.85/87.95	   redex_g(f(x0))
168.85/87.95	   redex_f(f(f(f(f(x0)))))
168.85/87.95	   check_g(result_g(x0))
168.85/87.95	   check_f(result_f(x0))
168.85/87.95	   check_g(redex_g(x0))
168.85/87.95	   check_f(redex_f(x0))
168.85/87.95	   in_f_1(go_up(x0))
168.85/87.95	   in_g_1(go_up(x0))
168.85/87.95	
168.85/87.95	We have to consider all minimal (P,Q,R)-chains.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(32) QDPOrderProof (EQUIVALENT)
168.85/87.95	We use the reduction pair processor [LPAR04,JAR06].
168.85/87.95	
168.85/87.95	
168.85/87.95	The following pairs can be oriented strictly and are deleted.
168.85/87.95	
168.85/87.95	   TOP(go_up(a)) -> TOP(go_up(f(a)))
168.85/87.95	The remaining pairs can at least be oriented weakly.
168.85/87.95	Used ordering:  Polynomial interpretation [POLO]:
168.85/87.95	
168.85/87.95	   POL(TOP(x_1)) = x_1
168.85/87.95	   POL(a) = 1
168.85/87.95	   POL(b) = 0
168.85/87.95	   POL(check_f(x_1)) = x_1
168.85/87.95	   POL(check_g(x_1)) = x_1
168.85/87.95	   POL(f(x_1)) = 0
168.85/87.95	   POL(g(x_1)) = 0
168.85/87.95	   POL(go_up(x_1)) = x_1
168.85/87.95	   POL(in_f_1(x_1)) = 0
168.85/87.95	   POL(in_g_1(x_1)) = 0
168.85/87.95	   POL(redex_f(x_1)) = 0
168.85/87.95	   POL(redex_g(x_1)) = 0
168.85/87.95	   POL(reduce(x_1)) = 0
168.85/87.95	   POL(result_f(x_1)) = x_1
168.85/87.95	   POL(result_g(x_1)) = x_1
168.85/87.95	
168.85/87.95	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
168.85/87.95	
168.85/87.95	   redex_g(f(x)) -> result_g(f(g(x)))
168.85/87.95	   check_g(result_g(x)) -> go_up(x)
168.85/87.95	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
168.85/87.95	   redex_f(f(f(f(f(x))))) -> result_f(b)
168.85/87.95	   check_f(result_f(x)) -> go_up(x)
168.85/87.95	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
168.85/87.95	   reduce(f(x_1)) -> check_f(redex_f(x_1))
168.85/87.95	   reduce(g(x_1)) -> check_g(redex_g(x_1))
168.85/87.95	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
168.85/87.95	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
168.85/87.95	
168.85/87.95	
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(33)
168.85/87.95	Obligation:
168.85/87.95	Q DP problem:
168.85/87.95	The TRS P consists of the following rules:
168.85/87.95	
168.85/87.95	   TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0)))
168.85/87.95	   TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0)))
168.85/87.95	
168.85/87.95	The TRS R consists of the following rules:
168.85/87.95	
168.85/87.95	   reduce(g(x_1)) -> check_g(redex_g(x_1))
168.85/87.95	   reduce(f(x_1)) -> check_f(redex_f(x_1))
168.85/87.95	   reduce(a) -> go_up(f(a))
168.85/87.95	   redex_f(f(f(f(f(x))))) -> result_f(b)
168.85/87.95	   check_f(result_f(x)) -> go_up(x)
168.85/87.95	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
168.85/87.95	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
168.85/87.95	   redex_g(f(x)) -> result_g(f(g(x)))
168.85/87.95	   check_g(result_g(x)) -> go_up(x)
168.85/87.95	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
168.85/87.95	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
168.85/87.95	
168.85/87.95	The set Q consists of the following terms:
168.85/87.95	
168.85/87.95	   reduce(g(x0))
168.85/87.95	   reduce(f(x0))
168.85/87.95	   reduce(a)
168.85/87.95	   redex_g(f(x0))
168.85/87.95	   redex_f(f(f(f(f(x0)))))
168.85/87.95	   check_g(result_g(x0))
168.85/87.95	   check_f(result_f(x0))
168.85/87.95	   check_g(redex_g(x0))
168.85/87.95	   check_f(redex_f(x0))
168.85/87.95	   in_f_1(go_up(x0))
168.85/87.95	   in_g_1(go_up(x0))
168.85/87.95	
168.85/87.95	We have to consider all minimal (P,Q,R)-chains.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(34) UsableRulesProof (EQUIVALENT)
168.85/87.95	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.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(35)
168.85/87.95	Obligation:
168.85/87.95	Q DP problem:
168.85/87.95	The TRS P consists of the following rules:
168.85/87.95	
168.85/87.95	   TOP(go_up(x)) -> TOP(reduce(x))
168.85/87.95	
168.85/87.95	The TRS R consists of the following rules:
168.85/87.95	
168.85/87.95	   reduce(g(x_1)) -> check_g(redex_g(x_1))
168.85/87.95	   reduce(f(x_1)) -> check_f(redex_f(x_1))
168.85/87.95	   reduce(a) -> go_up(f(a))
168.85/87.95	   redex_f(f(f(f(f(x))))) -> result_f(b)
168.85/87.95	   check_f(result_f(x)) -> go_up(x)
168.85/87.95	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
168.85/87.95	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
168.85/87.95	   redex_g(f(x)) -> result_g(f(g(x)))
168.85/87.95	   check_g(result_g(x)) -> go_up(x)
168.85/87.95	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
168.85/87.95	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
168.85/87.95	
168.85/87.95	The set Q consists of the following terms:
168.85/87.95	
168.85/87.95	   top(go_up(x0))
168.85/87.95	   reduce(g(x0))
168.85/87.95	   reduce(f(x0))
168.85/87.95	   reduce(a)
168.85/87.95	   redex_g(f(x0))
168.85/87.95	   redex_f(f(f(f(f(x0)))))
168.85/87.95	   check_g(result_g(x0))
168.85/87.95	   check_f(result_f(x0))
168.85/87.95	   check_g(redex_g(x0))
168.85/87.95	   check_f(redex_f(x0))
168.85/87.95	   in_f_1(go_up(x0))
168.85/87.95	   in_g_1(go_up(x0))
168.85/87.95	
168.85/87.95	We have to consider all minimal (P,Q,R)-chains.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(36) QReductionProof (EQUIVALENT)
168.85/87.95	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
168.85/87.95	
168.85/87.95	   top(go_up(x0))
168.85/87.95	
168.85/87.95	
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(37)
168.85/87.95	Obligation:
168.85/87.95	Q DP problem:
168.85/87.95	The TRS P consists of the following rules:
168.85/87.95	
168.85/87.95	   TOP(go_up(x)) -> TOP(reduce(x))
168.85/87.95	
168.85/87.95	The TRS R consists of the following rules:
168.85/87.95	
168.85/87.95	   reduce(g(x_1)) -> check_g(redex_g(x_1))
168.85/87.95	   reduce(f(x_1)) -> check_f(redex_f(x_1))
168.85/87.95	   reduce(a) -> go_up(f(a))
168.85/87.95	   redex_f(f(f(f(f(x))))) -> result_f(b)
168.85/87.95	   check_f(result_f(x)) -> go_up(x)
168.85/87.95	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
168.85/87.95	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
168.85/87.95	   redex_g(f(x)) -> result_g(f(g(x)))
168.85/87.95	   check_g(result_g(x)) -> go_up(x)
168.85/87.95	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
168.85/87.95	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
168.85/87.95	
168.85/87.95	The set Q consists of the following terms:
168.85/87.95	
168.85/87.95	   reduce(g(x0))
168.85/87.95	   reduce(f(x0))
168.85/87.95	   reduce(a)
168.85/87.95	   redex_g(f(x0))
168.85/87.95	   redex_f(f(f(f(f(x0)))))
168.85/87.95	   check_g(result_g(x0))
168.85/87.95	   check_f(result_f(x0))
168.85/87.95	   check_g(redex_g(x0))
168.85/87.95	   check_f(redex_f(x0))
168.85/87.95	   in_f_1(go_up(x0))
168.85/87.95	   in_g_1(go_up(x0))
168.85/87.95	
168.85/87.95	We have to consider all minimal (P,Q,R)-chains.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(38) Trivial-Transformation (SOUND)
168.85/87.95	We applied the Trivial transformation  to transform the outermost TRS to a standard TRS.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(39)
168.85/87.95	Obligation:
168.85/87.95	Q restricted rewrite system:
168.85/87.95	The TRS R consists of the following rules:
168.85/87.95	
168.85/87.95	   a -> f(a)
168.85/87.95	   g(f(x)) -> f(g(x))
168.85/87.95	   f(f(f(f(f(x))))) -> b
168.85/87.95	
168.85/87.95	Q is empty.
168.85/87.95	
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(40) DependencyPairsProof (EQUIVALENT)
168.85/87.95	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
168.85/87.95	----------------------------------------
168.85/87.95	
168.85/87.95	(41)
168.85/87.95	Obligation:
168.85/87.95	Q DP problem:
168.85/87.95	The TRS P consists of the following rules:
168.85/87.95	
168.85/87.95	   A -> F(a)
168.85/87.95	   A -> A
168.85/87.95	   G(f(x)) -> F(g(x))
168.85/87.96	   G(f(x)) -> G(x)
168.85/87.96	
168.85/87.96	The TRS R consists of the following rules:
168.85/87.96	
168.85/87.96	   a -> f(a)
168.85/87.96	   g(f(x)) -> f(g(x))
168.85/87.96	   f(f(f(f(f(x))))) -> b
168.85/87.96	
168.85/87.96	Q is empty.
168.85/87.96	We have to consider all minimal (P,Q,R)-chains.
168.85/87.96	----------------------------------------
168.85/87.96	
168.85/87.96	(42) DependencyGraphProof (EQUIVALENT)
168.85/87.96	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.
168.85/87.96	----------------------------------------
168.85/87.96	
168.85/87.96	(43)
168.85/87.96	Complex Obligation (AND)
168.85/87.96	
168.85/87.96	----------------------------------------
168.85/87.96	
168.85/87.96	(44)
168.85/87.96	Obligation:
168.85/87.96	Q DP problem:
168.85/87.96	The TRS P consists of the following rules:
168.85/87.96	
168.85/87.96	   G(f(x)) -> G(x)
168.85/87.96	
168.85/87.96	The TRS R consists of the following rules:
168.85/87.96	
168.85/87.96	   a -> f(a)
168.85/87.96	   g(f(x)) -> f(g(x))
168.85/87.96	   f(f(f(f(f(x))))) -> b
168.85/87.96	
168.85/87.96	Q is empty.
168.85/87.96	We have to consider all minimal (P,Q,R)-chains.
168.85/87.96	----------------------------------------
168.85/87.96	
168.85/87.96	(45) UsableRulesProof (EQUIVALENT)
168.85/87.96	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.
168.85/87.96	----------------------------------------
168.85/87.96	
168.85/87.96	(46)
168.85/87.96	Obligation:
168.85/87.96	Q DP problem:
168.85/87.96	The TRS P consists of the following rules:
168.85/87.96	
168.85/87.96	   G(f(x)) -> G(x)
168.85/87.96	
168.85/87.96	R is empty.
168.85/87.96	Q is empty.
168.85/87.96	We have to consider all minimal (P,Q,R)-chains.
168.85/87.96	----------------------------------------
168.85/87.96	
168.85/87.96	(47) QDPSizeChangeProof (EQUIVALENT)
168.85/87.96	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
168.85/87.96	
168.85/87.96	From the DPs we obtained the following set of size-change graphs:
168.85/87.96	*G(f(x)) -> G(x)
168.85/87.96	The graph contains the following edges 1 > 1
168.85/87.96	
168.85/87.96	
168.85/87.96	----------------------------------------
168.85/87.96	
168.85/87.96	(48)
168.85/87.96	YES
168.85/87.96	
168.85/87.96	----------------------------------------
168.85/87.96	
168.85/87.96	(49)
168.85/87.96	Obligation:
168.85/87.96	Q DP problem:
168.85/87.96	The TRS P consists of the following rules:
168.85/87.96	
168.85/87.96	   A -> A
168.85/87.96	
168.85/87.96	The TRS R consists of the following rules:
168.85/87.96	
168.85/87.96	   a -> f(a)
168.85/87.96	   g(f(x)) -> f(g(x))
168.85/87.96	   f(f(f(f(f(x))))) -> b
168.85/87.96	
168.85/87.96	Q is empty.
168.85/87.96	We have to consider all minimal (P,Q,R)-chains.
168.85/87.96	----------------------------------------
168.85/87.96	
168.85/87.96	(50) UsableRulesProof (EQUIVALENT)
168.85/87.96	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.
168.85/87.96	----------------------------------------
168.85/87.96	
168.85/87.96	(51)
168.85/87.96	Obligation:
168.85/87.96	Q DP problem:
168.85/87.96	The TRS P consists of the following rules:
168.85/87.96	
168.85/87.96	   A -> A
168.85/87.96	
168.85/87.96	R is empty.
168.85/87.96	Q is empty.
168.85/87.96	We have to consider all minimal (P,Q,R)-chains.
168.85/87.96	----------------------------------------
168.85/87.96	
168.85/87.96	(52) NonTerminationLoopProof (COMPLETE)
168.85/87.96	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
168.85/87.96	Found a loop by semiunifying a rule from P directly.
168.85/87.96	
168.85/87.96	s = A evaluates to  t =A
168.85/87.96	
168.85/87.96	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
168.85/87.96	* Matcher: [ ]
168.85/87.96	* Semiunifier: [ ]
168.85/87.96	
168.85/87.96	--------------------------------------------------------------------------------
168.85/87.96	Rewriting sequence
168.85/87.96	
168.85/87.96	The DP semiunifies directly so there is only one rewrite step from A to A.
168.85/87.96	
168.85/87.96	
168.85/87.96	
168.85/87.96	
168.85/87.96	----------------------------------------
168.85/87.96	
168.85/87.96	(53)
168.85/87.96	NO
168.85/87.96	
168.85/87.96	----------------------------------------
168.85/87.96	
168.85/87.96	(54) Raffelsieper-Zantema-Transformation (SOUND)
168.85/87.96	We applied the Raffelsieper-Zantema transformation  to transform the outermost TRS to a standard TRS.
168.85/87.96	----------------------------------------
168.85/87.96	
168.85/87.96	(55)
168.85/87.96	Obligation:
168.85/87.96	Q restricted rewrite system:
168.85/87.96	The TRS R consists of the following rules:
168.85/87.96	
168.85/87.96	   down(a) -> up(f(a))
168.85/87.96	   down(g(f(x))) -> up(f(g(x)))
168.85/87.96	   down(f(f(f(f(f(x)))))) -> up(b)
168.85/87.96	   top(up(x)) -> top(down(x))
168.85/87.96	   down(f(a)) -> f_flat(down(a))
168.85/87.96	   down(f(g(y4))) -> f_flat(down(g(y4)))
168.85/87.96	   down(f(b)) -> f_flat(down(b))
168.85/87.96	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
168.85/87.96	   down(g(a)) -> g_flat(down(a))
168.85/87.96	   down(g(g(y7))) -> g_flat(down(g(y7)))
168.85/87.96	   down(g(b)) -> g_flat(down(b))
168.85/87.96	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
168.85/87.96	   down(f(f(a))) -> f_flat(down(f(a)))
168.85/87.96	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
168.85/87.96	   down(f(f(b))) -> f_flat(down(f(b)))
168.85/87.96	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
168.85/87.96	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
168.85/87.96	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
168.85/87.96	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
168.85/87.96	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
168.85/87.96	   down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a)))))
168.85/87.96	   down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16))))))
168.85/87.96	   down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b)))))
168.85/87.96	   down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant)))))
168.85/87.96	   f_flat(up(x_1)) -> up(f(x_1))
168.85/87.96	   g_flat(up(x_1)) -> up(g(x_1))
168.85/87.96	
168.85/87.96	Q is empty.
168.85/87.96	
168.85/87.96	----------------------------------------
168.85/87.96	
168.85/87.96	(56) AAECC Innermost (EQUIVALENT)
168.85/87.96	We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is 
168.85/87.96	   down(f(a)) -> f_flat(down(a))
168.85/87.96	   down(f(g(y4))) -> f_flat(down(g(y4)))
168.85/87.96	   down(f(b)) -> f_flat(down(b))
168.85/87.96	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
168.85/87.96	   down(g(a)) -> g_flat(down(a))
168.85/87.96	   down(g(g(y7))) -> g_flat(down(g(y7)))
168.85/87.96	   down(g(b)) -> g_flat(down(b))
168.85/87.96	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
168.85/87.96	   down(f(f(a))) -> f_flat(down(f(a)))
168.85/87.96	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
168.85/87.96	   down(f(f(b))) -> f_flat(down(f(b)))
168.85/87.96	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
168.85/87.96	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
168.85/87.96	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
168.85/87.96	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
168.85/87.96	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
168.85/87.96	   down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a)))))
168.85/87.96	   down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16))))))
168.85/87.96	   down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b)))))
168.85/87.96	   down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant)))))
168.85/87.96	   f_flat(up(x_1)) -> up(f(x_1))
168.85/87.96	   g_flat(up(x_1)) -> up(g(x_1))
168.85/87.96	   down(a) -> up(f(a))
168.85/87.96	   down(g(f(x))) -> up(f(g(x)))
168.85/87.96	   down(f(f(f(f(f(x)))))) -> up(b)
168.85/87.96	
168.85/87.96	The TRS R 2 is 
168.85/87.96	   top(up(x)) -> top(down(x))
168.85/87.96	
168.85/87.96	The signature Sigma is {top_1}
168.85/87.96	----------------------------------------
168.85/87.96	
168.85/87.96	(57)
168.85/87.96	Obligation:
168.85/87.96	Q restricted rewrite system:
168.85/87.96	The TRS R consists of the following rules:
168.85/87.96	
168.85/87.96	   down(a) -> up(f(a))
168.85/87.96	   down(g(f(x))) -> up(f(g(x)))
168.85/87.96	   down(f(f(f(f(f(x)))))) -> up(b)
168.85/87.96	   top(up(x)) -> top(down(x))
168.85/87.96	   down(f(a)) -> f_flat(down(a))
168.85/87.96	   down(f(g(y4))) -> f_flat(down(g(y4)))
168.85/87.96	   down(f(b)) -> f_flat(down(b))
168.85/87.96	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
168.85/87.96	   down(g(a)) -> g_flat(down(a))
168.85/87.96	   down(g(g(y7))) -> g_flat(down(g(y7)))
168.85/87.96	   down(g(b)) -> g_flat(down(b))
168.85/87.96	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
168.85/87.96	   down(f(f(a))) -> f_flat(down(f(a)))
168.85/87.96	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
168.85/87.96	   down(f(f(b))) -> f_flat(down(f(b)))
168.85/87.96	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
168.85/87.96	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
168.85/87.96	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
168.85/87.96	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
168.85/87.96	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
168.85/87.96	   down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a)))))
168.85/87.96	   down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16))))))
168.85/87.96	   down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b)))))
168.85/87.96	   down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant)))))
168.85/87.96	   f_flat(up(x_1)) -> up(f(x_1))
168.85/87.96	   g_flat(up(x_1)) -> up(g(x_1))
168.85/87.96	
168.85/87.96	The set Q consists of the following terms:
168.85/87.96	
168.85/87.96	   down(a)
168.85/87.96	   down(g(f(x0)))
168.85/87.96	   down(f(f(f(f(f(x0))))))
168.85/87.96	   top(up(x0))
168.85/87.96	   down(f(a))
168.85/87.96	   down(f(g(x0)))
168.85/87.96	   down(f(b))
168.85/87.96	   down(f(fresh_constant))
168.85/87.96	   down(g(a))
168.85/87.96	   down(g(g(x0)))
168.85/87.96	   down(g(b))
168.85/87.96	   down(g(fresh_constant))
168.85/87.96	   down(f(f(a)))
168.85/87.96	   down(f(f(g(x0))))
168.85/87.96	   down(f(f(b)))
168.85/87.96	   down(f(f(fresh_constant)))
168.85/87.96	   down(f(f(f(a))))
168.85/87.96	   down(f(f(f(g(x0)))))
168.85/87.96	   down(f(f(f(b))))
168.85/87.96	   down(f(f(f(fresh_constant))))
168.85/87.96	   down(f(f(f(f(a)))))
168.85/87.96	   down(f(f(f(f(g(x0))))))
168.85/87.96	   down(f(f(f(f(b)))))
168.85/87.96	   down(f(f(f(f(fresh_constant)))))
168.85/87.96	   f_flat(up(x0))
168.85/87.96	   g_flat(up(x0))
168.85/87.96	
168.85/87.96	
168.85/87.96	----------------------------------------
168.85/87.96	
168.85/87.96	(58) DependencyPairsProof (EQUIVALENT)
168.85/87.96	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
168.85/87.96	----------------------------------------
168.85/87.96	
168.85/87.96	(59)
168.85/87.96	Obligation:
168.85/87.96	Q DP problem:
168.85/87.96	The TRS P consists of the following rules:
168.85/87.96	
168.85/87.96	   TOP(up(x)) -> TOP(down(x))
168.85/87.96	   TOP(up(x)) -> DOWN(x)
168.85/87.96	   DOWN(f(a)) -> F_FLAT(down(a))
168.85/87.96	   DOWN(f(a)) -> DOWN(a)
168.85/87.96	   DOWN(f(g(y4))) -> F_FLAT(down(g(y4)))
168.85/87.96	   DOWN(f(g(y4))) -> DOWN(g(y4))
168.85/87.96	   DOWN(f(b)) -> F_FLAT(down(b))
168.85/87.96	   DOWN(f(b)) -> DOWN(b)
168.85/87.96	   DOWN(f(fresh_constant)) -> F_FLAT(down(fresh_constant))
168.85/87.96	   DOWN(f(fresh_constant)) -> DOWN(fresh_constant)
168.85/87.96	   DOWN(g(a)) -> G_FLAT(down(a))
168.85/87.96	   DOWN(g(a)) -> DOWN(a)
168.85/87.96	   DOWN(g(g(y7))) -> G_FLAT(down(g(y7)))
168.85/87.96	   DOWN(g(g(y7))) -> DOWN(g(y7))
168.85/87.96	   DOWN(g(b)) -> G_FLAT(down(b))
168.85/87.96	   DOWN(g(b)) -> DOWN(b)
168.85/87.96	   DOWN(g(fresh_constant)) -> G_FLAT(down(fresh_constant))
168.85/87.96	   DOWN(g(fresh_constant)) -> DOWN(fresh_constant)
168.85/87.96	   DOWN(f(f(a))) -> F_FLAT(down(f(a)))
168.85/87.96	   DOWN(f(f(a))) -> DOWN(f(a))
168.85/87.96	   DOWN(f(f(g(y10)))) -> F_FLAT(down(f(g(y10))))
168.85/87.96	   DOWN(f(f(g(y10)))) -> DOWN(f(g(y10)))
168.85/87.96	   DOWN(f(f(b))) -> F_FLAT(down(f(b)))
168.85/87.96	   DOWN(f(f(b))) -> DOWN(f(b))
168.85/87.96	   DOWN(f(f(fresh_constant))) -> F_FLAT(down(f(fresh_constant)))
168.85/87.96	   DOWN(f(f(fresh_constant))) -> DOWN(f(fresh_constant))
168.85/87.96	   DOWN(f(f(f(a)))) -> F_FLAT(down(f(f(a))))
168.85/87.96	   DOWN(f(f(f(a)))) -> DOWN(f(f(a)))
168.85/87.96	   DOWN(f(f(f(g(y13))))) -> F_FLAT(down(f(f(g(y13)))))
168.85/87.96	   DOWN(f(f(f(g(y13))))) -> DOWN(f(f(g(y13))))
168.85/87.96	   DOWN(f(f(f(b)))) -> F_FLAT(down(f(f(b))))
168.85/87.96	   DOWN(f(f(f(b)))) -> DOWN(f(f(b)))
168.85/87.96	   DOWN(f(f(f(fresh_constant)))) -> F_FLAT(down(f(f(fresh_constant))))
168.85/87.96	   DOWN(f(f(f(fresh_constant)))) -> DOWN(f(f(fresh_constant)))
168.85/87.96	   DOWN(f(f(f(f(a))))) -> F_FLAT(down(f(f(f(a)))))
168.85/87.96	   DOWN(f(f(f(f(a))))) -> DOWN(f(f(f(a))))
168.85/87.96	   DOWN(f(f(f(f(g(y16)))))) -> F_FLAT(down(f(f(f(g(y16))))))
168.85/87.96	   DOWN(f(f(f(f(g(y16)))))) -> DOWN(f(f(f(g(y16)))))
168.85/87.96	   DOWN(f(f(f(f(b))))) -> F_FLAT(down(f(f(f(b)))))
168.85/87.96	   DOWN(f(f(f(f(b))))) -> DOWN(f(f(f(b))))
168.85/87.96	   DOWN(f(f(f(f(fresh_constant))))) -> F_FLAT(down(f(f(f(fresh_constant)))))
168.85/87.96	   DOWN(f(f(f(f(fresh_constant))))) -> DOWN(f(f(f(fresh_constant))))
168.85/87.96	
168.85/87.96	The TRS R consists of the following rules:
168.85/87.96	
168.85/87.96	   down(a) -> up(f(a))
168.85/87.96	   down(g(f(x))) -> up(f(g(x)))
168.85/87.96	   down(f(f(f(f(f(x)))))) -> up(b)
168.85/87.96	   top(up(x)) -> top(down(x))
168.85/87.96	   down(f(a)) -> f_flat(down(a))
168.85/87.96	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.97	   down(f(b)) -> f_flat(down(b))
169.21/87.97	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.97	   down(g(a)) -> g_flat(down(a))
169.21/87.97	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.97	   down(g(b)) -> g_flat(down(b))
169.21/87.97	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.97	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.97	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.97	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.97	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.97	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.97	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.97	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.97	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.97	   down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a)))))
169.21/87.97	   down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16))))))
169.21/87.97	   down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b)))))
169.21/87.97	   down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant)))))
169.21/87.97	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.97	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.97	
169.21/87.97	The set Q consists of the following terms:
169.21/87.97	
169.21/87.97	   down(a)
169.21/87.97	   down(g(f(x0)))
169.21/87.97	   down(f(f(f(f(f(x0))))))
169.21/87.97	   top(up(x0))
169.21/87.97	   down(f(a))
169.21/87.97	   down(f(g(x0)))
169.21/87.97	   down(f(b))
169.21/87.97	   down(f(fresh_constant))
169.21/87.97	   down(g(a))
169.21/87.97	   down(g(g(x0)))
169.21/87.97	   down(g(b))
169.21/87.97	   down(g(fresh_constant))
169.21/87.97	   down(f(f(a)))
169.21/87.97	   down(f(f(g(x0))))
169.21/87.97	   down(f(f(b)))
169.21/87.97	   down(f(f(fresh_constant)))
169.21/87.97	   down(f(f(f(a))))
169.21/87.97	   down(f(f(f(g(x0)))))
169.21/87.97	   down(f(f(f(b))))
169.21/87.97	   down(f(f(f(fresh_constant))))
169.21/87.97	   down(f(f(f(f(a)))))
169.21/87.97	   down(f(f(f(f(g(x0))))))
169.21/87.97	   down(f(f(f(f(b)))))
169.21/87.97	   down(f(f(f(f(fresh_constant)))))
169.21/87.97	   f_flat(up(x0))
169.21/87.97	   g_flat(up(x0))
169.21/87.97	
169.21/87.97	We have to consider all minimal (P,Q,R)-chains.
169.21/87.97	----------------------------------------
169.21/87.97	
169.21/87.97	(60) DependencyGraphProof (EQUIVALENT)
169.21/87.97	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 40 less nodes.
169.21/87.97	----------------------------------------
169.21/87.97	
169.21/87.97	(61)
169.21/87.97	Complex Obligation (AND)
169.21/87.97	
169.21/87.97	----------------------------------------
169.21/87.97	
169.21/87.97	(62)
169.21/87.97	Obligation:
169.21/87.97	Q DP problem:
169.21/87.97	The TRS P consists of the following rules:
169.21/87.97	
169.21/87.97	   DOWN(g(g(y7))) -> DOWN(g(y7))
169.21/87.97	
169.21/87.97	The TRS R consists of the following rules:
169.21/87.97	
169.21/87.97	   down(a) -> up(f(a))
169.21/87.97	   down(g(f(x))) -> up(f(g(x)))
169.21/87.97	   down(f(f(f(f(f(x)))))) -> up(b)
169.21/87.97	   top(up(x)) -> top(down(x))
169.21/87.97	   down(f(a)) -> f_flat(down(a))
169.21/87.97	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.97	   down(f(b)) -> f_flat(down(b))
169.21/87.97	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.97	   down(g(a)) -> g_flat(down(a))
169.21/87.97	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.97	   down(g(b)) -> g_flat(down(b))
169.21/87.97	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.97	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.97	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.97	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.97	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.97	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.97	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.97	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.97	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.97	   down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a)))))
169.21/87.97	   down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16))))))
169.21/87.97	   down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b)))))
169.21/87.97	   down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant)))))
169.21/87.97	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.97	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.97	
169.21/87.97	The set Q consists of the following terms:
169.21/87.97	
169.21/87.97	   down(a)
169.21/87.97	   down(g(f(x0)))
169.21/87.97	   down(f(f(f(f(f(x0))))))
169.21/87.97	   top(up(x0))
169.21/87.97	   down(f(a))
169.21/87.97	   down(f(g(x0)))
169.21/87.97	   down(f(b))
169.21/87.97	   down(f(fresh_constant))
169.21/87.97	   down(g(a))
169.21/87.97	   down(g(g(x0)))
169.21/87.97	   down(g(b))
169.21/87.97	   down(g(fresh_constant))
169.21/87.97	   down(f(f(a)))
169.21/87.97	   down(f(f(g(x0))))
169.21/87.97	   down(f(f(b)))
169.21/87.97	   down(f(f(fresh_constant)))
169.21/87.97	   down(f(f(f(a))))
169.21/87.97	   down(f(f(f(g(x0)))))
169.21/87.97	   down(f(f(f(b))))
169.21/87.97	   down(f(f(f(fresh_constant))))
169.21/87.97	   down(f(f(f(f(a)))))
169.21/87.97	   down(f(f(f(f(g(x0))))))
169.21/87.97	   down(f(f(f(f(b)))))
169.21/87.97	   down(f(f(f(f(fresh_constant)))))
169.21/87.97	   f_flat(up(x0))
169.21/87.97	   g_flat(up(x0))
169.21/87.97	
169.21/87.97	We have to consider all minimal (P,Q,R)-chains.
169.21/87.97	----------------------------------------
169.21/87.97	
169.21/87.97	(63) UsableRulesProof (EQUIVALENT)
169.21/87.97	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.
169.21/87.97	----------------------------------------
169.21/87.97	
169.21/87.97	(64)
169.21/87.97	Obligation:
169.21/87.97	Q DP problem:
169.21/87.97	The TRS P consists of the following rules:
169.21/87.97	
169.21/87.97	   DOWN(g(g(y7))) -> DOWN(g(y7))
169.21/87.97	
169.21/87.97	R is empty.
169.21/87.97	The set Q consists of the following terms:
169.21/87.97	
169.21/87.97	   down(a)
169.21/87.97	   down(g(f(x0)))
169.21/87.97	   down(f(f(f(f(f(x0))))))
169.21/87.97	   top(up(x0))
169.21/87.97	   down(f(a))
169.21/87.97	   down(f(g(x0)))
169.21/87.97	   down(f(b))
169.21/87.97	   down(f(fresh_constant))
169.21/87.97	   down(g(a))
169.21/87.97	   down(g(g(x0)))
169.21/87.97	   down(g(b))
169.21/87.97	   down(g(fresh_constant))
169.21/87.97	   down(f(f(a)))
169.21/87.97	   down(f(f(g(x0))))
169.21/87.97	   down(f(f(b)))
169.21/87.97	   down(f(f(fresh_constant)))
169.21/87.97	   down(f(f(f(a))))
169.21/87.97	   down(f(f(f(g(x0)))))
169.21/87.97	   down(f(f(f(b))))
169.21/87.97	   down(f(f(f(fresh_constant))))
169.21/87.97	   down(f(f(f(f(a)))))
169.21/87.97	   down(f(f(f(f(g(x0))))))
169.21/87.97	   down(f(f(f(f(b)))))
169.21/87.97	   down(f(f(f(f(fresh_constant)))))
169.21/87.97	   f_flat(up(x0))
169.21/87.97	   g_flat(up(x0))
169.21/87.97	
169.21/87.97	We have to consider all minimal (P,Q,R)-chains.
169.21/87.97	----------------------------------------
169.21/87.97	
169.21/87.97	(65) QReductionProof (EQUIVALENT)
169.21/87.97	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
169.21/87.98	
169.21/87.98	   down(a)
169.21/87.98	   down(g(f(x0)))
169.21/87.98	   down(f(f(f(f(f(x0))))))
169.21/87.98	   top(up(x0))
169.21/87.98	   down(f(a))
169.21/87.98	   down(f(g(x0)))
169.21/87.98	   down(f(b))
169.21/87.98	   down(f(fresh_constant))
169.21/87.98	   down(g(a))
169.21/87.98	   down(g(g(x0)))
169.21/87.98	   down(g(b))
169.21/87.98	   down(g(fresh_constant))
169.21/87.98	   down(f(f(a)))
169.21/87.98	   down(f(f(g(x0))))
169.21/87.98	   down(f(f(b)))
169.21/87.98	   down(f(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a))))
169.21/87.98	   down(f(f(f(g(x0)))))
169.21/87.98	   down(f(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(x0))))))
169.21/87.98	   down(f(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x0))
169.21/87.98	   g_flat(up(x0))
169.21/87.98	
169.21/87.98	
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(66)
169.21/87.98	Obligation:
169.21/87.98	Q DP problem:
169.21/87.98	The TRS P consists of the following rules:
169.21/87.98	
169.21/87.98	   DOWN(g(g(y7))) -> DOWN(g(y7))
169.21/87.98	
169.21/87.98	R is empty.
169.21/87.98	Q is empty.
169.21/87.98	We have to consider all minimal (P,Q,R)-chains.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(67) QDPSizeChangeProof (EQUIVALENT)
169.21/87.98	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
169.21/87.98	
169.21/87.98	From the DPs we obtained the following set of size-change graphs:
169.21/87.98	*DOWN(g(g(y7))) -> DOWN(g(y7))
169.21/87.98	The graph contains the following edges 1 > 1
169.21/87.98	
169.21/87.98	
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(68)
169.21/87.98	YES
169.21/87.98	
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(69)
169.21/87.98	Obligation:
169.21/87.98	Q DP problem:
169.21/87.98	The TRS P consists of the following rules:
169.21/87.98	
169.21/87.98	   TOP(up(x)) -> TOP(down(x))
169.21/87.98	
169.21/87.98	The TRS R consists of the following rules:
169.21/87.98	
169.21/87.98	   down(a) -> up(f(a))
169.21/87.98	   down(g(f(x))) -> up(f(g(x)))
169.21/87.98	   down(f(f(f(f(f(x)))))) -> up(b)
169.21/87.98	   top(up(x)) -> top(down(x))
169.21/87.98	   down(f(a)) -> f_flat(down(a))
169.21/87.98	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.98	   down(f(b)) -> f_flat(down(b))
169.21/87.98	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.98	   down(g(a)) -> g_flat(down(a))
169.21/87.98	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.98	   down(g(b)) -> g_flat(down(b))
169.21/87.98	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.98	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.98	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.98	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.98	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.98	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.98	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16))))))
169.21/87.98	   down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.98	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.98	
169.21/87.98	The set Q consists of the following terms:
169.21/87.98	
169.21/87.98	   down(a)
169.21/87.98	   down(g(f(x0)))
169.21/87.98	   down(f(f(f(f(f(x0))))))
169.21/87.98	   top(up(x0))
169.21/87.98	   down(f(a))
169.21/87.98	   down(f(g(x0)))
169.21/87.98	   down(f(b))
169.21/87.98	   down(f(fresh_constant))
169.21/87.98	   down(g(a))
169.21/87.98	   down(g(g(x0)))
169.21/87.98	   down(g(b))
169.21/87.98	   down(g(fresh_constant))
169.21/87.98	   down(f(f(a)))
169.21/87.98	   down(f(f(g(x0))))
169.21/87.98	   down(f(f(b)))
169.21/87.98	   down(f(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a))))
169.21/87.98	   down(f(f(f(g(x0)))))
169.21/87.98	   down(f(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(x0))))))
169.21/87.98	   down(f(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x0))
169.21/87.98	   g_flat(up(x0))
169.21/87.98	
169.21/87.98	We have to consider all minimal (P,Q,R)-chains.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(70) UsableRulesProof (EQUIVALENT)
169.21/87.98	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.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(71)
169.21/87.98	Obligation:
169.21/87.98	Q DP problem:
169.21/87.98	The TRS P consists of the following rules:
169.21/87.98	
169.21/87.98	   TOP(up(x)) -> TOP(down(x))
169.21/87.98	
169.21/87.98	The TRS R consists of the following rules:
169.21/87.98	
169.21/87.98	   down(a) -> up(f(a))
169.21/87.98	   down(g(f(x))) -> up(f(g(x)))
169.21/87.98	   down(f(f(f(f(f(x)))))) -> up(b)
169.21/87.98	   down(f(a)) -> f_flat(down(a))
169.21/87.98	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.98	   down(f(b)) -> f_flat(down(b))
169.21/87.98	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.98	   down(g(a)) -> g_flat(down(a))
169.21/87.98	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.98	   down(g(b)) -> g_flat(down(b))
169.21/87.98	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.98	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.98	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.98	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.98	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.98	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.98	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16))))))
169.21/87.98	   down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.98	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.98	
169.21/87.98	The set Q consists of the following terms:
169.21/87.98	
169.21/87.98	   down(a)
169.21/87.98	   down(g(f(x0)))
169.21/87.98	   down(f(f(f(f(f(x0))))))
169.21/87.98	   top(up(x0))
169.21/87.98	   down(f(a))
169.21/87.98	   down(f(g(x0)))
169.21/87.98	   down(f(b))
169.21/87.98	   down(f(fresh_constant))
169.21/87.98	   down(g(a))
169.21/87.98	   down(g(g(x0)))
169.21/87.98	   down(g(b))
169.21/87.98	   down(g(fresh_constant))
169.21/87.98	   down(f(f(a)))
169.21/87.98	   down(f(f(g(x0))))
169.21/87.98	   down(f(f(b)))
169.21/87.98	   down(f(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a))))
169.21/87.98	   down(f(f(f(g(x0)))))
169.21/87.98	   down(f(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(x0))))))
169.21/87.98	   down(f(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x0))
169.21/87.98	   g_flat(up(x0))
169.21/87.98	
169.21/87.98	We have to consider all minimal (P,Q,R)-chains.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(72) QReductionProof (EQUIVALENT)
169.21/87.98	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
169.21/87.98	
169.21/87.98	   top(up(x0))
169.21/87.98	
169.21/87.98	
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(73)
169.21/87.98	Obligation:
169.21/87.98	Q DP problem:
169.21/87.98	The TRS P consists of the following rules:
169.21/87.98	
169.21/87.98	   TOP(up(x)) -> TOP(down(x))
169.21/87.98	
169.21/87.98	The TRS R consists of the following rules:
169.21/87.98	
169.21/87.98	   down(a) -> up(f(a))
169.21/87.98	   down(g(f(x))) -> up(f(g(x)))
169.21/87.98	   down(f(f(f(f(f(x)))))) -> up(b)
169.21/87.98	   down(f(a)) -> f_flat(down(a))
169.21/87.98	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.98	   down(f(b)) -> f_flat(down(b))
169.21/87.98	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.98	   down(g(a)) -> g_flat(down(a))
169.21/87.98	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.98	   down(g(b)) -> g_flat(down(b))
169.21/87.98	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.98	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.98	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.98	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.98	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.98	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.98	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16))))))
169.21/87.98	   down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.98	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.98	
169.21/87.98	The set Q consists of the following terms:
169.21/87.98	
169.21/87.98	   down(a)
169.21/87.98	   down(g(f(x0)))
169.21/87.98	   down(f(f(f(f(f(x0))))))
169.21/87.98	   down(f(a))
169.21/87.98	   down(f(g(x0)))
169.21/87.98	   down(f(b))
169.21/87.98	   down(f(fresh_constant))
169.21/87.98	   down(g(a))
169.21/87.98	   down(g(g(x0)))
169.21/87.98	   down(g(b))
169.21/87.98	   down(g(fresh_constant))
169.21/87.98	   down(f(f(a)))
169.21/87.98	   down(f(f(g(x0))))
169.21/87.98	   down(f(f(b)))
169.21/87.98	   down(f(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a))))
169.21/87.98	   down(f(f(f(g(x0)))))
169.21/87.98	   down(f(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(x0))))))
169.21/87.98	   down(f(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x0))
169.21/87.98	   g_flat(up(x0))
169.21/87.98	
169.21/87.98	We have to consider all minimal (P,Q,R)-chains.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(74) TransformationProof (EQUIVALENT)
169.21/87.98	By narrowing [LPAR04] the rule TOP(up(x)) -> TOP(down(x)) at position [0] we obtained the following new rules [LPAR04]:
169.21/87.98	
169.21/87.98	   (TOP(up(a)) -> TOP(up(f(a))),TOP(up(a)) -> TOP(up(f(a))))
169.21/87.98	   (TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))),TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))))
169.21/87.98	   (TOP(up(f(f(f(f(f(x0))))))) -> TOP(up(b)),TOP(up(f(f(f(f(f(x0))))))) -> TOP(up(b)))
169.21/87.98	   (TOP(up(f(a))) -> TOP(f_flat(down(a))),TOP(up(f(a))) -> TOP(f_flat(down(a))))
169.21/87.98	   (TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))),TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))))
169.21/87.98	   (TOP(up(f(b))) -> TOP(f_flat(down(b))),TOP(up(f(b))) -> TOP(f_flat(down(b))))
169.21/87.98	   (TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))),TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))))
169.21/87.98	   (TOP(up(g(a))) -> TOP(g_flat(down(a))),TOP(up(g(a))) -> TOP(g_flat(down(a))))
169.21/87.98	   (TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))),TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))))
169.21/87.98	   (TOP(up(g(b))) -> TOP(g_flat(down(b))),TOP(up(g(b))) -> TOP(g_flat(down(b))))
169.21/87.98	   (TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))),TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))))
169.21/87.98	   (TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))),TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))))
169.21/87.98	   (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))))))
169.21/87.98	   (TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))),TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))))
169.21/87.98	   (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)))))
169.21/87.98	   (TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))))
169.21/87.98	   (TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))),TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))))
169.21/87.98	   (TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))),TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))))
169.21/87.98	   (TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))),TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))))
169.21/87.98	   (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))))
169.21/87.98	   (TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))),TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))))
169.21/87.98	   (TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))),TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))))
169.21/87.98	   (TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))),TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))))
169.21/87.98	
169.21/87.98	
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(75)
169.21/87.98	Obligation:
169.21/87.98	Q DP problem:
169.21/87.98	The TRS P consists of the following rules:
169.21/87.98	
169.21/87.98	   TOP(up(a)) -> TOP(up(f(a)))
169.21/87.98	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.98	   TOP(up(f(f(f(f(f(x0))))))) -> TOP(up(b))
169.21/87.98	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
169.21/87.98	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.98	   TOP(up(f(b))) -> TOP(f_flat(down(b)))
169.21/87.98	   TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant)))
169.21/87.98	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
169.21/87.98	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.98	   TOP(up(g(b))) -> TOP(g_flat(down(b)))
169.21/87.98	   TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant)))
169.21/87.98	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
169.21/87.98	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
169.21/87.98	   TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b))))
169.21/87.98	   TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant))))
169.21/87.98	   TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a)))))
169.21/87.98	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0))))))
169.21/87.98	   TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b)))))
169.21/87.98	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant)))))
169.21/87.98	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a))))))
169.21/87.98	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))
169.21/87.98	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))
169.21/87.98	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.98	
169.21/87.98	The TRS R consists of the following rules:
169.21/87.98	
169.21/87.98	   down(a) -> up(f(a))
169.21/87.98	   down(g(f(x))) -> up(f(g(x)))
169.21/87.98	   down(f(f(f(f(f(x)))))) -> up(b)
169.21/87.98	   down(f(a)) -> f_flat(down(a))
169.21/87.98	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.98	   down(f(b)) -> f_flat(down(b))
169.21/87.98	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.98	   down(g(a)) -> g_flat(down(a))
169.21/87.98	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.98	   down(g(b)) -> g_flat(down(b))
169.21/87.98	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.98	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.98	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.98	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.98	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.98	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.98	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16))))))
169.21/87.98	   down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.98	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.98	
169.21/87.98	The set Q consists of the following terms:
169.21/87.98	
169.21/87.98	   down(a)
169.21/87.98	   down(g(f(x0)))
169.21/87.98	   down(f(f(f(f(f(x0))))))
169.21/87.98	   down(f(a))
169.21/87.98	   down(f(g(x0)))
169.21/87.98	   down(f(b))
169.21/87.98	   down(f(fresh_constant))
169.21/87.98	   down(g(a))
169.21/87.98	   down(g(g(x0)))
169.21/87.98	   down(g(b))
169.21/87.98	   down(g(fresh_constant))
169.21/87.98	   down(f(f(a)))
169.21/87.98	   down(f(f(g(x0))))
169.21/87.98	   down(f(f(b)))
169.21/87.98	   down(f(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a))))
169.21/87.98	   down(f(f(f(g(x0)))))
169.21/87.98	   down(f(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(x0))))))
169.21/87.98	   down(f(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x0))
169.21/87.98	   g_flat(up(x0))
169.21/87.98	
169.21/87.98	We have to consider all minimal (P,Q,R)-chains.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(76) DependencyGraphProof (EQUIVALENT)
169.21/87.98	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(77)
169.21/87.98	Obligation:
169.21/87.98	Q DP problem:
169.21/87.98	The TRS P consists of the following rules:
169.21/87.98	
169.21/87.98	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.98	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.98	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
169.21/87.98	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
169.21/87.98	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
169.21/87.98	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
169.21/87.98	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.98	   TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b))))
169.21/87.98	   TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant))))
169.21/87.98	   TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a)))))
169.21/87.98	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0))))))
169.21/87.98	   TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b)))))
169.21/87.98	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant)))))
169.21/87.98	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a))))))
169.21/87.98	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))
169.21/87.98	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))
169.21/87.98	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.98	
169.21/87.98	The TRS R consists of the following rules:
169.21/87.98	
169.21/87.98	   down(a) -> up(f(a))
169.21/87.98	   down(g(f(x))) -> up(f(g(x)))
169.21/87.98	   down(f(f(f(f(f(x)))))) -> up(b)
169.21/87.98	   down(f(a)) -> f_flat(down(a))
169.21/87.98	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.98	   down(f(b)) -> f_flat(down(b))
169.21/87.98	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.98	   down(g(a)) -> g_flat(down(a))
169.21/87.98	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.98	   down(g(b)) -> g_flat(down(b))
169.21/87.98	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.98	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.98	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.98	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.98	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.98	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.98	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16))))))
169.21/87.98	   down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.98	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.98	
169.21/87.98	The set Q consists of the following terms:
169.21/87.98	
169.21/87.98	   down(a)
169.21/87.98	   down(g(f(x0)))
169.21/87.98	   down(f(f(f(f(f(x0))))))
169.21/87.98	   down(f(a))
169.21/87.98	   down(f(g(x0)))
169.21/87.98	   down(f(b))
169.21/87.98	   down(f(fresh_constant))
169.21/87.98	   down(g(a))
169.21/87.98	   down(g(g(x0)))
169.21/87.98	   down(g(b))
169.21/87.98	   down(g(fresh_constant))
169.21/87.98	   down(f(f(a)))
169.21/87.98	   down(f(f(g(x0))))
169.21/87.98	   down(f(f(b)))
169.21/87.98	   down(f(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a))))
169.21/87.98	   down(f(f(f(g(x0)))))
169.21/87.98	   down(f(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(x0))))))
169.21/87.98	   down(f(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x0))
169.21/87.98	   g_flat(up(x0))
169.21/87.98	
169.21/87.98	We have to consider all minimal (P,Q,R)-chains.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(78) UsableRulesProof (EQUIVALENT)
169.21/87.98	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.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(79)
169.21/87.98	Obligation:
169.21/87.98	Q DP problem:
169.21/87.98	The TRS P consists of the following rules:
169.21/87.98	
169.21/87.98	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.98	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.98	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
169.21/87.98	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
169.21/87.98	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
169.21/87.98	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
169.21/87.98	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.98	   TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b))))
169.21/87.98	   TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant))))
169.21/87.98	   TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a)))))
169.21/87.98	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0))))))
169.21/87.98	   TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b)))))
169.21/87.98	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant)))))
169.21/87.98	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a))))))
169.21/87.98	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))
169.21/87.98	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))
169.21/87.98	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.98	
169.21/87.98	The TRS R consists of the following rules:
169.21/87.98	
169.21/87.98	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.98	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.98	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.98	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.98	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.98	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.98	   down(f(b)) -> f_flat(down(b))
169.21/87.98	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.98	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.98	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.98	   down(g(f(x))) -> up(f(g(x)))
169.21/87.98	   down(g(a)) -> g_flat(down(a))
169.21/87.98	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.98	   down(g(b)) -> g_flat(down(b))
169.21/87.98	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.98	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.98	   down(a) -> up(f(a))
169.21/87.98	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.98	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.98	   down(f(a)) -> f_flat(down(a))
169.21/87.98	
169.21/87.98	The set Q consists of the following terms:
169.21/87.98	
169.21/87.98	   down(a)
169.21/87.98	   down(g(f(x0)))
169.21/87.98	   down(f(f(f(f(f(x0))))))
169.21/87.98	   down(f(a))
169.21/87.98	   down(f(g(x0)))
169.21/87.98	   down(f(b))
169.21/87.98	   down(f(fresh_constant))
169.21/87.98	   down(g(a))
169.21/87.98	   down(g(g(x0)))
169.21/87.98	   down(g(b))
169.21/87.98	   down(g(fresh_constant))
169.21/87.98	   down(f(f(a)))
169.21/87.98	   down(f(f(g(x0))))
169.21/87.98	   down(f(f(b)))
169.21/87.98	   down(f(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a))))
169.21/87.98	   down(f(f(f(g(x0)))))
169.21/87.98	   down(f(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(x0))))))
169.21/87.98	   down(f(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x0))
169.21/87.98	   g_flat(up(x0))
169.21/87.98	
169.21/87.98	We have to consider all minimal (P,Q,R)-chains.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(80) TransformationProof (EQUIVALENT)
169.21/87.98	By rewriting [LPAR04] the rule TOP(up(f(a))) -> TOP(f_flat(down(a))) at position [0,0] we obtained the following new rules [LPAR04]:
169.21/87.98	
169.21/87.98	   (TOP(up(f(a))) -> TOP(f_flat(up(f(a)))),TOP(up(f(a))) -> TOP(f_flat(up(f(a)))))
169.21/87.98	
169.21/87.98	
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(81)
169.21/87.98	Obligation:
169.21/87.98	Q DP problem:
169.21/87.98	The TRS P consists of the following rules:
169.21/87.98	
169.21/87.98	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.98	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.98	   TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))
169.21/87.98	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
169.21/87.98	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
169.21/87.98	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.98	   TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b))))
169.21/87.98	   TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant))))
169.21/87.98	   TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a)))))
169.21/87.98	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0))))))
169.21/87.98	   TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b)))))
169.21/87.98	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant)))))
169.21/87.98	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a))))))
169.21/87.98	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))
169.21/87.98	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))
169.21/87.98	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.98	   TOP(up(f(a))) -> TOP(f_flat(up(f(a))))
169.21/87.98	
169.21/87.98	The TRS R consists of the following rules:
169.21/87.98	
169.21/87.98	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.98	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.98	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.98	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.98	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.98	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.98	   down(f(b)) -> f_flat(down(b))
169.21/87.98	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.98	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.98	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.98	   down(g(f(x))) -> up(f(g(x)))
169.21/87.98	   down(g(a)) -> g_flat(down(a))
169.21/87.98	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.98	   down(g(b)) -> g_flat(down(b))
169.21/87.98	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.98	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.98	   down(a) -> up(f(a))
169.21/87.98	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.98	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.98	   down(f(a)) -> f_flat(down(a))
169.21/87.98	
169.21/87.98	The set Q consists of the following terms:
169.21/87.98	
169.21/87.98	   down(a)
169.21/87.98	   down(g(f(x0)))
169.21/87.98	   down(f(f(f(f(f(x0))))))
169.21/87.98	   down(f(a))
169.21/87.98	   down(f(g(x0)))
169.21/87.98	   down(f(b))
169.21/87.98	   down(f(fresh_constant))
169.21/87.98	   down(g(a))
169.21/87.98	   down(g(g(x0)))
169.21/87.98	   down(g(b))
169.21/87.98	   down(g(fresh_constant))
169.21/87.98	   down(f(f(a)))
169.21/87.98	   down(f(f(g(x0))))
169.21/87.98	   down(f(f(b)))
169.21/87.98	   down(f(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a))))
169.21/87.98	   down(f(f(f(g(x0)))))
169.21/87.98	   down(f(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(x0))))))
169.21/87.98	   down(f(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x0))
169.21/87.98	   g_flat(up(x0))
169.21/87.98	
169.21/87.98	We have to consider all minimal (P,Q,R)-chains.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(82) TransformationProof (EQUIVALENT)
169.21/87.98	By rewriting [LPAR04] the rule TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) at position [0,0] we obtained the following new rules [LPAR04]:
169.21/87.98	
169.21/87.98	   (TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))),TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))))
169.21/87.98	
169.21/87.98	
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(83)
169.21/87.98	Obligation:
169.21/87.98	Q DP problem:
169.21/87.98	The TRS P consists of the following rules:
169.21/87.98	
169.21/87.98	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.98	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.98	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))
169.21/87.98	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
169.21/87.98	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.98	   TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b))))
169.21/87.98	   TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant))))
169.21/87.98	   TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a)))))
169.21/87.98	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0))))))
169.21/87.98	   TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b)))))
169.21/87.98	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant)))))
169.21/87.98	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a))))))
169.21/87.98	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))
169.21/87.98	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))
169.21/87.98	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.98	   TOP(up(f(a))) -> TOP(f_flat(up(f(a))))
169.21/87.98	   TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a))))
169.21/87.98	
169.21/87.98	The TRS R consists of the following rules:
169.21/87.98	
169.21/87.98	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.98	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.98	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.98	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.98	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.98	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.98	   down(f(b)) -> f_flat(down(b))
169.21/87.98	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.98	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.98	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.98	   down(g(f(x))) -> up(f(g(x)))
169.21/87.98	   down(g(a)) -> g_flat(down(a))
169.21/87.98	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.98	   down(g(b)) -> g_flat(down(b))
169.21/87.98	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.98	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.98	   down(a) -> up(f(a))
169.21/87.98	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.98	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.98	   down(f(a)) -> f_flat(down(a))
169.21/87.98	
169.21/87.98	The set Q consists of the following terms:
169.21/87.98	
169.21/87.98	   down(a)
169.21/87.98	   down(g(f(x0)))
169.21/87.98	   down(f(f(f(f(f(x0))))))
169.21/87.98	   down(f(a))
169.21/87.98	   down(f(g(x0)))
169.21/87.98	   down(f(b))
169.21/87.98	   down(f(fresh_constant))
169.21/87.98	   down(g(a))
169.21/87.98	   down(g(g(x0)))
169.21/87.98	   down(g(b))
169.21/87.98	   down(g(fresh_constant))
169.21/87.98	   down(f(f(a)))
169.21/87.98	   down(f(f(g(x0))))
169.21/87.98	   down(f(f(b)))
169.21/87.98	   down(f(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a))))
169.21/87.98	   down(f(f(f(g(x0)))))
169.21/87.98	   down(f(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(x0))))))
169.21/87.98	   down(f(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x0))
169.21/87.98	   g_flat(up(x0))
169.21/87.98	
169.21/87.98	We have to consider all minimal (P,Q,R)-chains.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(84) TransformationProof (EQUIVALENT)
169.21/87.98	By rewriting [LPAR04] the rule TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) at position [0,0] we obtained the following new rules [LPAR04]:
169.21/87.98	
169.21/87.98	   (TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))),TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))))
169.21/87.98	
169.21/87.98	
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(85)
169.21/87.98	Obligation:
169.21/87.98	Q DP problem:
169.21/87.98	The TRS P consists of the following rules:
169.21/87.98	
169.21/87.98	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.98	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.98	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
169.21/87.98	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.98	   TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b))))
169.21/87.98	   TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant))))
169.21/87.98	   TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a)))))
169.21/87.98	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0))))))
169.21/87.98	   TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b)))))
169.21/87.98	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant)))))
169.21/87.98	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a))))))
169.21/87.98	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))
169.21/87.98	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))
169.21/87.98	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.98	   TOP(up(f(a))) -> TOP(f_flat(up(f(a))))
169.21/87.98	   TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a))))
169.21/87.98	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.98	
169.21/87.98	The TRS R consists of the following rules:
169.21/87.98	
169.21/87.98	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.98	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.98	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.98	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.98	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.98	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.98	   down(f(b)) -> f_flat(down(b))
169.21/87.98	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.98	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.98	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.98	   down(g(f(x))) -> up(f(g(x)))
169.21/87.98	   down(g(a)) -> g_flat(down(a))
169.21/87.98	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.98	   down(g(b)) -> g_flat(down(b))
169.21/87.98	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.98	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.98	   down(a) -> up(f(a))
169.21/87.98	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.98	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.98	   down(f(a)) -> f_flat(down(a))
169.21/87.98	
169.21/87.98	The set Q consists of the following terms:
169.21/87.98	
169.21/87.98	   down(a)
169.21/87.98	   down(g(f(x0)))
169.21/87.98	   down(f(f(f(f(f(x0))))))
169.21/87.98	   down(f(a))
169.21/87.98	   down(f(g(x0)))
169.21/87.98	   down(f(b))
169.21/87.98	   down(f(fresh_constant))
169.21/87.98	   down(g(a))
169.21/87.98	   down(g(g(x0)))
169.21/87.98	   down(g(b))
169.21/87.98	   down(g(fresh_constant))
169.21/87.98	   down(f(f(a)))
169.21/87.98	   down(f(f(g(x0))))
169.21/87.98	   down(f(f(b)))
169.21/87.98	   down(f(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a))))
169.21/87.98	   down(f(f(f(g(x0)))))
169.21/87.98	   down(f(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(x0))))))
169.21/87.98	   down(f(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x0))
169.21/87.98	   g_flat(up(x0))
169.21/87.98	
169.21/87.98	We have to consider all minimal (P,Q,R)-chains.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(86) TransformationProof (EQUIVALENT)
169.21/87.98	By rewriting [LPAR04] the rule TOP(up(g(a))) -> TOP(g_flat(down(a))) at position [0,0] we obtained the following new rules [LPAR04]:
169.21/87.98	
169.21/87.98	   (TOP(up(g(a))) -> TOP(g_flat(up(f(a)))),TOP(up(g(a))) -> TOP(g_flat(up(f(a)))))
169.21/87.98	
169.21/87.98	
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(87)
169.21/87.98	Obligation:
169.21/87.98	Q DP problem:
169.21/87.98	The TRS P consists of the following rules:
169.21/87.98	
169.21/87.98	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.98	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.98	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.98	   TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b))))
169.21/87.98	   TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant))))
169.21/87.98	   TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a)))))
169.21/87.98	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0))))))
169.21/87.98	   TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b)))))
169.21/87.98	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant)))))
169.21/87.98	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a))))))
169.21/87.98	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))
169.21/87.98	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))
169.21/87.98	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.98	   TOP(up(f(a))) -> TOP(f_flat(up(f(a))))
169.21/87.98	   TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a))))
169.21/87.98	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.98	   TOP(up(g(a))) -> TOP(g_flat(up(f(a))))
169.21/87.98	
169.21/87.98	The TRS R consists of the following rules:
169.21/87.98	
169.21/87.98	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.98	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.98	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.98	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.98	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.98	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.98	   down(f(b)) -> f_flat(down(b))
169.21/87.98	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.98	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.98	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.98	   down(g(f(x))) -> up(f(g(x)))
169.21/87.98	   down(g(a)) -> g_flat(down(a))
169.21/87.98	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.98	   down(g(b)) -> g_flat(down(b))
169.21/87.98	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.98	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.98	   down(a) -> up(f(a))
169.21/87.98	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.98	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.98	   down(f(a)) -> f_flat(down(a))
169.21/87.98	
169.21/87.98	The set Q consists of the following terms:
169.21/87.98	
169.21/87.98	   down(a)
169.21/87.98	   down(g(f(x0)))
169.21/87.98	   down(f(f(f(f(f(x0))))))
169.21/87.98	   down(f(a))
169.21/87.98	   down(f(g(x0)))
169.21/87.98	   down(f(b))
169.21/87.98	   down(f(fresh_constant))
169.21/87.98	   down(g(a))
169.21/87.98	   down(g(g(x0)))
169.21/87.98	   down(g(b))
169.21/87.98	   down(g(fresh_constant))
169.21/87.98	   down(f(f(a)))
169.21/87.98	   down(f(f(g(x0))))
169.21/87.98	   down(f(f(b)))
169.21/87.98	   down(f(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a))))
169.21/87.98	   down(f(f(f(g(x0)))))
169.21/87.98	   down(f(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(x0))))))
169.21/87.98	   down(f(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x0))
169.21/87.98	   g_flat(up(x0))
169.21/87.98	
169.21/87.98	We have to consider all minimal (P,Q,R)-chains.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(88) TransformationProof (EQUIVALENT)
169.21/87.98	By rewriting [LPAR04] the rule TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) at position [0,0] we obtained the following new rules [LPAR04]:
169.21/87.98	
169.21/87.98	   (TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))),TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))))
169.21/87.98	
169.21/87.98	
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(89)
169.21/87.98	Obligation:
169.21/87.98	Q DP problem:
169.21/87.98	The TRS P consists of the following rules:
169.21/87.98	
169.21/87.98	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.98	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.98	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.98	   TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant))))
169.21/87.98	   TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a)))))
169.21/87.98	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0))))))
169.21/87.98	   TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b)))))
169.21/87.98	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant)))))
169.21/87.98	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a))))))
169.21/87.98	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))
169.21/87.98	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))
169.21/87.98	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.98	   TOP(up(f(a))) -> TOP(f_flat(up(f(a))))
169.21/87.98	   TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a))))
169.21/87.98	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.98	   TOP(up(g(a))) -> TOP(g_flat(up(f(a))))
169.21/87.98	   TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b))))
169.21/87.98	
169.21/87.98	The TRS R consists of the following rules:
169.21/87.98	
169.21/87.98	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.98	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.98	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.98	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.98	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.98	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.98	   down(f(b)) -> f_flat(down(b))
169.21/87.98	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.98	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.98	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.98	   down(g(f(x))) -> up(f(g(x)))
169.21/87.98	   down(g(a)) -> g_flat(down(a))
169.21/87.98	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.98	   down(g(b)) -> g_flat(down(b))
169.21/87.98	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.98	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.98	   down(a) -> up(f(a))
169.21/87.98	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.98	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.98	   down(f(a)) -> f_flat(down(a))
169.21/87.98	
169.21/87.98	The set Q consists of the following terms:
169.21/87.98	
169.21/87.98	   down(a)
169.21/87.98	   down(g(f(x0)))
169.21/87.98	   down(f(f(f(f(f(x0))))))
169.21/87.98	   down(f(a))
169.21/87.98	   down(f(g(x0)))
169.21/87.98	   down(f(b))
169.21/87.98	   down(f(fresh_constant))
169.21/87.98	   down(g(a))
169.21/87.98	   down(g(g(x0)))
169.21/87.98	   down(g(b))
169.21/87.98	   down(g(fresh_constant))
169.21/87.98	   down(f(f(a)))
169.21/87.98	   down(f(f(g(x0))))
169.21/87.98	   down(f(f(b)))
169.21/87.98	   down(f(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a))))
169.21/87.98	   down(f(f(f(g(x0)))))
169.21/87.98	   down(f(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(x0))))))
169.21/87.98	   down(f(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x0))
169.21/87.98	   g_flat(up(x0))
169.21/87.98	
169.21/87.98	We have to consider all minimal (P,Q,R)-chains.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(90) DependencyGraphProof (EQUIVALENT)
169.21/87.98	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(91)
169.21/87.98	Obligation:
169.21/87.98	Q DP problem:
169.21/87.98	The TRS P consists of the following rules:
169.21/87.98	
169.21/87.98	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.98	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.98	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.98	   TOP(up(f(a))) -> TOP(f_flat(up(f(a))))
169.21/87.98	   TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant))))
169.21/87.98	   TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a)))))
169.21/87.98	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0))))))
169.21/87.98	   TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b)))))
169.21/87.98	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant)))))
169.21/87.98	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a))))))
169.21/87.98	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))
169.21/87.98	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))
169.21/87.98	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.98	   TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a))))
169.21/87.98	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.98	   TOP(up(g(a))) -> TOP(g_flat(up(f(a))))
169.21/87.98	
169.21/87.98	The TRS R consists of the following rules:
169.21/87.98	
169.21/87.98	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.98	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.98	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.98	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.98	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.98	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.98	   down(f(b)) -> f_flat(down(b))
169.21/87.98	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.98	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.98	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.98	   down(g(f(x))) -> up(f(g(x)))
169.21/87.98	   down(g(a)) -> g_flat(down(a))
169.21/87.98	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.98	   down(g(b)) -> g_flat(down(b))
169.21/87.98	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.98	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.98	   down(a) -> up(f(a))
169.21/87.98	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.98	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.98	   down(f(a)) -> f_flat(down(a))
169.21/87.98	
169.21/87.98	The set Q consists of the following terms:
169.21/87.98	
169.21/87.98	   down(a)
169.21/87.98	   down(g(f(x0)))
169.21/87.98	   down(f(f(f(f(f(x0))))))
169.21/87.98	   down(f(a))
169.21/87.98	   down(f(g(x0)))
169.21/87.98	   down(f(b))
169.21/87.98	   down(f(fresh_constant))
169.21/87.98	   down(g(a))
169.21/87.98	   down(g(g(x0)))
169.21/87.98	   down(g(b))
169.21/87.98	   down(g(fresh_constant))
169.21/87.98	   down(f(f(a)))
169.21/87.98	   down(f(f(g(x0))))
169.21/87.98	   down(f(f(b)))
169.21/87.98	   down(f(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a))))
169.21/87.98	   down(f(f(f(g(x0)))))
169.21/87.98	   down(f(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(x0))))))
169.21/87.98	   down(f(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x0))
169.21/87.98	   g_flat(up(x0))
169.21/87.98	
169.21/87.98	We have to consider all minimal (P,Q,R)-chains.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(92) TransformationProof (EQUIVALENT)
169.21/87.98	By rewriting [LPAR04] the rule TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) at position [0] we obtained the following new rules [LPAR04]:
169.21/87.98	
169.21/87.98	   (TOP(up(f(a))) -> TOP(up(f(f(a)))),TOP(up(f(a))) -> TOP(up(f(f(a)))))
169.21/87.98	
169.21/87.98	
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(93)
169.21/87.98	Obligation:
169.21/87.98	Q DP problem:
169.21/87.98	The TRS P consists of the following rules:
169.21/87.98	
169.21/87.98	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.98	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.98	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.98	   TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant))))
169.21/87.98	   TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a)))))
169.21/87.98	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0))))))
169.21/87.98	   TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b)))))
169.21/87.98	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant)))))
169.21/87.98	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a))))))
169.21/87.98	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))
169.21/87.98	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))
169.21/87.98	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.98	   TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a))))
169.21/87.98	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.98	   TOP(up(g(a))) -> TOP(g_flat(up(f(a))))
169.21/87.98	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.98	
169.21/87.98	The TRS R consists of the following rules:
169.21/87.98	
169.21/87.98	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.98	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.98	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.98	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.98	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.98	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.98	   down(f(b)) -> f_flat(down(b))
169.21/87.98	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.98	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.98	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.98	   down(g(f(x))) -> up(f(g(x)))
169.21/87.98	   down(g(a)) -> g_flat(down(a))
169.21/87.98	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.98	   down(g(b)) -> g_flat(down(b))
169.21/87.98	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.98	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.98	   down(a) -> up(f(a))
169.21/87.98	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.98	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.98	   down(f(a)) -> f_flat(down(a))
169.21/87.98	
169.21/87.98	The set Q consists of the following terms:
169.21/87.98	
169.21/87.98	   down(a)
169.21/87.98	   down(g(f(x0)))
169.21/87.98	   down(f(f(f(f(f(x0))))))
169.21/87.98	   down(f(a))
169.21/87.98	   down(f(g(x0)))
169.21/87.98	   down(f(b))
169.21/87.98	   down(f(fresh_constant))
169.21/87.98	   down(g(a))
169.21/87.98	   down(g(g(x0)))
169.21/87.98	   down(g(b))
169.21/87.98	   down(g(fresh_constant))
169.21/87.98	   down(f(f(a)))
169.21/87.98	   down(f(f(g(x0))))
169.21/87.98	   down(f(f(b)))
169.21/87.98	   down(f(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a))))
169.21/87.98	   down(f(f(f(g(x0)))))
169.21/87.98	   down(f(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(x0))))))
169.21/87.98	   down(f(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x0))
169.21/87.98	   g_flat(up(x0))
169.21/87.98	
169.21/87.98	We have to consider all minimal (P,Q,R)-chains.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(94) TransformationProof (EQUIVALENT)
169.21/87.98	By rewriting [LPAR04] the rule TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) at position [0,0] we obtained the following new rules [LPAR04]:
169.21/87.98	
169.21/87.98	   (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)))))
169.21/87.98	
169.21/87.98	
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(95)
169.21/87.98	Obligation:
169.21/87.98	Q DP problem:
169.21/87.98	The TRS P consists of the following rules:
169.21/87.98	
169.21/87.98	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.98	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.98	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.98	   TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a)))))
169.21/87.98	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0))))))
169.21/87.98	   TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b)))))
169.21/87.98	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant)))))
169.21/87.98	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a))))))
169.21/87.98	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))
169.21/87.98	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))
169.21/87.98	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.98	   TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a))))
169.21/87.98	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.98	   TOP(up(g(a))) -> TOP(g_flat(up(f(a))))
169.21/87.98	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.98	   TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(f_flat(down(fresh_constant))))
169.21/87.98	
169.21/87.98	The TRS R consists of the following rules:
169.21/87.98	
169.21/87.98	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.98	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.98	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.98	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.98	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.98	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.98	   down(f(b)) -> f_flat(down(b))
169.21/87.98	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.98	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.98	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.98	   down(g(f(x))) -> up(f(g(x)))
169.21/87.98	   down(g(a)) -> g_flat(down(a))
169.21/87.98	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.98	   down(g(b)) -> g_flat(down(b))
169.21/87.98	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.98	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.98	   down(a) -> up(f(a))
169.21/87.98	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.98	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.98	   down(f(a)) -> f_flat(down(a))
169.21/87.98	
169.21/87.98	The set Q consists of the following terms:
169.21/87.98	
169.21/87.98	   down(a)
169.21/87.98	   down(g(f(x0)))
169.21/87.98	   down(f(f(f(f(f(x0))))))
169.21/87.98	   down(f(a))
169.21/87.98	   down(f(g(x0)))
169.21/87.98	   down(f(b))
169.21/87.98	   down(f(fresh_constant))
169.21/87.98	   down(g(a))
169.21/87.98	   down(g(g(x0)))
169.21/87.98	   down(g(b))
169.21/87.98	   down(g(fresh_constant))
169.21/87.98	   down(f(f(a)))
169.21/87.98	   down(f(f(g(x0))))
169.21/87.98	   down(f(f(b)))
169.21/87.98	   down(f(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a))))
169.21/87.98	   down(f(f(f(g(x0)))))
169.21/87.98	   down(f(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(x0))))))
169.21/87.98	   down(f(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x0))
169.21/87.98	   g_flat(up(x0))
169.21/87.98	
169.21/87.98	We have to consider all minimal (P,Q,R)-chains.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(96) DependencyGraphProof (EQUIVALENT)
169.21/87.98	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(97)
169.21/87.98	Obligation:
169.21/87.98	Q DP problem:
169.21/87.98	The TRS P consists of the following rules:
169.21/87.98	
169.21/87.98	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.98	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.98	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.98	   TOP(up(g(a))) -> TOP(g_flat(up(f(a))))
169.21/87.98	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.98	   TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a))))
169.21/87.98	   TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a)))))
169.21/87.98	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0))))))
169.21/87.98	   TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b)))))
169.21/87.98	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant)))))
169.21/87.98	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a))))))
169.21/87.98	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))
169.21/87.98	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))
169.21/87.98	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.98	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.98	
169.21/87.98	The TRS R consists of the following rules:
169.21/87.98	
169.21/87.98	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.98	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.98	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.98	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.98	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.98	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.98	   down(f(b)) -> f_flat(down(b))
169.21/87.98	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.98	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.98	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.98	   down(g(f(x))) -> up(f(g(x)))
169.21/87.98	   down(g(a)) -> g_flat(down(a))
169.21/87.98	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.98	   down(g(b)) -> g_flat(down(b))
169.21/87.98	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.98	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.98	   down(a) -> up(f(a))
169.21/87.98	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.98	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.98	   down(f(a)) -> f_flat(down(a))
169.21/87.98	
169.21/87.98	The set Q consists of the following terms:
169.21/87.98	
169.21/87.98	   down(a)
169.21/87.98	   down(g(f(x0)))
169.21/87.98	   down(f(f(f(f(f(x0))))))
169.21/87.98	   down(f(a))
169.21/87.98	   down(f(g(x0)))
169.21/87.98	   down(f(b))
169.21/87.98	   down(f(fresh_constant))
169.21/87.98	   down(g(a))
169.21/87.98	   down(g(g(x0)))
169.21/87.98	   down(g(b))
169.21/87.98	   down(g(fresh_constant))
169.21/87.98	   down(f(f(a)))
169.21/87.98	   down(f(f(g(x0))))
169.21/87.98	   down(f(f(b)))
169.21/87.98	   down(f(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a))))
169.21/87.98	   down(f(f(f(g(x0)))))
169.21/87.98	   down(f(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(x0))))))
169.21/87.98	   down(f(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x0))
169.21/87.98	   g_flat(up(x0))
169.21/87.98	
169.21/87.98	We have to consider all minimal (P,Q,R)-chains.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(98) TransformationProof (EQUIVALENT)
169.21/87.98	By rewriting [LPAR04] the rule TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) at position [0] we obtained the following new rules [LPAR04]:
169.21/87.98	
169.21/87.98	   (TOP(up(g(a))) -> TOP(up(g(f(a)))),TOP(up(g(a))) -> TOP(up(g(f(a)))))
169.21/87.98	
169.21/87.98	
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(99)
169.21/87.98	Obligation:
169.21/87.98	Q DP problem:
169.21/87.98	The TRS P consists of the following rules:
169.21/87.98	
169.21/87.98	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.98	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.98	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.98	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.98	   TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a))))
169.21/87.98	   TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a)))))
169.21/87.98	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0))))))
169.21/87.98	   TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b)))))
169.21/87.98	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant)))))
169.21/87.98	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a))))))
169.21/87.98	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))
169.21/87.98	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))
169.21/87.98	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.98	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.98	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.98	
169.21/87.98	The TRS R consists of the following rules:
169.21/87.98	
169.21/87.98	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.98	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.98	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.98	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.98	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.98	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.98	   down(f(b)) -> f_flat(down(b))
169.21/87.98	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.98	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.98	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.98	   down(g(f(x))) -> up(f(g(x)))
169.21/87.98	   down(g(a)) -> g_flat(down(a))
169.21/87.98	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.98	   down(g(b)) -> g_flat(down(b))
169.21/87.98	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.98	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.98	   down(a) -> up(f(a))
169.21/87.98	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.98	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.98	   down(f(a)) -> f_flat(down(a))
169.21/87.98	
169.21/87.98	The set Q consists of the following terms:
169.21/87.98	
169.21/87.98	   down(a)
169.21/87.98	   down(g(f(x0)))
169.21/87.98	   down(f(f(f(f(f(x0))))))
169.21/87.98	   down(f(a))
169.21/87.98	   down(f(g(x0)))
169.21/87.98	   down(f(b))
169.21/87.98	   down(f(fresh_constant))
169.21/87.98	   down(g(a))
169.21/87.98	   down(g(g(x0)))
169.21/87.98	   down(g(b))
169.21/87.98	   down(g(fresh_constant))
169.21/87.98	   down(f(f(a)))
169.21/87.98	   down(f(f(g(x0))))
169.21/87.98	   down(f(f(b)))
169.21/87.98	   down(f(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a))))
169.21/87.98	   down(f(f(f(g(x0)))))
169.21/87.98	   down(f(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(x0))))))
169.21/87.98	   down(f(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x0))
169.21/87.98	   g_flat(up(x0))
169.21/87.98	
169.21/87.98	We have to consider all minimal (P,Q,R)-chains.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(100) TransformationProof (EQUIVALENT)
169.21/87.98	By rewriting [LPAR04] the rule TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) at position [0,0,0] we obtained the following new rules [LPAR04]:
169.21/87.98	
169.21/87.98	   (TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))),TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))))
169.21/87.98	
169.21/87.98	
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(101)
169.21/87.98	Obligation:
169.21/87.98	Q DP problem:
169.21/87.98	The TRS P consists of the following rules:
169.21/87.98	
169.21/87.98	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.98	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.98	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.98	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.98	   TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a)))))
169.21/87.98	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0))))))
169.21/87.98	   TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b)))))
169.21/87.98	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant)))))
169.21/87.98	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a))))))
169.21/87.98	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))
169.21/87.98	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))
169.21/87.98	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.98	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.98	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.98	   TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a)))))
169.21/87.98	
169.21/87.98	The TRS R consists of the following rules:
169.21/87.98	
169.21/87.98	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.98	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.98	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.98	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.98	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.98	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.98	   down(f(b)) -> f_flat(down(b))
169.21/87.98	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.98	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.98	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.98	   down(g(f(x))) -> up(f(g(x)))
169.21/87.98	   down(g(a)) -> g_flat(down(a))
169.21/87.98	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.98	   down(g(b)) -> g_flat(down(b))
169.21/87.98	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.98	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.98	   down(a) -> up(f(a))
169.21/87.98	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.98	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.98	   down(f(a)) -> f_flat(down(a))
169.21/87.98	
169.21/87.98	The set Q consists of the following terms:
169.21/87.98	
169.21/87.98	   down(a)
169.21/87.98	   down(g(f(x0)))
169.21/87.98	   down(f(f(f(f(f(x0))))))
169.21/87.98	   down(f(a))
169.21/87.98	   down(f(g(x0)))
169.21/87.98	   down(f(b))
169.21/87.98	   down(f(fresh_constant))
169.21/87.98	   down(g(a))
169.21/87.98	   down(g(g(x0)))
169.21/87.98	   down(g(b))
169.21/87.98	   down(g(fresh_constant))
169.21/87.98	   down(f(f(a)))
169.21/87.98	   down(f(f(g(x0))))
169.21/87.98	   down(f(f(b)))
169.21/87.98	   down(f(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a))))
169.21/87.98	   down(f(f(f(g(x0)))))
169.21/87.98	   down(f(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(x0))))))
169.21/87.98	   down(f(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x0))
169.21/87.98	   g_flat(up(x0))
169.21/87.98	
169.21/87.98	We have to consider all minimal (P,Q,R)-chains.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(102) TransformationProof (EQUIVALENT)
169.21/87.98	By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) at position [0,0] we obtained the following new rules [LPAR04]:
169.21/87.98	
169.21/87.98	   (TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))))
169.21/87.98	
169.21/87.98	
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(103)
169.21/87.98	Obligation:
169.21/87.98	Q DP problem:
169.21/87.98	The TRS P consists of the following rules:
169.21/87.98	
169.21/87.98	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.98	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.98	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.98	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.98	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0))))))
169.21/87.98	   TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b)))))
169.21/87.98	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant)))))
169.21/87.98	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a))))))
169.21/87.98	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))
169.21/87.98	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))
169.21/87.98	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.98	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.98	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.98	   TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a)))))
169.21/87.98	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a)))))
169.21/87.98	
169.21/87.98	The TRS R consists of the following rules:
169.21/87.98	
169.21/87.98	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.98	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.98	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.98	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.98	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.98	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.98	   down(f(b)) -> f_flat(down(b))
169.21/87.98	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.98	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.98	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.98	   down(g(f(x))) -> up(f(g(x)))
169.21/87.98	   down(g(a)) -> g_flat(down(a))
169.21/87.98	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.98	   down(g(b)) -> g_flat(down(b))
169.21/87.98	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.98	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.98	   down(a) -> up(f(a))
169.21/87.98	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.98	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.98	   down(f(a)) -> f_flat(down(a))
169.21/87.98	
169.21/87.98	The set Q consists of the following terms:
169.21/87.98	
169.21/87.98	   down(a)
169.21/87.98	   down(g(f(x0)))
169.21/87.98	   down(f(f(f(f(f(x0))))))
169.21/87.98	   down(f(a))
169.21/87.98	   down(f(g(x0)))
169.21/87.98	   down(f(b))
169.21/87.98	   down(f(fresh_constant))
169.21/87.98	   down(g(a))
169.21/87.98	   down(g(g(x0)))
169.21/87.98	   down(g(b))
169.21/87.98	   down(g(fresh_constant))
169.21/87.98	   down(f(f(a)))
169.21/87.98	   down(f(f(g(x0))))
169.21/87.98	   down(f(f(b)))
169.21/87.98	   down(f(f(fresh_constant)))
169.21/87.98	   down(f(f(f(a))))
169.21/87.98	   down(f(f(f(g(x0)))))
169.21/87.98	   down(f(f(f(b))))
169.21/87.98	   down(f(f(f(fresh_constant))))
169.21/87.98	   down(f(f(f(f(a)))))
169.21/87.98	   down(f(f(f(f(g(x0))))))
169.21/87.98	   down(f(f(f(f(b)))))
169.21/87.98	   down(f(f(f(f(fresh_constant)))))
169.21/87.98	   f_flat(up(x0))
169.21/87.98	   g_flat(up(x0))
169.21/87.98	
169.21/87.98	We have to consider all minimal (P,Q,R)-chains.
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(104) TransformationProof (EQUIVALENT)
169.21/87.98	By rewriting [LPAR04] the rule TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) at position [0,0] we obtained the following new rules [LPAR04]:
169.21/87.98	
169.21/87.98	   (TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))),TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))))
169.21/87.98	
169.21/87.98	
169.21/87.98	----------------------------------------
169.21/87.98	
169.21/87.98	(105)
169.21/87.98	Obligation:
169.21/87.98	Q DP problem:
169.21/87.98	The TRS P consists of the following rules:
169.21/87.98	
169.21/87.98	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.98	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.98	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.98	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.98	   TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b)))))
169.21/87.98	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant)))))
169.21/87.98	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a))))))
169.21/87.98	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))
169.21/87.98	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))
169.21/87.98	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.98	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.98	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.98	   TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a)))))
169.21/87.98	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a)))))
169.21/87.98	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0))))))
169.21/87.98	
169.21/87.98	The TRS R consists of the following rules:
169.21/87.98	
169.21/87.98	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.98	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.98	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.98	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.98	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.98	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.98	   down(f(b)) -> f_flat(down(b))
169.21/87.98	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.98	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.98	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.98	   down(g(f(x))) -> up(f(g(x)))
169.21/87.98	   down(g(a)) -> g_flat(down(a))
169.21/87.98	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.98	   down(g(b)) -> g_flat(down(b))
169.21/87.98	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.98	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.98	   down(a) -> up(f(a))
169.21/87.98	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.98	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.98	   down(f(a)) -> f_flat(down(a))
169.21/87.98	
169.21/87.98	The set Q consists of the following terms:
169.21/87.98	
169.21/87.98	   down(a)
169.21/87.98	   down(g(f(x0)))
169.21/87.98	   down(f(f(f(f(f(x0))))))
169.21/87.98	   down(f(a))
169.21/87.98	   down(f(g(x0)))
169.21/87.98	   down(f(b))
169.21/87.98	   down(f(fresh_constant))
169.21/87.98	   down(g(a))
169.21/87.98	   down(g(g(x0)))
169.21/87.98	   down(g(b))
169.21/87.98	   down(g(fresh_constant))
169.21/87.98	   down(f(f(a)))
169.21/87.98	   down(f(f(g(x0))))
169.21/87.98	   down(f(f(b)))
169.21/87.98	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(106) TransformationProof (EQUIVALENT)
169.21/87.99	By rewriting [LPAR04] the rule TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) at position [0,0] we obtained the following new rules [LPAR04]:
169.21/87.99	
169.21/87.99	   (TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))),TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))))
169.21/87.99	
169.21/87.99	
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(107)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant)))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a)))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a)))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0))))))
169.21/87.99	   TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b)))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.99	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.99	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(108) TransformationProof (EQUIVALENT)
169.21/87.99	By rewriting [LPAR04] the rule TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) at position [0,0] we obtained the following new rules [LPAR04]:
169.21/87.99	
169.21/87.99	   (TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))),TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))))
169.21/87.99	
169.21/87.99	
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(109)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a)))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a)))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0))))))
169.21/87.99	   TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b)))))
169.21/87.99	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant)))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.99	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.99	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(110) TransformationProof (EQUIVALENT)
169.21/87.99	By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) at position [0,0] we obtained the following new rules [LPAR04]:
169.21/87.99	
169.21/87.99	   (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))))
169.21/87.99	
169.21/87.99	
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(111)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a)))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a)))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0))))))
169.21/87.99	   TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b)))))
169.21/87.99	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant)))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a))))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.99	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(f(f(a)))) -> f_flat(down(f(f(a))))
169.21/87.99	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(112) UsableRulesProof (EQUIVALENT)
169.21/87.99	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.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(113)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a)))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a)))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0))))))
169.21/87.99	   TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b)))))
169.21/87.99	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant)))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a))))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.99	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(114) TransformationProof (EQUIVALENT)
169.21/87.99	By rewriting [LPAR04] the rule TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) at position [0,0] we obtained the following new rules [LPAR04]:
169.21/87.99	
169.21/87.99	   (TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))),TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))))
169.21/87.99	
169.21/87.99	
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(115)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a)))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a)))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0))))))
169.21/87.99	   TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b)))))
169.21/87.99	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant)))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0)))))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	   down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13)))))
169.21/87.99	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(116) UsableRulesProof (EQUIVALENT)
169.21/87.99	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.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(117)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a)))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a)))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0))))))
169.21/87.99	   TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b)))))
169.21/87.99	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant)))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0)))))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(118) TransformationProof (EQUIVALENT)
169.21/87.99	By rewriting [LPAR04] the rule TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) at position [0,0] we obtained the following new rules [LPAR04]:
169.21/87.99	
169.21/87.99	   (TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))),TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))))
169.21/87.99	
169.21/87.99	
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(119)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a)))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a)))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0))))))
169.21/87.99	   TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b)))))
169.21/87.99	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant)))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0)))))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b))))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   down(f(f(f(b)))) -> f_flat(down(f(f(b))))
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(120) UsableRulesProof (EQUIVALENT)
169.21/87.99	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.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(121)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a)))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a)))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0))))))
169.21/87.99	   TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b)))))
169.21/87.99	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant)))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0)))))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b))))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(122) TransformationProof (EQUIVALENT)
169.21/87.99	By rewriting [LPAR04] the rule TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) at position [0,0] we obtained the following new rules [LPAR04]:
169.21/87.99	
169.21/87.99	   (TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))),TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))))
169.21/87.99	
169.21/87.99	
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(123)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a)))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a)))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0))))))
169.21/87.99	   TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b)))))
169.21/87.99	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant)))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0)))))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b))))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant))))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant))))
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(124) UsableRulesProof (EQUIVALENT)
169.21/87.99	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.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(125)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a)))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a)))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0))))))
169.21/87.99	   TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b)))))
169.21/87.99	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant)))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0)))))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b))))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant))))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(126) TransformationProof (EQUIVALENT)
169.21/87.99	By rewriting [LPAR04] the rule TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) at position [0,0] we obtained the following new rules [LPAR04]:
169.21/87.99	
169.21/87.99	   (TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))),TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))))
169.21/87.99	
169.21/87.99	
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(127)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a)))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0))))))
169.21/87.99	   TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b)))))
169.21/87.99	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant)))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0)))))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b))))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a)))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(128) TransformationProof (EQUIVALENT)
169.21/87.99	By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) at position [0,0,0] we obtained the following new rules [LPAR04]:
169.21/87.99	
169.21/87.99	   (TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))))
169.21/87.99	
169.21/87.99	
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(129)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0))))))
169.21/87.99	   TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b)))))
169.21/87.99	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant)))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0)))))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b))))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a)))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a)))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(130) TransformationProof (EQUIVALENT)
169.21/87.99	By rewriting [LPAR04] the rule TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) at position [0,0,0] we obtained the following new rules [LPAR04]:
169.21/87.99	
169.21/87.99	   (TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))),TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))))
169.21/87.99	
169.21/87.99	
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(131)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b)))))
169.21/87.99	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant)))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0)))))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b))))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a)))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a)))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(132) TransformationProof (EQUIVALENT)
169.21/87.99	By rewriting [LPAR04] the rule TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) at position [0,0,0] we obtained the following new rules [LPAR04]:
169.21/87.99	
169.21/87.99	   (TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))),TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))))
169.21/87.99	
169.21/87.99	
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(133)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant)))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0)))))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b))))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a)))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a)))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/87.99	   TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b)))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(134) DependencyGraphProof (EQUIVALENT)
169.21/87.99	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(135)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a)))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant)))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0)))))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b))))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a)))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(136) TransformationProof (EQUIVALENT)
169.21/87.99	By rewriting [LPAR04] the rule TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) at position [0] we obtained the following new rules [LPAR04]:
169.21/87.99	
169.21/87.99	   (TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))),TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))))
169.21/87.99	
169.21/87.99	
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(137)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant)))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0)))))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b))))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a)))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(138) TransformationProof (EQUIVALENT)
169.21/87.99	By rewriting [LPAR04] the rule TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) at position [0,0,0] we obtained the following new rules [LPAR04]:
169.21/87.99	
169.21/87.99	   (TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(f_flat(down(fresh_constant))))),TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(f_flat(down(fresh_constant))))))
169.21/87.99	
169.21/87.99	
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(139)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0)))))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b))))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a)))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/87.99	   TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(f_flat(down(fresh_constant)))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(140) DependencyGraphProof (EQUIVALENT)
169.21/87.99	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(141)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a)))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0)))))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b))))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(142) TransformationProof (EQUIVALENT)
169.21/87.99	By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:
169.21/87.99	
169.21/87.99	   (TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))))
169.21/87.99	
169.21/87.99	
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(143)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0)))))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b))))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a))))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(144) TransformationProof (EQUIVALENT)
169.21/87.99	By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) at position [0,0,0] we obtained the following new rules [LPAR04]:
169.21/87.99	
169.21/87.99	   (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))))
169.21/87.99	
169.21/87.99	
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(145)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0)))))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b))))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a))))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(f(a))) -> f_flat(down(f(a)))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(146) UsableRulesProof (EQUIVALENT)
169.21/87.99	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.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(147)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0)))))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b))))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a))))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(148) TransformationProof (EQUIVALENT)
169.21/87.99	By rewriting [LPAR04] the rule TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) at position [0,0,0] we obtained the following new rules [LPAR04]:
169.21/87.99	
169.21/87.99	   (TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))),TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))))
169.21/87.99	
169.21/87.99	
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(149)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b))))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0)))))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   down(f(f(g(y10)))) -> f_flat(down(f(g(y10))))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(150) UsableRulesProof (EQUIVALENT)
169.21/87.99	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.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(151)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b))))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0)))))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(152) TransformationProof (EQUIVALENT)
169.21/87.99	By rewriting [LPAR04] the rule TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) at position [0,0,0] we obtained the following new rules [LPAR04]:
169.21/87.99	
169.21/87.99	   (TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))),TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))))
169.21/87.99	
169.21/87.99	
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(153)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0)))))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b))))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	   down(f(f(b))) -> f_flat(down(f(b)))
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(154) UsableRulesProof (EQUIVALENT)
169.21/87.99	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.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(155)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant))))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0)))))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b))))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(156) TransformationProof (EQUIVALENT)
169.21/87.99	By rewriting [LPAR04] the rule TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) at position [0,0,0] we obtained the following new rules [LPAR04]:
169.21/87.99	
169.21/87.99	   (TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))),TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))
169.21/87.99	
169.21/87.99	
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(157)
169.21/87.99	Obligation:
169.21/87.99	Q DP problem:
169.21/87.99	The TRS P consists of the following rules:
169.21/87.99	
169.21/87.99	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/87.99	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/87.99	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/87.99	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/87.99	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/87.99	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/87.99	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/87.99	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/87.99	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a))))))
169.21/87.99	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0)))))))
169.21/87.99	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b))))))
169.21/87.99	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant))))))
169.21/87.99	
169.21/87.99	The TRS R consists of the following rules:
169.21/87.99	
169.21/87.99	   down(f(b)) -> f_flat(down(b))
169.21/87.99	   f_flat(up(x_1)) -> up(f(x_1))
169.21/87.99	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/87.99	   down(g(f(x))) -> up(f(g(x)))
169.21/87.99	   down(g(a)) -> g_flat(down(a))
169.21/87.99	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/87.99	   down(g(b)) -> g_flat(down(b))
169.21/87.99	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/87.99	   g_flat(up(x_1)) -> up(g(x_1))
169.21/87.99	   down(a) -> up(f(a))
169.21/87.99	   down(f(a)) -> f_flat(down(a))
169.21/87.99	   down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant)))
169.21/87.99	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/87.99	
169.21/87.99	The set Q consists of the following terms:
169.21/87.99	
169.21/87.99	   down(a)
169.21/87.99	   down(g(f(x0)))
169.21/87.99	   down(f(f(f(f(f(x0))))))
169.21/87.99	   down(f(a))
169.21/87.99	   down(f(g(x0)))
169.21/87.99	   down(f(b))
169.21/87.99	   down(f(fresh_constant))
169.21/87.99	   down(g(a))
169.21/87.99	   down(g(g(x0)))
169.21/87.99	   down(g(b))
169.21/87.99	   down(g(fresh_constant))
169.21/87.99	   down(f(f(a)))
169.21/87.99	   down(f(f(g(x0))))
169.21/87.99	   down(f(f(b)))
169.21/87.99	   down(f(f(fresh_constant)))
169.21/87.99	   down(f(f(f(a))))
169.21/87.99	   down(f(f(f(g(x0)))))
169.21/87.99	   down(f(f(f(b))))
169.21/87.99	   down(f(f(f(fresh_constant))))
169.21/87.99	   down(f(f(f(f(a)))))
169.21/87.99	   down(f(f(f(f(g(x0))))))
169.21/87.99	   down(f(f(f(f(b)))))
169.21/87.99	   down(f(f(f(f(fresh_constant)))))
169.21/87.99	   f_flat(up(x0))
169.21/87.99	   g_flat(up(x0))
169.21/87.99	
169.21/87.99	We have to consider all minimal (P,Q,R)-chains.
169.21/87.99	----------------------------------------
169.21/87.99	
169.21/87.99	(158) UsableRulesProof (EQUIVALENT)
169.21/87.99	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.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(159)
169.21/88.00	Obligation:
169.21/88.00	Q DP problem:
169.21/88.00	The TRS P consists of the following rules:
169.21/88.00	
169.21/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/88.00	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/88.00	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/88.00	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/88.00	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/88.00	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a))))))
169.21/88.00	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a))))))
169.21/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0)))))))
169.21/88.00	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b))))))
169.21/88.00	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant))))))
169.21/88.00	
169.21/88.00	The TRS R consists of the following rules:
169.21/88.00	
169.21/88.00	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.21/88.00	   down(f(b)) -> f_flat(down(b))
169.21/88.00	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/88.00	   down(g(f(x))) -> up(f(g(x)))
169.21/88.00	   down(g(a)) -> g_flat(down(a))
169.21/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/88.00	   down(g(b)) -> g_flat(down(b))
169.21/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.21/88.00	   down(a) -> up(f(a))
169.21/88.00	   down(f(a)) -> f_flat(down(a))
169.21/88.00	
169.21/88.00	The set Q consists of the following terms:
169.21/88.00	
169.21/88.00	   down(a)
169.21/88.00	   down(g(f(x0)))
169.21/88.00	   down(f(f(f(f(f(x0))))))
169.21/88.00	   down(f(a))
169.21/88.00	   down(f(g(x0)))
169.21/88.00	   down(f(b))
169.21/88.00	   down(f(fresh_constant))
169.21/88.00	   down(g(a))
169.21/88.00	   down(g(g(x0)))
169.21/88.00	   down(g(b))
169.21/88.00	   down(g(fresh_constant))
169.21/88.00	   down(f(f(a)))
169.21/88.00	   down(f(f(g(x0))))
169.21/88.00	   down(f(f(b)))
169.21/88.00	   down(f(f(fresh_constant)))
169.21/88.00	   down(f(f(f(a))))
169.21/88.00	   down(f(f(f(g(x0)))))
169.21/88.00	   down(f(f(f(b))))
169.21/88.00	   down(f(f(f(fresh_constant))))
169.21/88.00	   down(f(f(f(f(a)))))
169.21/88.00	   down(f(f(f(f(g(x0))))))
169.21/88.00	   down(f(f(f(f(b)))))
169.21/88.00	   down(f(f(f(f(fresh_constant)))))
169.21/88.00	   f_flat(up(x0))
169.21/88.00	   g_flat(up(x0))
169.21/88.00	
169.21/88.00	We have to consider all minimal (P,Q,R)-chains.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(160) TransformationProof (EQUIVALENT)
169.21/88.00	By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) at position [0,0,0] we obtained the following new rules [LPAR04]:
169.21/88.00	
169.21/88.00	   (TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))))
169.21/88.00	
169.21/88.00	
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(161)
169.21/88.00	Obligation:
169.21/88.00	Q DP problem:
169.21/88.00	The TRS P consists of the following rules:
169.21/88.00	
169.21/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/88.00	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/88.00	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/88.00	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/88.00	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/88.00	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a))))))
169.21/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0)))))))
169.21/88.00	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b))))))
169.21/88.00	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant))))))
169.21/88.00	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a))))))
169.21/88.00	
169.21/88.00	The TRS R consists of the following rules:
169.21/88.00	
169.21/88.00	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.21/88.00	   down(f(b)) -> f_flat(down(b))
169.21/88.00	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/88.00	   down(g(f(x))) -> up(f(g(x)))
169.21/88.00	   down(g(a)) -> g_flat(down(a))
169.21/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/88.00	   down(g(b)) -> g_flat(down(b))
169.21/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.21/88.00	   down(a) -> up(f(a))
169.21/88.00	   down(f(a)) -> f_flat(down(a))
169.21/88.00	
169.21/88.00	The set Q consists of the following terms:
169.21/88.00	
169.21/88.00	   down(a)
169.21/88.00	   down(g(f(x0)))
169.21/88.00	   down(f(f(f(f(f(x0))))))
169.21/88.00	   down(f(a))
169.21/88.00	   down(f(g(x0)))
169.21/88.00	   down(f(b))
169.21/88.00	   down(f(fresh_constant))
169.21/88.00	   down(g(a))
169.21/88.00	   down(g(g(x0)))
169.21/88.00	   down(g(b))
169.21/88.00	   down(g(fresh_constant))
169.21/88.00	   down(f(f(a)))
169.21/88.00	   down(f(f(g(x0))))
169.21/88.00	   down(f(f(b)))
169.21/88.00	   down(f(f(fresh_constant)))
169.21/88.00	   down(f(f(f(a))))
169.21/88.00	   down(f(f(f(g(x0)))))
169.21/88.00	   down(f(f(f(b))))
169.21/88.00	   down(f(f(f(fresh_constant))))
169.21/88.00	   down(f(f(f(f(a)))))
169.21/88.00	   down(f(f(f(f(g(x0))))))
169.21/88.00	   down(f(f(f(f(b)))))
169.21/88.00	   down(f(f(f(f(fresh_constant)))))
169.21/88.00	   f_flat(up(x0))
169.21/88.00	   g_flat(up(x0))
169.21/88.00	
169.21/88.00	We have to consider all minimal (P,Q,R)-chains.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(162) TransformationProof (EQUIVALENT)
169.21/88.00	By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:
169.21/88.00	
169.21/88.00	   (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))))
169.21/88.00	
169.21/88.00	
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(163)
169.21/88.00	Obligation:
169.21/88.00	Q DP problem:
169.21/88.00	The TRS P consists of the following rules:
169.21/88.00	
169.21/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/88.00	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/88.00	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/88.00	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/88.00	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0)))))))
169.21/88.00	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b))))))
169.21/88.00	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant))))))
169.21/88.00	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a))))))
169.21/88.00	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a))))))
169.21/88.00	
169.21/88.00	The TRS R consists of the following rules:
169.21/88.00	
169.21/88.00	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.21/88.00	   down(f(b)) -> f_flat(down(b))
169.21/88.00	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/88.00	   down(g(f(x))) -> up(f(g(x)))
169.21/88.00	   down(g(a)) -> g_flat(down(a))
169.21/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/88.00	   down(g(b)) -> g_flat(down(b))
169.21/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.21/88.00	   down(a) -> up(f(a))
169.21/88.00	   down(f(a)) -> f_flat(down(a))
169.21/88.00	
169.21/88.00	The set Q consists of the following terms:
169.21/88.00	
169.21/88.00	   down(a)
169.21/88.00	   down(g(f(x0)))
169.21/88.00	   down(f(f(f(f(f(x0))))))
169.21/88.00	   down(f(a))
169.21/88.00	   down(f(g(x0)))
169.21/88.00	   down(f(b))
169.21/88.00	   down(f(fresh_constant))
169.21/88.00	   down(g(a))
169.21/88.00	   down(g(g(x0)))
169.21/88.00	   down(g(b))
169.21/88.00	   down(g(fresh_constant))
169.21/88.00	   down(f(f(a)))
169.21/88.00	   down(f(f(g(x0))))
169.21/88.00	   down(f(f(b)))
169.21/88.00	   down(f(f(fresh_constant)))
169.21/88.00	   down(f(f(f(a))))
169.21/88.00	   down(f(f(f(g(x0)))))
169.21/88.00	   down(f(f(f(b))))
169.21/88.00	   down(f(f(f(fresh_constant))))
169.21/88.00	   down(f(f(f(f(a)))))
169.21/88.00	   down(f(f(f(f(g(x0))))))
169.21/88.00	   down(f(f(f(f(b)))))
169.21/88.00	   down(f(f(f(f(fresh_constant)))))
169.21/88.00	   f_flat(up(x0))
169.21/88.00	   g_flat(up(x0))
169.21/88.00	
169.21/88.00	We have to consider all minimal (P,Q,R)-chains.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(164) UsableRulesProof (EQUIVALENT)
169.21/88.00	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.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(165)
169.21/88.00	Obligation:
169.21/88.00	Q DP problem:
169.21/88.00	The TRS P consists of the following rules:
169.21/88.00	
169.21/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/88.00	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/88.00	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/88.00	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/88.00	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0)))))))
169.21/88.00	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b))))))
169.21/88.00	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant))))))
169.21/88.00	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a))))))
169.21/88.00	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a))))))
169.21/88.00	
169.21/88.00	The TRS R consists of the following rules:
169.21/88.00	
169.21/88.00	   down(a) -> up(f(a))
169.21/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.21/88.00	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/88.00	   down(f(b)) -> f_flat(down(b))
169.21/88.00	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/88.00	   down(g(f(x))) -> up(f(g(x)))
169.21/88.00	   down(g(a)) -> g_flat(down(a))
169.21/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/88.00	   down(g(b)) -> g_flat(down(b))
169.21/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.21/88.00	
169.21/88.00	The set Q consists of the following terms:
169.21/88.00	
169.21/88.00	   down(a)
169.21/88.00	   down(g(f(x0)))
169.21/88.00	   down(f(f(f(f(f(x0))))))
169.21/88.00	   down(f(a))
169.21/88.00	   down(f(g(x0)))
169.21/88.00	   down(f(b))
169.21/88.00	   down(f(fresh_constant))
169.21/88.00	   down(g(a))
169.21/88.00	   down(g(g(x0)))
169.21/88.00	   down(g(b))
169.21/88.00	   down(g(fresh_constant))
169.21/88.00	   down(f(f(a)))
169.21/88.00	   down(f(f(g(x0))))
169.21/88.00	   down(f(f(b)))
169.21/88.00	   down(f(f(fresh_constant)))
169.21/88.00	   down(f(f(f(a))))
169.21/88.00	   down(f(f(f(g(x0)))))
169.21/88.00	   down(f(f(f(b))))
169.21/88.00	   down(f(f(f(fresh_constant))))
169.21/88.00	   down(f(f(f(f(a)))))
169.21/88.00	   down(f(f(f(f(g(x0))))))
169.21/88.00	   down(f(f(f(f(b)))))
169.21/88.00	   down(f(f(f(f(fresh_constant)))))
169.21/88.00	   f_flat(up(x0))
169.21/88.00	   g_flat(up(x0))
169.21/88.00	
169.21/88.00	We have to consider all minimal (P,Q,R)-chains.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(166) TransformationProof (EQUIVALENT)
169.21/88.00	By rewriting [LPAR04] the rule TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:
169.21/88.00	
169.21/88.00	   (TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))),TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))
169.21/88.00	
169.21/88.00	
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(167)
169.21/88.00	Obligation:
169.21/88.00	Q DP problem:
169.21/88.00	The TRS P consists of the following rules:
169.21/88.00	
169.21/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/88.00	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/88.00	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/88.00	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/88.00	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/88.00	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b))))))
169.21/88.00	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant))))))
169.21/88.00	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a))))))
169.21/88.00	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a))))))
169.21/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.21/88.00	
169.21/88.00	The TRS R consists of the following rules:
169.21/88.00	
169.21/88.00	   down(a) -> up(f(a))
169.21/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.21/88.00	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/88.00	   down(f(b)) -> f_flat(down(b))
169.21/88.00	   down(f(g(y4))) -> f_flat(down(g(y4)))
169.21/88.00	   down(g(f(x))) -> up(f(g(x)))
169.21/88.00	   down(g(a)) -> g_flat(down(a))
169.21/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/88.00	   down(g(b)) -> g_flat(down(b))
169.21/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.21/88.00	
169.21/88.00	The set Q consists of the following terms:
169.21/88.00	
169.21/88.00	   down(a)
169.21/88.00	   down(g(f(x0)))
169.21/88.00	   down(f(f(f(f(f(x0))))))
169.21/88.00	   down(f(a))
169.21/88.00	   down(f(g(x0)))
169.21/88.00	   down(f(b))
169.21/88.00	   down(f(fresh_constant))
169.21/88.00	   down(g(a))
169.21/88.00	   down(g(g(x0)))
169.21/88.00	   down(g(b))
169.21/88.00	   down(g(fresh_constant))
169.21/88.00	   down(f(f(a)))
169.21/88.00	   down(f(f(g(x0))))
169.21/88.00	   down(f(f(b)))
169.21/88.00	   down(f(f(fresh_constant)))
169.21/88.00	   down(f(f(f(a))))
169.21/88.00	   down(f(f(f(g(x0)))))
169.21/88.00	   down(f(f(f(b))))
169.21/88.00	   down(f(f(f(fresh_constant))))
169.21/88.00	   down(f(f(f(f(a)))))
169.21/88.00	   down(f(f(f(f(g(x0))))))
169.21/88.00	   down(f(f(f(f(b)))))
169.21/88.00	   down(f(f(f(f(fresh_constant)))))
169.21/88.00	   f_flat(up(x0))
169.21/88.00	   g_flat(up(x0))
169.21/88.00	
169.21/88.00	We have to consider all minimal (P,Q,R)-chains.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(168) UsableRulesProof (EQUIVALENT)
169.21/88.00	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.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(169)
169.21/88.00	Obligation:
169.21/88.00	Q DP problem:
169.21/88.00	The TRS P consists of the following rules:
169.21/88.00	
169.21/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/88.00	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/88.00	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/88.00	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/88.00	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/88.00	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b))))))
169.21/88.00	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant))))))
169.21/88.00	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a))))))
169.21/88.00	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a))))))
169.21/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.21/88.00	
169.21/88.00	The TRS R consists of the following rules:
169.21/88.00	
169.21/88.00	   down(g(f(x))) -> up(f(g(x)))
169.21/88.00	   down(g(a)) -> g_flat(down(a))
169.21/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/88.00	   down(g(b)) -> g_flat(down(b))
169.21/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.21/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.21/88.00	   down(a) -> up(f(a))
169.21/88.00	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/88.00	   down(f(b)) -> f_flat(down(b))
169.21/88.00	
169.21/88.00	The set Q consists of the following terms:
169.21/88.00	
169.21/88.00	   down(a)
169.21/88.00	   down(g(f(x0)))
169.21/88.00	   down(f(f(f(f(f(x0))))))
169.21/88.00	   down(f(a))
169.21/88.00	   down(f(g(x0)))
169.21/88.00	   down(f(b))
169.21/88.00	   down(f(fresh_constant))
169.21/88.00	   down(g(a))
169.21/88.00	   down(g(g(x0)))
169.21/88.00	   down(g(b))
169.21/88.00	   down(g(fresh_constant))
169.21/88.00	   down(f(f(a)))
169.21/88.00	   down(f(f(g(x0))))
169.21/88.00	   down(f(f(b)))
169.21/88.00	   down(f(f(fresh_constant)))
169.21/88.00	   down(f(f(f(a))))
169.21/88.00	   down(f(f(f(g(x0)))))
169.21/88.00	   down(f(f(f(b))))
169.21/88.00	   down(f(f(f(fresh_constant))))
169.21/88.00	   down(f(f(f(f(a)))))
169.21/88.00	   down(f(f(f(f(g(x0))))))
169.21/88.00	   down(f(f(f(f(b)))))
169.21/88.00	   down(f(f(f(f(fresh_constant)))))
169.21/88.00	   f_flat(up(x0))
169.21/88.00	   g_flat(up(x0))
169.21/88.00	
169.21/88.00	We have to consider all minimal (P,Q,R)-chains.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(170) TransformationProof (EQUIVALENT)
169.21/88.00	By rewriting [LPAR04] the rule TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:
169.21/88.00	
169.21/88.00	   (TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))),TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))))
169.21/88.00	
169.21/88.00	
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(171)
169.21/88.00	Obligation:
169.21/88.00	Q DP problem:
169.21/88.00	The TRS P consists of the following rules:
169.21/88.00	
169.21/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/88.00	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/88.00	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/88.00	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/88.00	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/88.00	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant))))))
169.21/88.00	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a))))))
169.21/88.00	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a))))))
169.21/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.21/88.00	   TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b))))))
169.21/88.00	
169.21/88.00	The TRS R consists of the following rules:
169.21/88.00	
169.21/88.00	   down(g(f(x))) -> up(f(g(x)))
169.21/88.00	   down(g(a)) -> g_flat(down(a))
169.21/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/88.00	   down(g(b)) -> g_flat(down(b))
169.21/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.21/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.21/88.00	   down(a) -> up(f(a))
169.21/88.00	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/88.00	   down(f(b)) -> f_flat(down(b))
169.21/88.00	
169.21/88.00	The set Q consists of the following terms:
169.21/88.00	
169.21/88.00	   down(a)
169.21/88.00	   down(g(f(x0)))
169.21/88.00	   down(f(f(f(f(f(x0))))))
169.21/88.00	   down(f(a))
169.21/88.00	   down(f(g(x0)))
169.21/88.00	   down(f(b))
169.21/88.00	   down(f(fresh_constant))
169.21/88.00	   down(g(a))
169.21/88.00	   down(g(g(x0)))
169.21/88.00	   down(g(b))
169.21/88.00	   down(g(fresh_constant))
169.21/88.00	   down(f(f(a)))
169.21/88.00	   down(f(f(g(x0))))
169.21/88.00	   down(f(f(b)))
169.21/88.00	   down(f(f(fresh_constant)))
169.21/88.00	   down(f(f(f(a))))
169.21/88.00	   down(f(f(f(g(x0)))))
169.21/88.00	   down(f(f(f(b))))
169.21/88.00	   down(f(f(f(fresh_constant))))
169.21/88.00	   down(f(f(f(f(a)))))
169.21/88.00	   down(f(f(f(f(g(x0))))))
169.21/88.00	   down(f(f(f(f(b)))))
169.21/88.00	   down(f(f(f(f(fresh_constant)))))
169.21/88.00	   f_flat(up(x0))
169.21/88.00	   g_flat(up(x0))
169.21/88.00	
169.21/88.00	We have to consider all minimal (P,Q,R)-chains.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(172) DependencyGraphProof (EQUIVALENT)
169.21/88.00	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(173)
169.21/88.00	Obligation:
169.21/88.00	Q DP problem:
169.21/88.00	The TRS P consists of the following rules:
169.21/88.00	
169.21/88.00	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/88.00	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/88.00	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/88.00	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a))))))
169.21/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/88.00	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/88.00	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant))))))
169.21/88.00	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a))))))
169.21/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.21/88.00	
169.21/88.00	The TRS R consists of the following rules:
169.21/88.00	
169.21/88.00	   down(g(f(x))) -> up(f(g(x)))
169.21/88.00	   down(g(a)) -> g_flat(down(a))
169.21/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/88.00	   down(g(b)) -> g_flat(down(b))
169.21/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.21/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.21/88.00	   down(a) -> up(f(a))
169.21/88.00	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/88.00	   down(f(b)) -> f_flat(down(b))
169.21/88.00	
169.21/88.00	The set Q consists of the following terms:
169.21/88.00	
169.21/88.00	   down(a)
169.21/88.00	   down(g(f(x0)))
169.21/88.00	   down(f(f(f(f(f(x0))))))
169.21/88.00	   down(f(a))
169.21/88.00	   down(f(g(x0)))
169.21/88.00	   down(f(b))
169.21/88.00	   down(f(fresh_constant))
169.21/88.00	   down(g(a))
169.21/88.00	   down(g(g(x0)))
169.21/88.00	   down(g(b))
169.21/88.00	   down(g(fresh_constant))
169.21/88.00	   down(f(f(a)))
169.21/88.00	   down(f(f(g(x0))))
169.21/88.00	   down(f(f(b)))
169.21/88.00	   down(f(f(fresh_constant)))
169.21/88.00	   down(f(f(f(a))))
169.21/88.00	   down(f(f(f(g(x0)))))
169.21/88.00	   down(f(f(f(b))))
169.21/88.00	   down(f(f(f(fresh_constant))))
169.21/88.00	   down(f(f(f(f(a)))))
169.21/88.00	   down(f(f(f(f(g(x0))))))
169.21/88.00	   down(f(f(f(f(b)))))
169.21/88.00	   down(f(f(f(f(fresh_constant)))))
169.21/88.00	   f_flat(up(x0))
169.21/88.00	   g_flat(up(x0))
169.21/88.00	
169.21/88.00	We have to consider all minimal (P,Q,R)-chains.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(174) UsableRulesProof (EQUIVALENT)
169.21/88.00	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.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(175)
169.21/88.00	Obligation:
169.21/88.00	Q DP problem:
169.21/88.00	The TRS P consists of the following rules:
169.21/88.00	
169.21/88.00	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/88.00	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/88.00	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/88.00	   TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a))))))
169.21/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/88.00	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/88.00	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant))))))
169.21/88.00	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a))))))
169.21/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.21/88.00	
169.21/88.00	The TRS R consists of the following rules:
169.21/88.00	
169.21/88.00	   down(g(f(x))) -> up(f(g(x)))
169.21/88.00	   down(g(a)) -> g_flat(down(a))
169.21/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/88.00	   down(g(b)) -> g_flat(down(b))
169.21/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.21/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.21/88.00	   down(a) -> up(f(a))
169.21/88.00	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/88.00	
169.21/88.00	The set Q consists of the following terms:
169.21/88.00	
169.21/88.00	   down(a)
169.21/88.00	   down(g(f(x0)))
169.21/88.00	   down(f(f(f(f(f(x0))))))
169.21/88.00	   down(f(a))
169.21/88.00	   down(f(g(x0)))
169.21/88.00	   down(f(b))
169.21/88.00	   down(f(fresh_constant))
169.21/88.00	   down(g(a))
169.21/88.00	   down(g(g(x0)))
169.21/88.00	   down(g(b))
169.21/88.00	   down(g(fresh_constant))
169.21/88.00	   down(f(f(a)))
169.21/88.00	   down(f(f(g(x0))))
169.21/88.00	   down(f(f(b)))
169.21/88.00	   down(f(f(fresh_constant)))
169.21/88.00	   down(f(f(f(a))))
169.21/88.00	   down(f(f(f(g(x0)))))
169.21/88.00	   down(f(f(f(b))))
169.21/88.00	   down(f(f(f(fresh_constant))))
169.21/88.00	   down(f(f(f(f(a)))))
169.21/88.00	   down(f(f(f(f(g(x0))))))
169.21/88.00	   down(f(f(f(f(b)))))
169.21/88.00	   down(f(f(f(f(fresh_constant)))))
169.21/88.00	   f_flat(up(x0))
169.21/88.00	   g_flat(up(x0))
169.21/88.00	
169.21/88.00	We have to consider all minimal (P,Q,R)-chains.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(176) TransformationProof (EQUIVALENT)
169.21/88.00	By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) at position [0,0] we obtained the following new rules [LPAR04]:
169.21/88.00	
169.21/88.00	   (TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a)))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a)))))))
169.21/88.00	
169.21/88.00	
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(177)
169.21/88.00	Obligation:
169.21/88.00	Q DP problem:
169.21/88.00	The TRS P consists of the following rules:
169.21/88.00	
169.21/88.00	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/88.00	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/88.00	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/88.00	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/88.00	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant))))))
169.21/88.00	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a))))))
169.21/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.21/88.00	   TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a))))))
169.21/88.00	
169.21/88.00	The TRS R consists of the following rules:
169.21/88.00	
169.21/88.00	   down(g(f(x))) -> up(f(g(x)))
169.21/88.00	   down(g(a)) -> g_flat(down(a))
169.21/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/88.00	   down(g(b)) -> g_flat(down(b))
169.21/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.21/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.21/88.00	   down(a) -> up(f(a))
169.21/88.00	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/88.00	
169.21/88.00	The set Q consists of the following terms:
169.21/88.00	
169.21/88.00	   down(a)
169.21/88.00	   down(g(f(x0)))
169.21/88.00	   down(f(f(f(f(f(x0))))))
169.21/88.00	   down(f(a))
169.21/88.00	   down(f(g(x0)))
169.21/88.00	   down(f(b))
169.21/88.00	   down(f(fresh_constant))
169.21/88.00	   down(g(a))
169.21/88.00	   down(g(g(x0)))
169.21/88.00	   down(g(b))
169.21/88.00	   down(g(fresh_constant))
169.21/88.00	   down(f(f(a)))
169.21/88.00	   down(f(f(g(x0))))
169.21/88.00	   down(f(f(b)))
169.21/88.00	   down(f(f(fresh_constant)))
169.21/88.00	   down(f(f(f(a))))
169.21/88.00	   down(f(f(f(g(x0)))))
169.21/88.00	   down(f(f(f(b))))
169.21/88.00	   down(f(f(f(fresh_constant))))
169.21/88.00	   down(f(f(f(f(a)))))
169.21/88.00	   down(f(f(f(f(g(x0))))))
169.21/88.00	   down(f(f(f(f(b)))))
169.21/88.00	   down(f(f(f(f(fresh_constant)))))
169.21/88.00	   f_flat(up(x0))
169.21/88.00	   g_flat(up(x0))
169.21/88.00	
169.21/88.00	We have to consider all minimal (P,Q,R)-chains.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(178) TransformationProof (EQUIVALENT)
169.21/88.00	By rewriting [LPAR04] the rule TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:
169.21/88.00	
169.21/88.00	   (TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(fresh_constant)))))),TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(fresh_constant)))))))
169.21/88.00	
169.21/88.00	
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(179)
169.21/88.00	Obligation:
169.21/88.00	Q DP problem:
169.21/88.00	The TRS P consists of the following rules:
169.21/88.00	
169.21/88.00	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/88.00	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/88.00	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/88.00	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/88.00	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a))))))
169.21/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.21/88.00	   TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a))))))
169.21/88.00	   TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(fresh_constant))))))
169.21/88.00	
169.21/88.00	The TRS R consists of the following rules:
169.21/88.00	
169.21/88.00	   down(g(f(x))) -> up(f(g(x)))
169.21/88.00	   down(g(a)) -> g_flat(down(a))
169.21/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/88.00	   down(g(b)) -> g_flat(down(b))
169.21/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.21/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.21/88.00	   down(a) -> up(f(a))
169.21/88.00	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/88.00	
169.21/88.00	The set Q consists of the following terms:
169.21/88.00	
169.21/88.00	   down(a)
169.21/88.00	   down(g(f(x0)))
169.21/88.00	   down(f(f(f(f(f(x0))))))
169.21/88.00	   down(f(a))
169.21/88.00	   down(f(g(x0)))
169.21/88.00	   down(f(b))
169.21/88.00	   down(f(fresh_constant))
169.21/88.00	   down(g(a))
169.21/88.00	   down(g(g(x0)))
169.21/88.00	   down(g(b))
169.21/88.00	   down(g(fresh_constant))
169.21/88.00	   down(f(f(a)))
169.21/88.00	   down(f(f(g(x0))))
169.21/88.00	   down(f(f(b)))
169.21/88.00	   down(f(f(fresh_constant)))
169.21/88.00	   down(f(f(f(a))))
169.21/88.00	   down(f(f(f(g(x0)))))
169.21/88.00	   down(f(f(f(b))))
169.21/88.00	   down(f(f(f(fresh_constant))))
169.21/88.00	   down(f(f(f(f(a)))))
169.21/88.00	   down(f(f(f(f(g(x0))))))
169.21/88.00	   down(f(f(f(f(b)))))
169.21/88.00	   down(f(f(f(f(fresh_constant)))))
169.21/88.00	   f_flat(up(x0))
169.21/88.00	   g_flat(up(x0))
169.21/88.00	
169.21/88.00	We have to consider all minimal (P,Q,R)-chains.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(180) DependencyGraphProof (EQUIVALENT)
169.21/88.00	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(181)
169.21/88.00	Obligation:
169.21/88.00	Q DP problem:
169.21/88.00	The TRS P consists of the following rules:
169.21/88.00	
169.21/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/88.00	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/88.00	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/88.00	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/88.00	   TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a))))))
169.21/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/88.00	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/88.00	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a))))))
169.21/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.21/88.00	
169.21/88.00	The TRS R consists of the following rules:
169.21/88.00	
169.21/88.00	   down(g(f(x))) -> up(f(g(x)))
169.21/88.00	   down(g(a)) -> g_flat(down(a))
169.21/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/88.00	   down(g(b)) -> g_flat(down(b))
169.21/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.21/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.21/88.00	   down(a) -> up(f(a))
169.21/88.00	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
169.21/88.00	
169.21/88.00	The set Q consists of the following terms:
169.21/88.00	
169.21/88.00	   down(a)
169.21/88.00	   down(g(f(x0)))
169.21/88.00	   down(f(f(f(f(f(x0))))))
169.21/88.00	   down(f(a))
169.21/88.00	   down(f(g(x0)))
169.21/88.00	   down(f(b))
169.21/88.00	   down(f(fresh_constant))
169.21/88.00	   down(g(a))
169.21/88.00	   down(g(g(x0)))
169.21/88.00	   down(g(b))
169.21/88.00	   down(g(fresh_constant))
169.21/88.00	   down(f(f(a)))
169.21/88.00	   down(f(f(g(x0))))
169.21/88.00	   down(f(f(b)))
169.21/88.00	   down(f(f(fresh_constant)))
169.21/88.00	   down(f(f(f(a))))
169.21/88.00	   down(f(f(f(g(x0)))))
169.21/88.00	   down(f(f(f(b))))
169.21/88.00	   down(f(f(f(fresh_constant))))
169.21/88.00	   down(f(f(f(f(a)))))
169.21/88.00	   down(f(f(f(f(g(x0))))))
169.21/88.00	   down(f(f(f(f(b)))))
169.21/88.00	   down(f(f(f(f(fresh_constant)))))
169.21/88.00	   f_flat(up(x0))
169.21/88.00	   g_flat(up(x0))
169.21/88.00	
169.21/88.00	We have to consider all minimal (P,Q,R)-chains.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(182) UsableRulesProof (EQUIVALENT)
169.21/88.00	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.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(183)
169.21/88.00	Obligation:
169.21/88.00	Q DP problem:
169.21/88.00	The TRS P consists of the following rules:
169.21/88.00	
169.21/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/88.00	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/88.00	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/88.00	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/88.00	   TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a))))))
169.21/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/88.00	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/88.00	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a))))))
169.21/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.21/88.00	
169.21/88.00	The TRS R consists of the following rules:
169.21/88.00	
169.21/88.00	   down(g(f(x))) -> up(f(g(x)))
169.21/88.00	   down(g(a)) -> g_flat(down(a))
169.21/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/88.00	   down(g(b)) -> g_flat(down(b))
169.21/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.21/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.21/88.00	   down(a) -> up(f(a))
169.21/88.00	
169.21/88.00	The set Q consists of the following terms:
169.21/88.00	
169.21/88.00	   down(a)
169.21/88.00	   down(g(f(x0)))
169.21/88.00	   down(f(f(f(f(f(x0))))))
169.21/88.00	   down(f(a))
169.21/88.00	   down(f(g(x0)))
169.21/88.00	   down(f(b))
169.21/88.00	   down(f(fresh_constant))
169.21/88.00	   down(g(a))
169.21/88.00	   down(g(g(x0)))
169.21/88.00	   down(g(b))
169.21/88.00	   down(g(fresh_constant))
169.21/88.00	   down(f(f(a)))
169.21/88.00	   down(f(f(g(x0))))
169.21/88.00	   down(f(f(b)))
169.21/88.00	   down(f(f(fresh_constant)))
169.21/88.00	   down(f(f(f(a))))
169.21/88.00	   down(f(f(f(g(x0)))))
169.21/88.00	   down(f(f(f(b))))
169.21/88.00	   down(f(f(f(fresh_constant))))
169.21/88.00	   down(f(f(f(f(a)))))
169.21/88.00	   down(f(f(f(f(g(x0))))))
169.21/88.00	   down(f(f(f(f(b)))))
169.21/88.00	   down(f(f(f(f(fresh_constant)))))
169.21/88.00	   f_flat(up(x0))
169.21/88.00	   g_flat(up(x0))
169.21/88.00	
169.21/88.00	We have to consider all minimal (P,Q,R)-chains.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(184) TransformationProof (EQUIVALENT)
169.21/88.00	By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a)))))) at position [0] we obtained the following new rules [LPAR04]:
169.21/88.00	
169.21/88.00	   (TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))),TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))))
169.21/88.00	
169.21/88.00	
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(185)
169.21/88.00	Obligation:
169.21/88.00	Q DP problem:
169.21/88.00	The TRS P consists of the following rules:
169.21/88.00	
169.21/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/88.00	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/88.00	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/88.00	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/88.00	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/88.00	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a))))))
169.21/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.21/88.00	   TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a))))))
169.21/88.00	
169.21/88.00	The TRS R consists of the following rules:
169.21/88.00	
169.21/88.00	   down(g(f(x))) -> up(f(g(x)))
169.21/88.00	   down(g(a)) -> g_flat(down(a))
169.21/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/88.00	   down(g(b)) -> g_flat(down(b))
169.21/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.21/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.21/88.00	   down(a) -> up(f(a))
169.21/88.00	
169.21/88.00	The set Q consists of the following terms:
169.21/88.00	
169.21/88.00	   down(a)
169.21/88.00	   down(g(f(x0)))
169.21/88.00	   down(f(f(f(f(f(x0))))))
169.21/88.00	   down(f(a))
169.21/88.00	   down(f(g(x0)))
169.21/88.00	   down(f(b))
169.21/88.00	   down(f(fresh_constant))
169.21/88.00	   down(g(a))
169.21/88.00	   down(g(g(x0)))
169.21/88.00	   down(g(b))
169.21/88.00	   down(g(fresh_constant))
169.21/88.00	   down(f(f(a)))
169.21/88.00	   down(f(f(g(x0))))
169.21/88.00	   down(f(f(b)))
169.21/88.00	   down(f(f(fresh_constant)))
169.21/88.00	   down(f(f(f(a))))
169.21/88.00	   down(f(f(f(g(x0)))))
169.21/88.00	   down(f(f(f(b))))
169.21/88.00	   down(f(f(f(fresh_constant))))
169.21/88.00	   down(f(f(f(f(a)))))
169.21/88.00	   down(f(f(f(f(g(x0))))))
169.21/88.00	   down(f(f(f(f(b)))))
169.21/88.00	   down(f(f(f(f(fresh_constant)))))
169.21/88.00	   f_flat(up(x0))
169.21/88.00	   g_flat(up(x0))
169.21/88.00	
169.21/88.00	We have to consider all minimal (P,Q,R)-chains.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(186) TransformationProof (EQUIVALENT)
169.21/88.00	By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]:
169.21/88.00	
169.21/88.00	   (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))
169.21/88.00	
169.21/88.00	
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(187)
169.21/88.00	Obligation:
169.21/88.00	Q DP problem:
169.21/88.00	The TRS P consists of the following rules:
169.21/88.00	
169.21/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/88.00	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/88.00	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/88.00	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/88.00	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.21/88.00	   TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a))))))
169.21/88.00	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))
169.21/88.00	
169.21/88.00	The TRS R consists of the following rules:
169.21/88.00	
169.21/88.00	   down(g(f(x))) -> up(f(g(x)))
169.21/88.00	   down(g(a)) -> g_flat(down(a))
169.21/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/88.00	   down(g(b)) -> g_flat(down(b))
169.21/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.21/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.21/88.00	   down(a) -> up(f(a))
169.21/88.00	
169.21/88.00	The set Q consists of the following terms:
169.21/88.00	
169.21/88.00	   down(a)
169.21/88.00	   down(g(f(x0)))
169.21/88.00	   down(f(f(f(f(f(x0))))))
169.21/88.00	   down(f(a))
169.21/88.00	   down(f(g(x0)))
169.21/88.00	   down(f(b))
169.21/88.00	   down(f(fresh_constant))
169.21/88.00	   down(g(a))
169.21/88.00	   down(g(g(x0)))
169.21/88.00	   down(g(b))
169.21/88.00	   down(g(fresh_constant))
169.21/88.00	   down(f(f(a)))
169.21/88.00	   down(f(f(g(x0))))
169.21/88.00	   down(f(f(b)))
169.21/88.00	   down(f(f(fresh_constant)))
169.21/88.00	   down(f(f(f(a))))
169.21/88.00	   down(f(f(f(g(x0)))))
169.21/88.00	   down(f(f(f(b))))
169.21/88.00	   down(f(f(f(fresh_constant))))
169.21/88.00	   down(f(f(f(f(a)))))
169.21/88.00	   down(f(f(f(f(g(x0))))))
169.21/88.00	   down(f(f(f(f(b)))))
169.21/88.00	   down(f(f(f(f(fresh_constant)))))
169.21/88.00	   f_flat(up(x0))
169.21/88.00	   g_flat(up(x0))
169.21/88.00	
169.21/88.00	We have to consider all minimal (P,Q,R)-chains.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(188) TransformationProof (EQUIVALENT)
169.21/88.00	By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:
169.21/88.00	
169.21/88.00	   (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(a))))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(a))))))))
169.21/88.00	
169.21/88.00	
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(189)
169.21/88.00	Obligation:
169.21/88.00	Q DP problem:
169.21/88.00	The TRS P consists of the following rules:
169.21/88.00	
169.21/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/88.00	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/88.00	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/88.00	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/88.00	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.21/88.00	   TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a))))))
169.21/88.00	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(a)))))))
169.21/88.00	
169.21/88.00	The TRS R consists of the following rules:
169.21/88.00	
169.21/88.00	   down(g(f(x))) -> up(f(g(x)))
169.21/88.00	   down(g(a)) -> g_flat(down(a))
169.21/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/88.00	   down(g(b)) -> g_flat(down(b))
169.21/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.21/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.21/88.00	   down(a) -> up(f(a))
169.21/88.00	
169.21/88.00	The set Q consists of the following terms:
169.21/88.00	
169.21/88.00	   down(a)
169.21/88.00	   down(g(f(x0)))
169.21/88.00	   down(f(f(f(f(f(x0))))))
169.21/88.00	   down(f(a))
169.21/88.00	   down(f(g(x0)))
169.21/88.00	   down(f(b))
169.21/88.00	   down(f(fresh_constant))
169.21/88.00	   down(g(a))
169.21/88.00	   down(g(g(x0)))
169.21/88.00	   down(g(b))
169.21/88.00	   down(g(fresh_constant))
169.21/88.00	   down(f(f(a)))
169.21/88.00	   down(f(f(g(x0))))
169.21/88.00	   down(f(f(b)))
169.21/88.00	   down(f(f(fresh_constant)))
169.21/88.00	   down(f(f(f(a))))
169.21/88.00	   down(f(f(f(g(x0)))))
169.21/88.00	   down(f(f(f(b))))
169.21/88.00	   down(f(f(f(fresh_constant))))
169.21/88.00	   down(f(f(f(f(a)))))
169.21/88.00	   down(f(f(f(f(g(x0))))))
169.21/88.00	   down(f(f(f(f(b)))))
169.21/88.00	   down(f(f(f(f(fresh_constant)))))
169.21/88.00	   f_flat(up(x0))
169.21/88.00	   g_flat(up(x0))
169.21/88.00	
169.21/88.00	We have to consider all minimal (P,Q,R)-chains.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(190) TransformationProof (EQUIVALENT)
169.21/88.00	By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(a))))))) at position [0,0,0] we obtained the following new rules [LPAR04]:
169.21/88.00	
169.21/88.00	   (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a))))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a))))))))
169.21/88.00	
169.21/88.00	
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(191)
169.21/88.00	Obligation:
169.21/88.00	Q DP problem:
169.21/88.00	The TRS P consists of the following rules:
169.21/88.00	
169.21/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/88.00	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/88.00	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/88.00	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/88.00	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.21/88.00	   TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a))))))
169.21/88.00	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a)))))))
169.21/88.00	
169.21/88.00	The TRS R consists of the following rules:
169.21/88.00	
169.21/88.00	   down(g(f(x))) -> up(f(g(x)))
169.21/88.00	   down(g(a)) -> g_flat(down(a))
169.21/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/88.00	   down(g(b)) -> g_flat(down(b))
169.21/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.21/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.21/88.00	   down(a) -> up(f(a))
169.21/88.00	
169.21/88.00	The set Q consists of the following terms:
169.21/88.00	
169.21/88.00	   down(a)
169.21/88.00	   down(g(f(x0)))
169.21/88.00	   down(f(f(f(f(f(x0))))))
169.21/88.00	   down(f(a))
169.21/88.00	   down(f(g(x0)))
169.21/88.00	   down(f(b))
169.21/88.00	   down(f(fresh_constant))
169.21/88.00	   down(g(a))
169.21/88.00	   down(g(g(x0)))
169.21/88.00	   down(g(b))
169.21/88.00	   down(g(fresh_constant))
169.21/88.00	   down(f(f(a)))
169.21/88.00	   down(f(f(g(x0))))
169.21/88.00	   down(f(f(b)))
169.21/88.00	   down(f(f(fresh_constant)))
169.21/88.00	   down(f(f(f(a))))
169.21/88.00	   down(f(f(f(g(x0)))))
169.21/88.00	   down(f(f(f(b))))
169.21/88.00	   down(f(f(f(fresh_constant))))
169.21/88.00	   down(f(f(f(f(a)))))
169.21/88.00	   down(f(f(f(f(g(x0))))))
169.21/88.00	   down(f(f(f(f(b)))))
169.21/88.00	   down(f(f(f(f(fresh_constant)))))
169.21/88.00	   f_flat(up(x0))
169.21/88.00	   g_flat(up(x0))
169.21/88.00	
169.21/88.00	We have to consider all minimal (P,Q,R)-chains.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(192) TransformationProof (EQUIVALENT)
169.21/88.00	By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a))))))) at position [0,0] we obtained the following new rules [LPAR04]:
169.21/88.00	
169.21/88.00	   (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(up(f(f(f(f(a))))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(up(f(f(f(f(a))))))))
169.21/88.00	
169.21/88.00	
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(193)
169.21/88.00	Obligation:
169.21/88.00	Q DP problem:
169.21/88.00	The TRS P consists of the following rules:
169.21/88.00	
169.21/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/88.00	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/88.00	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/88.00	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/88.00	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.21/88.00	   TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a))))))
169.21/88.00	   TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(up(f(f(f(f(a)))))))
169.21/88.00	
169.21/88.00	The TRS R consists of the following rules:
169.21/88.00	
169.21/88.00	   down(g(f(x))) -> up(f(g(x)))
169.21/88.00	   down(g(a)) -> g_flat(down(a))
169.21/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/88.00	   down(g(b)) -> g_flat(down(b))
169.21/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.21/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.21/88.00	   down(a) -> up(f(a))
169.21/88.00	
169.21/88.00	The set Q consists of the following terms:
169.21/88.00	
169.21/88.00	   down(a)
169.21/88.00	   down(g(f(x0)))
169.21/88.00	   down(f(f(f(f(f(x0))))))
169.21/88.00	   down(f(a))
169.21/88.00	   down(f(g(x0)))
169.21/88.00	   down(f(b))
169.21/88.00	   down(f(fresh_constant))
169.21/88.00	   down(g(a))
169.21/88.00	   down(g(g(x0)))
169.21/88.00	   down(g(b))
169.21/88.00	   down(g(fresh_constant))
169.21/88.00	   down(f(f(a)))
169.21/88.00	   down(f(f(g(x0))))
169.21/88.00	   down(f(f(b)))
169.21/88.00	   down(f(f(fresh_constant)))
169.21/88.00	   down(f(f(f(a))))
169.21/88.00	   down(f(f(f(g(x0)))))
169.21/88.00	   down(f(f(f(b))))
169.21/88.00	   down(f(f(f(fresh_constant))))
169.21/88.00	   down(f(f(f(f(a)))))
169.21/88.00	   down(f(f(f(f(g(x0))))))
169.21/88.00	   down(f(f(f(f(b)))))
169.21/88.00	   down(f(f(f(f(fresh_constant)))))
169.21/88.00	   f_flat(up(x0))
169.21/88.00	   g_flat(up(x0))
169.21/88.00	
169.21/88.00	We have to consider all minimal (P,Q,R)-chains.
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(194) TransformationProof (EQUIVALENT)
169.21/88.00	By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(up(f(f(f(f(a))))))) at position [0] we obtained the following new rules [LPAR04]:
169.21/88.00	
169.21/88.00	   (TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))),TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))))
169.21/88.00	
169.21/88.00	
169.21/88.00	----------------------------------------
169.21/88.00	
169.21/88.00	(195)
169.21/88.00	Obligation:
169.21/88.00	Q DP problem:
169.21/88.00	The TRS P consists of the following rules:
169.21/88.00	
169.21/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.21/88.00	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.21/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.21/88.00	   TOP(up(f(a))) -> TOP(up(f(f(a))))
169.21/88.00	   TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))
169.21/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.21/88.00	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.21/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.21/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.21/88.00	   TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a))))))
169.21/88.00	   TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a)))))))
169.21/88.00	
169.21/88.00	The TRS R consists of the following rules:
169.21/88.00	
169.21/88.00	   down(g(f(x))) -> up(f(g(x)))
169.21/88.00	   down(g(a)) -> g_flat(down(a))
169.21/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.21/88.00	   down(g(b)) -> g_flat(down(b))
169.21/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.21/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.21/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.21/88.00	   down(a) -> up(f(a))
169.21/88.00	
169.21/88.00	The set Q consists of the following terms:
169.36/88.00	
169.36/88.00	   down(a)
169.36/88.00	   down(g(f(x0)))
169.36/88.00	   down(f(f(f(f(f(x0))))))
169.36/88.00	   down(f(a))
169.36/88.00	   down(f(g(x0)))
169.36/88.00	   down(f(b))
169.36/88.00	   down(f(fresh_constant))
169.36/88.00	   down(g(a))
169.36/88.00	   down(g(g(x0)))
169.36/88.00	   down(g(b))
169.36/88.00	   down(g(fresh_constant))
169.36/88.00	   down(f(f(a)))
169.36/88.00	   down(f(f(g(x0))))
169.36/88.00	   down(f(f(b)))
169.36/88.00	   down(f(f(fresh_constant)))
169.36/88.00	   down(f(f(f(a))))
169.36/88.00	   down(f(f(f(g(x0)))))
169.36/88.00	   down(f(f(f(b))))
169.36/88.00	   down(f(f(f(fresh_constant))))
169.36/88.00	   down(f(f(f(f(a)))))
169.36/88.00	   down(f(f(f(f(g(x0))))))
169.36/88.00	   down(f(f(f(f(b)))))
169.36/88.00	   down(f(f(f(f(fresh_constant)))))
169.36/88.00	   f_flat(up(x0))
169.36/88.00	   g_flat(up(x0))
169.36/88.00	
169.36/88.00	We have to consider all minimal (P,Q,R)-chains.
169.36/88.00	----------------------------------------
169.36/88.00	
169.36/88.00	(196) DependencyGraphProof (EQUIVALENT)
169.36/88.00	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
169.36/88.00	----------------------------------------
169.36/88.00	
169.36/88.00	(197)
169.36/88.00	Obligation:
169.36/88.00	Q DP problem:
169.36/88.00	The TRS P consists of the following rules:
169.36/88.00	
169.36/88.00	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.36/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.36/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.36/88.00	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.36/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.36/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.36/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.36/88.00	
169.36/88.00	The TRS R consists of the following rules:
169.36/88.00	
169.36/88.00	   down(g(f(x))) -> up(f(g(x)))
169.36/88.00	   down(g(a)) -> g_flat(down(a))
169.36/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.36/88.00	   down(g(b)) -> g_flat(down(b))
169.36/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.36/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.36/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.36/88.00	   down(a) -> up(f(a))
169.36/88.00	
169.36/88.00	The set Q consists of the following terms:
169.36/88.00	
169.36/88.00	   down(a)
169.36/88.00	   down(g(f(x0)))
169.36/88.00	   down(f(f(f(f(f(x0))))))
169.36/88.00	   down(f(a))
169.36/88.00	   down(f(g(x0)))
169.36/88.00	   down(f(b))
169.36/88.00	   down(f(fresh_constant))
169.36/88.00	   down(g(a))
169.36/88.00	   down(g(g(x0)))
169.36/88.00	   down(g(b))
169.36/88.00	   down(g(fresh_constant))
169.36/88.00	   down(f(f(a)))
169.36/88.00	   down(f(f(g(x0))))
169.36/88.00	   down(f(f(b)))
169.36/88.00	   down(f(f(fresh_constant)))
169.36/88.00	   down(f(f(f(a))))
169.36/88.00	   down(f(f(f(g(x0)))))
169.36/88.00	   down(f(f(f(b))))
169.36/88.00	   down(f(f(f(fresh_constant))))
169.36/88.00	   down(f(f(f(f(a)))))
169.36/88.00	   down(f(f(f(f(g(x0))))))
169.36/88.00	   down(f(f(f(f(b)))))
169.36/88.00	   down(f(f(f(f(fresh_constant)))))
169.36/88.00	   f_flat(up(x0))
169.36/88.00	   g_flat(up(x0))
169.36/88.00	
169.36/88.00	We have to consider all minimal (P,Q,R)-chains.
169.36/88.00	----------------------------------------
169.36/88.00	
169.36/88.00	(198) QDPOrderProof (EQUIVALENT)
169.36/88.00	We use the reduction pair processor [LPAR04,JAR06].
169.36/88.00	
169.36/88.00	
169.36/88.00	The following pairs can be oriented strictly and are deleted.
169.36/88.00	
169.36/88.00	   TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))
169.36/88.00	The remaining pairs can at least be oriented weakly.
169.36/88.00	Used ordering:  Polynomial interpretation [POLO]:
169.36/88.00	
169.36/88.00	   POL(TOP(x_1)) = x_1
169.36/88.00	   POL(a) = 0
169.36/88.00	   POL(b) = 0
169.36/88.00	   POL(down(x_1)) = 0
169.36/88.00	   POL(f(x_1)) = 0
169.36/88.00	   POL(f_flat(x_1)) = 0
169.36/88.00	   POL(fresh_constant) = 0
169.36/88.00	   POL(g(x_1)) = 1
169.36/88.00	   POL(g_flat(x_1)) = 1
169.36/88.00	   POL(up(x_1)) = x_1
169.36/88.00	
169.36/88.00	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
169.36/88.00	
169.36/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.36/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.36/88.00	
169.36/88.00	
169.36/88.00	----------------------------------------
169.36/88.00	
169.36/88.00	(199)
169.36/88.00	Obligation:
169.36/88.00	Q DP problem:
169.36/88.00	The TRS P consists of the following rules:
169.36/88.00	
169.36/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.36/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.36/88.00	   TOP(up(g(a))) -> TOP(up(g(f(a))))
169.36/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.36/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.36/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.36/88.00	
169.36/88.00	The TRS R consists of the following rules:
169.36/88.00	
169.36/88.00	   down(g(f(x))) -> up(f(g(x)))
169.36/88.00	   down(g(a)) -> g_flat(down(a))
169.36/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.36/88.00	   down(g(b)) -> g_flat(down(b))
169.36/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.36/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.36/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.36/88.00	   down(a) -> up(f(a))
169.36/88.00	
169.36/88.00	The set Q consists of the following terms:
169.36/88.00	
169.36/88.00	   down(a)
169.36/88.00	   down(g(f(x0)))
169.36/88.00	   down(f(f(f(f(f(x0))))))
169.36/88.00	   down(f(a))
169.36/88.00	   down(f(g(x0)))
169.36/88.00	   down(f(b))
169.36/88.00	   down(f(fresh_constant))
169.36/88.00	   down(g(a))
169.36/88.00	   down(g(g(x0)))
169.36/88.00	   down(g(b))
169.36/88.00	   down(g(fresh_constant))
169.36/88.00	   down(f(f(a)))
169.36/88.00	   down(f(f(g(x0))))
169.36/88.00	   down(f(f(b)))
169.36/88.00	   down(f(f(fresh_constant)))
169.36/88.00	   down(f(f(f(a))))
169.36/88.00	   down(f(f(f(g(x0)))))
169.36/88.00	   down(f(f(f(b))))
169.36/88.00	   down(f(f(f(fresh_constant))))
169.36/88.00	   down(f(f(f(f(a)))))
169.36/88.00	   down(f(f(f(f(g(x0))))))
169.36/88.00	   down(f(f(f(f(b)))))
169.36/88.00	   down(f(f(f(f(fresh_constant)))))
169.36/88.00	   f_flat(up(x0))
169.36/88.00	   g_flat(up(x0))
169.36/88.00	
169.36/88.00	We have to consider all minimal (P,Q,R)-chains.
169.36/88.00	----------------------------------------
169.36/88.00	
169.36/88.00	(200) DependencyGraphProof (EQUIVALENT)
169.36/88.00	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
169.36/88.00	----------------------------------------
169.36/88.00	
169.36/88.00	(201)
169.36/88.00	Obligation:
169.36/88.00	Q DP problem:
169.36/88.00	The TRS P consists of the following rules:
169.36/88.00	
169.36/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.36/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.36/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.36/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.36/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.36/88.00	
169.36/88.00	The TRS R consists of the following rules:
169.36/88.00	
169.36/88.00	   down(g(f(x))) -> up(f(g(x)))
169.36/88.00	   down(g(a)) -> g_flat(down(a))
169.36/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.36/88.00	   down(g(b)) -> g_flat(down(b))
169.36/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.36/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.36/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.36/88.00	   down(a) -> up(f(a))
169.36/88.00	
169.36/88.00	The set Q consists of the following terms:
169.36/88.00	
169.36/88.00	   down(a)
169.36/88.00	   down(g(f(x0)))
169.36/88.00	   down(f(f(f(f(f(x0))))))
169.36/88.00	   down(f(a))
169.36/88.00	   down(f(g(x0)))
169.36/88.00	   down(f(b))
169.36/88.00	   down(f(fresh_constant))
169.36/88.00	   down(g(a))
169.36/88.00	   down(g(g(x0)))
169.36/88.00	   down(g(b))
169.36/88.00	   down(g(fresh_constant))
169.36/88.00	   down(f(f(a)))
169.36/88.00	   down(f(f(g(x0))))
169.36/88.00	   down(f(f(b)))
169.36/88.00	   down(f(f(fresh_constant)))
169.36/88.00	   down(f(f(f(a))))
169.36/88.00	   down(f(f(f(g(x0)))))
169.36/88.00	   down(f(f(f(b))))
169.36/88.00	   down(f(f(f(fresh_constant))))
169.36/88.00	   down(f(f(f(f(a)))))
169.36/88.00	   down(f(f(f(f(g(x0))))))
169.36/88.00	   down(f(f(f(f(b)))))
169.36/88.00	   down(f(f(f(f(fresh_constant)))))
169.36/88.00	   f_flat(up(x0))
169.36/88.00	   g_flat(up(x0))
169.36/88.00	
169.36/88.00	We have to consider all minimal (P,Q,R)-chains.
169.36/88.00	----------------------------------------
169.36/88.00	
169.36/88.00	(202) QDPOrderProof (EQUIVALENT)
169.36/88.00	We use the reduction pair processor [LPAR04,JAR06].
169.36/88.00	
169.36/88.00	
169.36/88.00	The following pairs can be oriented strictly and are deleted.
169.36/88.00	
169.36/88.00	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
169.36/88.00	The remaining pairs can at least be oriented weakly.
169.36/88.00	Used ordering:  Polynomial interpretation [POLO]:
169.36/88.00	
169.36/88.00	   POL(TOP(x_1)) = x_1
169.36/88.00	   POL(a) = 1
169.36/88.00	   POL(b) = 0
169.36/88.00	   POL(down(x_1)) = x_1
169.36/88.00	   POL(f(x_1)) = 0
169.36/88.00	   POL(f_flat(x_1)) = 1
169.36/88.00	   POL(fresh_constant) = 0
169.36/88.00	   POL(g(x_1)) = 1 + x_1
169.36/88.00	   POL(g_flat(x_1)) = 1 + x_1
169.36/88.00	   POL(up(x_1)) = 1 + x_1
169.36/88.00	
169.36/88.00	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
169.36/88.00	
169.36/88.00	   down(g(f(x))) -> up(f(g(x)))
169.36/88.00	   down(g(a)) -> g_flat(down(a))
169.36/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.36/88.00	   down(g(b)) -> g_flat(down(b))
169.36/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.36/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.36/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.36/88.00	   down(a) -> up(f(a))
169.36/88.00	
169.36/88.00	
169.36/88.00	----------------------------------------
169.36/88.00	
169.36/88.00	(203)
169.36/88.00	Obligation:
169.36/88.00	Q DP problem:
169.36/88.00	The TRS P consists of the following rules:
169.36/88.00	
169.36/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.36/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.36/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.36/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.36/88.00	
169.36/88.00	The TRS R consists of the following rules:
169.36/88.00	
169.36/88.00	   down(g(f(x))) -> up(f(g(x)))
169.36/88.00	   down(g(a)) -> g_flat(down(a))
169.36/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.36/88.00	   down(g(b)) -> g_flat(down(b))
169.36/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.36/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.36/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.36/88.00	   down(a) -> up(f(a))
169.36/88.00	
169.36/88.00	The set Q consists of the following terms:
169.36/88.00	
169.36/88.00	   down(a)
169.36/88.00	   down(g(f(x0)))
169.36/88.00	   down(f(f(f(f(f(x0))))))
169.36/88.00	   down(f(a))
169.36/88.00	   down(f(g(x0)))
169.36/88.00	   down(f(b))
169.36/88.00	   down(f(fresh_constant))
169.36/88.00	   down(g(a))
169.36/88.00	   down(g(g(x0)))
169.36/88.00	   down(g(b))
169.36/88.00	   down(g(fresh_constant))
169.36/88.00	   down(f(f(a)))
169.36/88.00	   down(f(f(g(x0))))
169.36/88.00	   down(f(f(b)))
169.36/88.00	   down(f(f(fresh_constant)))
169.36/88.00	   down(f(f(f(a))))
169.36/88.00	   down(f(f(f(g(x0)))))
169.36/88.00	   down(f(f(f(b))))
169.36/88.00	   down(f(f(f(fresh_constant))))
169.36/88.00	   down(f(f(f(f(a)))))
169.36/88.00	   down(f(f(f(f(g(x0))))))
169.36/88.00	   down(f(f(f(f(b)))))
169.36/88.00	   down(f(f(f(f(fresh_constant)))))
169.36/88.00	   f_flat(up(x0))
169.36/88.00	   g_flat(up(x0))
169.36/88.00	
169.36/88.00	We have to consider all minimal (P,Q,R)-chains.
169.36/88.00	----------------------------------------
169.36/88.00	
169.36/88.00	(204) QDPOrderProof (EQUIVALENT)
169.36/88.00	We use the reduction pair processor [LPAR04,JAR06].
169.36/88.00	
169.36/88.00	
169.36/88.00	The following pairs can be oriented strictly and are deleted.
169.36/88.00	
169.36/88.00	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
169.36/88.00	The remaining pairs can at least be oriented weakly.
169.36/88.00	Used ordering:  Matrix interpretation [MATRO]:
169.36/88.00	
169.36/88.00	Non-tuple symbols: 
169.36/88.00	<<<
169.36/88.00	 M( a ) =  	[[1], [1]]
169.36/88.00	>>>
169.36/88.00	
169.36/88.00	<<<
169.36/88.00	 M( b ) =  	[[0], [0]]
169.36/88.00	>>>
169.36/88.00	
169.36/88.00	<<<
169.36/88.00	 M( down_1(x_1) ) =  	[[0], [1]] 	+ 	[[1, 0], [1, 1]] 	* 	x_1
169.36/88.00	>>>
169.36/88.00	
169.36/88.00	<<<
169.36/88.00	 M( f_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [1, 0]] 	* 	x_1
169.36/88.00	>>>
169.36/88.00	
169.36/88.00	<<<
169.36/88.00	 M( fresh_constant ) =  	[[1], [0]]
169.36/88.00	>>>
169.36/88.00	
169.36/88.00	<<<
169.36/88.00	 M( up_1(x_1) ) =  	[[1], [1]] 	+ 	[[1, 0], [0, 1]] 	* 	x_1
169.36/88.00	>>>
169.36/88.00	
169.36/88.00	<<<
169.36/88.00	 M( f_flat_1(x_1) ) =  	[[1], [0]] 	+ 	[[0, 0], [1, 0]] 	* 	x_1
169.36/88.00	>>>
169.36/88.00	
169.36/88.00	<<<
169.36/88.00	 M( g_1(x_1) ) =  	[[1], [0]] 	+ 	[[1, 0], [0, 1]] 	* 	x_1
169.36/88.00	>>>
169.36/88.00	
169.36/88.00	<<<
169.36/88.00	 M( g_flat_1(x_1) ) =  	[[1], [1]] 	+ 	[[1, 0], [0, 1]] 	* 	x_1
169.36/88.00	>>>
169.36/88.00	
169.36/88.00	Tuple symbols: 
169.36/88.00	<<<
169.36/88.00	 M( TOP_1(x_1) ) =  	[[0]] 	+ 	[[0, 1]] 	* 	x_1
169.36/88.00	>>>
169.36/88.00	
169.36/88.00	
169.36/88.00	
169.36/88.00	Matrix type: 
169.36/88.00	
169.36/88.00	We used a basic matrix type which is not further parametrizeable.
169.36/88.00	
169.36/88.00	
169.36/88.00	
169.36/88.00	
169.36/88.00	
169.36/88.00	As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
169.36/88.00	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
169.36/88.00	
169.36/88.00	   down(g(f(x))) -> up(f(g(x)))
169.36/88.00	   down(g(a)) -> g_flat(down(a))
169.36/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.36/88.00	   down(g(b)) -> g_flat(down(b))
169.36/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.36/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.36/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.36/88.00	   down(a) -> up(f(a))
169.36/88.00	
169.36/88.00	
169.36/88.00	----------------------------------------
169.36/88.00	
169.36/88.00	(205)
169.36/88.00	Obligation:
169.36/88.00	Q DP problem:
169.36/88.00	The TRS P consists of the following rules:
169.36/88.00	
169.36/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.36/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.36/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.36/88.00	
169.36/88.00	The TRS R consists of the following rules:
169.36/88.00	
169.36/88.00	   down(g(f(x))) -> up(f(g(x)))
169.36/88.00	   down(g(a)) -> g_flat(down(a))
169.36/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.36/88.00	   down(g(b)) -> g_flat(down(b))
169.36/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.36/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.36/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.36/88.00	   down(a) -> up(f(a))
169.36/88.00	
169.36/88.00	The set Q consists of the following terms:
169.36/88.00	
169.36/88.00	   down(a)
169.36/88.00	   down(g(f(x0)))
169.36/88.00	   down(f(f(f(f(f(x0))))))
169.36/88.00	   down(f(a))
169.36/88.00	   down(f(g(x0)))
169.36/88.00	   down(f(b))
169.36/88.00	   down(f(fresh_constant))
169.36/88.00	   down(g(a))
169.36/88.00	   down(g(g(x0)))
169.36/88.00	   down(g(b))
169.36/88.00	   down(g(fresh_constant))
169.36/88.00	   down(f(f(a)))
169.36/88.00	   down(f(f(g(x0))))
169.36/88.00	   down(f(f(b)))
169.36/88.00	   down(f(f(fresh_constant)))
169.36/88.00	   down(f(f(f(a))))
169.36/88.00	   down(f(f(f(g(x0)))))
169.36/88.00	   down(f(f(f(b))))
169.36/88.00	   down(f(f(f(fresh_constant))))
169.36/88.00	   down(f(f(f(f(a)))))
169.36/88.00	   down(f(f(f(f(g(x0))))))
169.36/88.00	   down(f(f(f(f(b)))))
169.36/88.00	   down(f(f(f(f(fresh_constant)))))
169.36/88.00	   f_flat(up(x0))
169.36/88.00	   g_flat(up(x0))
169.36/88.00	
169.36/88.00	We have to consider all minimal (P,Q,R)-chains.
169.36/88.00	----------------------------------------
169.36/88.00	
169.36/88.00	(206) SplitQDPProof (EQUIVALENT)
169.36/88.00	We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem
169.36/88.00	
169.36/88.00	----------------------------------------
169.36/88.00	
169.36/88.00	(207)
169.36/88.00	Complex Obligation (AND)
169.36/88.00	
169.36/88.00	----------------------------------------
169.36/88.00	
169.36/88.00	(208)
169.36/88.00	Obligation:
169.36/88.00	Q DP problem:
169.36/88.00	The TRS P consists of the following rules:
169.36/88.00	
169.36/88.00	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.36/88.00	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.36/88.00	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.36/88.00	
169.36/88.00	The TRS R consists of the following rules:
169.36/88.00	
169.36/88.00	   down(g(f(x))) -> up(f(g(x)))
169.36/88.00	   down(g(a)) -> g_flat(down(a))
169.36/88.00	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.36/88.00	   down(g(b)) -> g_flat(down(b))
169.36/88.00	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
169.36/88.00	   f_flat(up(x_1)) -> up(f(x_1))
169.36/88.00	   g_flat(up(x_1)) -> up(g(x_1))
169.36/88.00	   down(a) -> up(f(a))
169.36/88.00	
169.36/88.00	The set Q consists of the following terms:
169.36/88.00	
169.36/88.00	   down(a)
169.36/88.00	   down(g(f(x0)))
169.36/88.00	   down(f(f(f(f(f(x0))))))
169.36/88.00	   down(f(a))
169.36/88.00	   down(f(g(x0)))
169.36/88.00	   down(f(b))
169.36/88.00	   down(f(fresh_constant))
169.36/88.00	   down(g(a))
169.36/88.00	   down(g(g(x0)))
169.36/88.00	   down(g(b))
169.36/88.00	   down(g(fresh_constant))
169.36/88.00	   down(f(f(a)))
169.36/88.00	   down(f(f(g(x0))))
169.36/88.00	   down(f(f(b)))
169.36/88.00	   down(f(f(fresh_constant)))
169.36/88.00	   down(f(f(f(a))))
169.36/88.00	   down(f(f(f(g(x0)))))
169.36/88.00	   down(f(f(f(b))))
169.36/88.00	   down(f(f(f(fresh_constant))))
169.36/88.00	   down(f(f(f(f(a)))))
169.36/88.00	   down(f(f(f(f(g(x0))))))
169.36/88.00	   down(f(f(f(f(b)))))
169.36/88.00	   down(f(f(f(f(fresh_constant)))))
169.36/88.00	   f_flat(up(x0))
169.36/88.00	   g_flat(up(x0))
169.36/88.00	
169.36/88.00	We have to consider all minimal (P,Q,R)-chains.
169.36/88.00	----------------------------------------
169.36/88.00	
169.36/88.00	(209) SemLabProof (SOUND)
169.36/88.00	We found the following model for the rules of the TRSs R and P.
169.36/88.00	Interpretation over the domain with elements from 0 to 1.
169.36/88.00	a: 0
169.36/88.00	b: 0
169.36/88.00	down: 0
169.36/88.00	f: 0
169.36/88.00	fresh_constant: 1
169.36/88.00	up: 0
169.36/88.00	f_flat: 0
169.36/88.00	TOP: 0
169.36/88.00	g_flat: 0
169.36/88.00	g: 0
169.36/88.00	By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
169.36/88.00	----------------------------------------
169.36/88.00	
169.36/88.00	(210)
169.36/88.00	Obligation:
169.36/88.00	Q DP problem:
169.36/88.00	The TRS P consists of the following rules:
169.36/88.00	
169.36/88.00	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))
169.36/88.00	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))
169.36/88.01	
169.36/88.01	The TRS R consists of the following rules:
169.36/88.01	
169.36/88.01	   down.0(g.0(f.0(x))) -> up.0(f.0(g.0(x)))
169.36/88.01	   down.0(g.0(f.1(x))) -> up.0(f.0(g.1(x)))
169.36/88.01	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
169.36/88.01	   down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7)))
169.36/88.01	   down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7)))
169.36/88.01	   down.0(g.0(b.)) -> g_flat.0(down.0(b.))
169.36/88.01	   down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.))
169.36/88.01	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
169.36/88.01	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
169.36/88.01	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
169.36/88.01	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
169.36/88.01	   down.0(a.) -> up.0(f.0(a.))
169.36/88.01	
169.36/88.01	The set Q consists of the following terms:
169.36/88.01	
169.36/88.01	   down.0(a.)
169.36/88.01	   down.0(g.0(f.0(x0)))
169.36/88.01	   down.0(g.0(f.1(x0)))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(f.0(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(f.1(x0))))))
169.36/88.01	   down.0(f.0(a.))
169.36/88.01	   down.0(f.0(g.0(x0)))
169.36/88.01	   down.0(f.0(g.1(x0)))
169.36/88.01	   down.0(f.0(b.))
169.36/88.01	   down.0(f.1(fresh_constant.))
169.36/88.01	   down.0(g.0(a.))
169.36/88.01	   down.0(g.0(g.0(x0)))
169.36/88.01	   down.0(g.0(g.1(x0)))
169.36/88.01	   down.0(g.0(b.))
169.36/88.01	   down.0(g.1(fresh_constant.))
169.36/88.01	   down.0(f.0(f.0(a.)))
169.36/88.01	   down.0(f.0(f.0(g.0(x0))))
169.36/88.01	   down.0(f.0(f.0(g.1(x0))))
169.36/88.01	   down.0(f.0(f.0(b.)))
169.36/88.01	   down.0(f.0(f.1(fresh_constant.)))
169.36/88.01	   down.0(f.0(f.0(f.0(a.))))
169.36/88.01	   down.0(f.0(f.0(f.0(g.0(x0)))))
169.36/88.01	   down.0(f.0(f.0(f.0(g.1(x0)))))
169.36/88.01	   down.0(f.0(f.0(f.0(b.))))
169.36/88.01	   down.0(f.0(f.0(f.1(fresh_constant.))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(a.)))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(g.0(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(g.1(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(b.)))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.1(fresh_constant.)))))
169.36/88.01	   f_flat.0(up.0(x0))
169.36/88.01	   f_flat.0(up.1(x0))
169.36/88.01	   g_flat.0(up.0(x0))
169.36/88.01	   g_flat.0(up.1(x0))
169.36/88.01	
169.36/88.01	We have to consider all minimal (P,Q,R)-chains.
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(211) UsableRulesReductionPairsProof (EQUIVALENT)
169.36/88.01	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.
169.36/88.01	
169.36/88.01	No dependency pairs are removed.
169.36/88.01	
169.36/88.01	The following rules are removed from R:
169.36/88.01	
169.36/88.01	   down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.))
169.36/88.01	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
169.36/88.01	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
169.36/88.01	Used ordering: POLO with Polynomial interpretation [POLO]:
169.36/88.01	
169.36/88.01	   POL(TOP.0(x_1)) = x_1
169.36/88.01	   POL(a.) = 0
169.36/88.01	   POL(b.) = 0
169.36/88.01	   POL(down.0(x_1)) = 1 + x_1
169.36/88.01	   POL(down.1(x_1)) = x_1
169.36/88.01	   POL(f.0(x_1)) = x_1
169.36/88.01	   POL(f.1(x_1)) = x_1
169.36/88.01	   POL(f_flat.0(x_1)) = x_1
169.36/88.01	   POL(fresh_constant.) = 0
169.36/88.01	   POL(g.0(x_1)) = 1 + x_1
169.36/88.01	   POL(g.1(x_1)) = 1 + x_1
169.36/88.01	   POL(g_flat.0(x_1)) = 1 + x_1
169.36/88.01	   POL(up.0(x_1)) = 1 + x_1
169.36/88.01	   POL(up.1(x_1)) = 1 + x_1
169.36/88.01	
169.36/88.01	
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(212)
169.36/88.01	Obligation:
169.36/88.01	Q DP problem:
169.36/88.01	The TRS P consists of the following rules:
169.36/88.01	
169.36/88.01	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))
169.36/88.01	
169.36/88.01	The TRS R consists of the following rules:
169.36/88.01	
169.36/88.01	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
169.36/88.01	   down.0(g.0(f.0(x))) -> up.0(f.0(g.0(x)))
169.36/88.01	   down.0(g.0(f.1(x))) -> up.0(f.0(g.1(x)))
169.36/88.01	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
169.36/88.01	   down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7)))
169.36/88.01	   down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7)))
169.36/88.01	   down.0(g.0(b.)) -> g_flat.0(down.0(b.))
169.36/88.01	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
169.36/88.01	   down.0(a.) -> up.0(f.0(a.))
169.36/88.01	
169.36/88.01	The set Q consists of the following terms:
169.36/88.01	
169.36/88.01	   down.0(a.)
169.36/88.01	   down.0(g.0(f.0(x0)))
169.36/88.01	   down.0(g.0(f.1(x0)))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(f.0(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(f.1(x0))))))
169.36/88.01	   down.0(f.0(a.))
169.36/88.01	   down.0(f.0(g.0(x0)))
169.36/88.01	   down.0(f.0(g.1(x0)))
169.36/88.01	   down.0(f.0(b.))
169.36/88.01	   down.0(f.1(fresh_constant.))
169.36/88.01	   down.0(g.0(a.))
169.36/88.01	   down.0(g.0(g.0(x0)))
169.36/88.01	   down.0(g.0(g.1(x0)))
169.36/88.01	   down.0(g.0(b.))
169.36/88.01	   down.0(g.1(fresh_constant.))
169.36/88.01	   down.0(f.0(f.0(a.)))
169.36/88.01	   down.0(f.0(f.0(g.0(x0))))
169.36/88.01	   down.0(f.0(f.0(g.1(x0))))
169.36/88.01	   down.0(f.0(f.0(b.)))
169.36/88.01	   down.0(f.0(f.1(fresh_constant.)))
169.36/88.01	   down.0(f.0(f.0(f.0(a.))))
169.36/88.01	   down.0(f.0(f.0(f.0(g.0(x0)))))
169.36/88.01	   down.0(f.0(f.0(f.0(g.1(x0)))))
169.36/88.01	   down.0(f.0(f.0(f.0(b.))))
169.36/88.01	   down.0(f.0(f.0(f.1(fresh_constant.))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(a.)))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(g.0(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(g.1(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(b.)))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.1(fresh_constant.)))))
169.36/88.01	   f_flat.0(up.0(x0))
169.36/88.01	   f_flat.0(up.1(x0))
169.36/88.01	   g_flat.0(up.0(x0))
169.36/88.01	   g_flat.0(up.1(x0))
169.36/88.01	
169.36/88.01	We have to consider all minimal (P,Q,R)-chains.
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(213) DependencyGraphProof (EQUIVALENT)
169.36/88.01	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(214)
169.36/88.01	Obligation:
169.36/88.01	Q DP problem:
169.36/88.01	The TRS P consists of the following rules:
169.36/88.01	
169.36/88.01	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))
169.36/88.01	
169.36/88.01	The TRS R consists of the following rules:
169.36/88.01	
169.36/88.01	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
169.36/88.01	   down.0(g.0(f.0(x))) -> up.0(f.0(g.0(x)))
169.36/88.01	   down.0(g.0(f.1(x))) -> up.0(f.0(g.1(x)))
169.36/88.01	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
169.36/88.01	   down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7)))
169.36/88.01	   down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7)))
169.36/88.01	   down.0(g.0(b.)) -> g_flat.0(down.0(b.))
169.36/88.01	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
169.36/88.01	   down.0(a.) -> up.0(f.0(a.))
169.36/88.01	
169.36/88.01	The set Q consists of the following terms:
169.36/88.01	
169.36/88.01	   down.0(a.)
169.36/88.01	   down.0(g.0(f.0(x0)))
169.36/88.01	   down.0(g.0(f.1(x0)))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(f.0(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(f.1(x0))))))
169.36/88.01	   down.0(f.0(a.))
169.36/88.01	   down.0(f.0(g.0(x0)))
169.36/88.01	   down.0(f.0(g.1(x0)))
169.36/88.01	   down.0(f.0(b.))
169.36/88.01	   down.0(f.1(fresh_constant.))
169.36/88.01	   down.0(g.0(a.))
169.36/88.01	   down.0(g.0(g.0(x0)))
169.36/88.01	   down.0(g.0(g.1(x0)))
169.36/88.01	   down.0(g.0(b.))
169.36/88.01	   down.0(g.1(fresh_constant.))
169.36/88.01	   down.0(f.0(f.0(a.)))
169.36/88.01	   down.0(f.0(f.0(g.0(x0))))
169.36/88.01	   down.0(f.0(f.0(g.1(x0))))
169.36/88.01	   down.0(f.0(f.0(b.)))
169.36/88.01	   down.0(f.0(f.1(fresh_constant.)))
169.36/88.01	   down.0(f.0(f.0(f.0(a.))))
169.36/88.01	   down.0(f.0(f.0(f.0(g.0(x0)))))
169.36/88.01	   down.0(f.0(f.0(f.0(g.1(x0)))))
169.36/88.01	   down.0(f.0(f.0(f.0(b.))))
169.36/88.01	   down.0(f.0(f.0(f.1(fresh_constant.))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(a.)))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(g.0(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(g.1(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(b.)))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.1(fresh_constant.)))))
169.36/88.01	   f_flat.0(up.0(x0))
169.36/88.01	   f_flat.0(up.1(x0))
169.36/88.01	   g_flat.0(up.0(x0))
169.36/88.01	   g_flat.0(up.1(x0))
169.36/88.01	
169.36/88.01	We have to consider all minimal (P,Q,R)-chains.
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(215) UsableRulesReductionPairsProof (EQUIVALENT)
169.36/88.01	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.
169.36/88.01	
169.36/88.01	No dependency pairs are removed.
169.36/88.01	
169.36/88.01	The following rules are removed from R:
169.36/88.01	
169.36/88.01	   down.0(g.0(f.1(x))) -> up.0(f.0(g.1(x)))
169.36/88.01	Used ordering: POLO with Polynomial interpretation [POLO]:
169.36/88.01	
169.36/88.01	   POL(TOP.0(x_1)) = x_1
169.36/88.01	   POL(a.) = 0
169.36/88.01	   POL(b.) = 0
169.36/88.01	   POL(down.0(x_1)) = 1 + x_1
169.36/88.01	   POL(f.0(x_1)) = x_1
169.36/88.01	   POL(f.1(x_1)) = 1 + x_1
169.36/88.01	   POL(f_flat.0(x_1)) = x_1
169.36/88.01	   POL(g.0(x_1)) = x_1
169.36/88.01	   POL(g.1(x_1)) = 1 + x_1
169.36/88.01	   POL(g_flat.0(x_1)) = x_1
169.36/88.01	   POL(up.0(x_1)) = 1 + x_1
169.36/88.01	
169.36/88.01	
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(216)
169.36/88.01	Obligation:
169.36/88.01	Q DP problem:
169.36/88.01	The TRS P consists of the following rules:
169.36/88.01	
169.36/88.01	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))
169.36/88.01	
169.36/88.01	The TRS R consists of the following rules:
169.36/88.01	
169.36/88.01	   down.0(g.0(f.0(x))) -> up.0(f.0(g.0(x)))
169.36/88.01	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
169.36/88.01	   down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7)))
169.36/88.01	   down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7)))
169.36/88.01	   down.0(g.0(b.)) -> g_flat.0(down.0(b.))
169.36/88.01	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
169.36/88.01	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
169.36/88.01	   down.0(a.) -> up.0(f.0(a.))
169.36/88.01	
169.36/88.01	The set Q consists of the following terms:
169.36/88.01	
169.36/88.01	   down.0(a.)
169.36/88.01	   down.0(g.0(f.0(x0)))
169.36/88.01	   down.0(g.0(f.1(x0)))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(f.0(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(f.1(x0))))))
169.36/88.01	   down.0(f.0(a.))
169.36/88.01	   down.0(f.0(g.0(x0)))
169.36/88.01	   down.0(f.0(g.1(x0)))
169.36/88.01	   down.0(f.0(b.))
169.36/88.01	   down.0(f.1(fresh_constant.))
169.36/88.01	   down.0(g.0(a.))
169.36/88.01	   down.0(g.0(g.0(x0)))
169.36/88.01	   down.0(g.0(g.1(x0)))
169.36/88.01	   down.0(g.0(b.))
169.36/88.01	   down.0(g.1(fresh_constant.))
169.36/88.01	   down.0(f.0(f.0(a.)))
169.36/88.01	   down.0(f.0(f.0(g.0(x0))))
169.36/88.01	   down.0(f.0(f.0(g.1(x0))))
169.36/88.01	   down.0(f.0(f.0(b.)))
169.36/88.01	   down.0(f.0(f.1(fresh_constant.)))
169.36/88.01	   down.0(f.0(f.0(f.0(a.))))
169.36/88.01	   down.0(f.0(f.0(f.0(g.0(x0)))))
169.36/88.01	   down.0(f.0(f.0(f.0(g.1(x0)))))
169.36/88.01	   down.0(f.0(f.0(f.0(b.))))
169.36/88.01	   down.0(f.0(f.0(f.1(fresh_constant.))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(a.)))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(g.0(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(g.1(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(b.)))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.1(fresh_constant.)))))
169.36/88.01	   f_flat.0(up.0(x0))
169.36/88.01	   f_flat.0(up.1(x0))
169.36/88.01	   g_flat.0(up.0(x0))
169.36/88.01	   g_flat.0(up.1(x0))
169.36/88.01	
169.36/88.01	We have to consider all minimal (P,Q,R)-chains.
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(217) PisEmptyProof (SOUND)
169.36/88.01	The TRS P is empty. Hence, there is no (P,Q,R) chain.
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(218)
169.36/88.01	TRUE
169.36/88.01	
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(219)
169.36/88.01	Obligation:
169.36/88.01	Q DP problem:
169.36/88.01	The TRS P consists of the following rules:
169.36/88.01	
169.36/88.01	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.36/88.01	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.36/88.01	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.36/88.01	
169.36/88.01	The TRS R consists of the following rules:
169.36/88.01	
169.36/88.01	   down(g(f(x))) -> up(f(g(x)))
169.36/88.01	   down(g(a)) -> g_flat(down(a))
169.36/88.01	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.36/88.01	   down(g(b)) -> g_flat(down(b))
169.36/88.01	   f_flat(up(x_1)) -> up(f(x_1))
169.36/88.01	   g_flat(up(x_1)) -> up(g(x_1))
169.36/88.01	   down(a) -> up(f(a))
169.36/88.01	
169.36/88.01	The set Q consists of the following terms:
169.36/88.01	
169.36/88.01	   down(a)
169.36/88.01	   down(g(f(x0)))
169.36/88.01	   down(f(f(f(f(f(x0))))))
169.36/88.01	   down(f(a))
169.36/88.01	   down(f(g(x0)))
169.36/88.01	   down(f(b))
169.36/88.01	   down(f(fresh_constant))
169.36/88.01	   down(g(a))
169.36/88.01	   down(g(g(x0)))
169.36/88.01	   down(g(b))
169.36/88.01	   down(g(fresh_constant))
169.36/88.01	   down(f(f(a)))
169.36/88.01	   down(f(f(g(x0))))
169.36/88.01	   down(f(f(b)))
169.36/88.01	   down(f(f(fresh_constant)))
169.36/88.01	   down(f(f(f(a))))
169.36/88.01	   down(f(f(f(g(x0)))))
169.36/88.01	   down(f(f(f(b))))
169.36/88.01	   down(f(f(f(fresh_constant))))
169.36/88.01	   down(f(f(f(f(a)))))
169.36/88.01	   down(f(f(f(f(g(x0))))))
169.36/88.01	   down(f(f(f(f(b)))))
169.36/88.01	   down(f(f(f(f(fresh_constant)))))
169.36/88.01	   f_flat(up(x0))
169.36/88.01	   g_flat(up(x0))
169.36/88.01	
169.36/88.01	We have to consider all minimal (P,Q,R)-chains.
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(220) QReductionProof (EQUIVALENT)
169.36/88.01	We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN].
169.36/88.01	
169.36/88.01	   down(f(fresh_constant))
169.36/88.01	   down(g(fresh_constant))
169.36/88.01	   down(f(f(fresh_constant)))
169.36/88.01	   down(f(f(f(fresh_constant))))
169.36/88.01	   down(f(f(f(f(fresh_constant)))))
169.36/88.01	
169.36/88.01	
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(221)
169.36/88.01	Obligation:
169.36/88.01	Q DP problem:
169.36/88.01	The TRS P consists of the following rules:
169.36/88.01	
169.36/88.01	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.36/88.01	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.36/88.01	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.36/88.01	
169.36/88.01	The TRS R consists of the following rules:
169.36/88.01	
169.36/88.01	   down(g(f(x))) -> up(f(g(x)))
169.36/88.01	   down(g(a)) -> g_flat(down(a))
169.36/88.01	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.36/88.01	   down(g(b)) -> g_flat(down(b))
169.36/88.01	   f_flat(up(x_1)) -> up(f(x_1))
169.36/88.01	   g_flat(up(x_1)) -> up(g(x_1))
169.36/88.01	   down(a) -> up(f(a))
169.36/88.01	
169.36/88.01	The set Q consists of the following terms:
169.36/88.01	
169.36/88.01	   down(a)
169.36/88.01	   down(g(f(x0)))
169.36/88.01	   down(f(f(f(f(f(x0))))))
169.36/88.01	   down(f(a))
169.36/88.01	   down(f(g(x0)))
169.36/88.01	   down(f(b))
169.36/88.01	   down(g(a))
169.36/88.01	   down(g(g(x0)))
169.36/88.01	   down(g(b))
169.36/88.01	   down(f(f(a)))
169.36/88.01	   down(f(f(g(x0))))
169.36/88.01	   down(f(f(b)))
169.36/88.01	   down(f(f(f(a))))
169.36/88.01	   down(f(f(f(g(x0)))))
169.36/88.01	   down(f(f(f(b))))
169.36/88.01	   down(f(f(f(f(a)))))
169.36/88.01	   down(f(f(f(f(g(x0))))))
169.36/88.01	   down(f(f(f(f(b)))))
169.36/88.01	   f_flat(up(x0))
169.36/88.01	   g_flat(up(x0))
169.36/88.01	
169.36/88.01	We have to consider all (P,Q,R)-chains.
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(222) SplitQDPProof (EQUIVALENT)
169.36/88.01	We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem
169.36/88.01	
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(223)
169.36/88.01	Complex Obligation (AND)
169.36/88.01	
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(224)
169.36/88.01	Obligation:
169.36/88.01	Q DP problem:
169.36/88.01	The TRS P consists of the following rules:
169.36/88.01	
169.36/88.01	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.36/88.01	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.36/88.01	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.36/88.01	
169.36/88.01	The TRS R consists of the following rules:
169.36/88.01	
169.36/88.01	   down(g(f(x))) -> up(f(g(x)))
169.36/88.01	   down(g(a)) -> g_flat(down(a))
169.36/88.01	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.36/88.01	   down(g(b)) -> g_flat(down(b))
169.36/88.01	   f_flat(up(x_1)) -> up(f(x_1))
169.36/88.01	   g_flat(up(x_1)) -> up(g(x_1))
169.36/88.01	   down(a) -> up(f(a))
169.36/88.01	
169.36/88.01	The set Q consists of the following terms:
169.36/88.01	
169.36/88.01	   down(a)
169.36/88.01	   down(g(f(x0)))
169.36/88.01	   down(f(f(f(f(f(x0))))))
169.36/88.01	   down(f(a))
169.36/88.01	   down(f(g(x0)))
169.36/88.01	   down(f(b))
169.36/88.01	   down(f(fresh_constant))
169.36/88.01	   down(g(a))
169.36/88.01	   down(g(g(x0)))
169.36/88.01	   down(g(b))
169.36/88.01	   down(g(fresh_constant))
169.36/88.01	   down(f(f(a)))
169.36/88.01	   down(f(f(g(x0))))
169.36/88.01	   down(f(f(b)))
169.36/88.01	   down(f(f(fresh_constant)))
169.36/88.01	   down(f(f(f(a))))
169.36/88.01	   down(f(f(f(g(x0)))))
169.36/88.01	   down(f(f(f(b))))
169.36/88.01	   down(f(f(f(fresh_constant))))
169.36/88.01	   down(f(f(f(f(a)))))
169.36/88.01	   down(f(f(f(f(g(x0))))))
169.36/88.01	   down(f(f(f(f(b)))))
169.36/88.01	   down(f(f(f(f(fresh_constant)))))
169.36/88.01	   f_flat(up(x0))
169.36/88.01	   g_flat(up(x0))
169.36/88.01	
169.36/88.01	We have to consider all minimal (P,Q,R)-chains.
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(225) SemLabProof (SOUND)
169.36/88.01	We found the following model for the rules of the TRSs R and P.
169.36/88.01	Interpretation over the domain with elements from 0 to 1.
169.36/88.01	a: 0
169.36/88.01	b: 1
169.36/88.01	down: 0
169.36/88.01	f: 0
169.36/88.01	fresh_constant: 0
169.36/88.01	up: 0
169.36/88.01	f_flat: 0
169.36/88.01	TOP: 0
169.36/88.01	g_flat: 0
169.36/88.01	g: 0
169.36/88.01	By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(226)
169.36/88.01	Obligation:
169.36/88.01	Q DP problem:
169.36/88.01	The TRS P consists of the following rules:
169.36/88.01	
169.36/88.01	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))
169.36/88.01	
169.36/88.01	The TRS R consists of the following rules:
169.36/88.01	
169.36/88.01	   down.0(g.0(f.0(x))) -> up.0(f.0(g.0(x)))
169.36/88.01	   down.0(g.0(f.1(x))) -> up.0(f.0(g.1(x)))
169.36/88.01	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
169.36/88.01	   down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7)))
169.36/88.01	   down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7)))
169.36/88.01	   down.0(g.1(b.)) -> g_flat.0(down.1(b.))
169.36/88.01	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
169.36/88.01	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
169.36/88.01	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
169.36/88.01	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
169.36/88.01	   down.0(a.) -> up.0(f.0(a.))
169.36/88.01	
169.36/88.01	The set Q consists of the following terms:
169.36/88.01	
169.36/88.01	   down.0(a.)
169.36/88.01	   down.0(g.0(f.0(x0)))
169.36/88.01	   down.0(g.0(f.1(x0)))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(f.0(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(f.1(x0))))))
169.36/88.01	   down.0(f.0(a.))
169.36/88.01	   down.0(f.0(g.0(x0)))
169.36/88.01	   down.0(f.0(g.1(x0)))
169.36/88.01	   down.0(f.1(b.))
169.36/88.01	   down.0(f.0(fresh_constant.))
169.36/88.01	   down.0(g.0(a.))
169.36/88.01	   down.0(g.0(g.0(x0)))
169.36/88.01	   down.0(g.0(g.1(x0)))
169.36/88.01	   down.0(g.1(b.))
169.36/88.01	   down.0(g.0(fresh_constant.))
169.36/88.01	   down.0(f.0(f.0(a.)))
169.36/88.01	   down.0(f.0(f.0(g.0(x0))))
169.36/88.01	   down.0(f.0(f.0(g.1(x0))))
169.36/88.01	   down.0(f.0(f.1(b.)))
169.36/88.01	   down.0(f.0(f.0(fresh_constant.)))
169.36/88.01	   down.0(f.0(f.0(f.0(a.))))
169.36/88.01	   down.0(f.0(f.0(f.0(g.0(x0)))))
169.36/88.01	   down.0(f.0(f.0(f.0(g.1(x0)))))
169.36/88.01	   down.0(f.0(f.0(f.1(b.))))
169.36/88.01	   down.0(f.0(f.0(f.0(fresh_constant.))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(a.)))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(g.0(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(g.1(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.1(b.)))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(fresh_constant.)))))
169.36/88.01	   f_flat.0(up.0(x0))
169.36/88.01	   f_flat.0(up.1(x0))
169.36/88.01	   g_flat.0(up.0(x0))
169.36/88.01	   g_flat.0(up.1(x0))
169.36/88.01	
169.36/88.01	We have to consider all minimal (P,Q,R)-chains.
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(227) UsableRulesReductionPairsProof (EQUIVALENT)
169.36/88.01	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.
169.36/88.01	
169.36/88.01	No dependency pairs are removed.
169.36/88.01	
169.36/88.01	The following rules are removed from R:
169.36/88.01	
169.36/88.01	   down.0(g.1(b.)) -> g_flat.0(down.1(b.))
169.36/88.01	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
169.36/88.01	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
169.36/88.01	Used ordering: POLO with Polynomial interpretation [POLO]:
169.36/88.01	
169.36/88.01	   POL(TOP.0(x_1)) = x_1
169.36/88.01	   POL(a.) = 0
169.36/88.01	   POL(b.) = 0
169.36/88.01	   POL(down.0(x_1)) = 1 + x_1
169.36/88.01	   POL(down.1(x_1)) = x_1
169.36/88.01	   POL(f.0(x_1)) = x_1
169.36/88.01	   POL(f.1(x_1)) = x_1
169.36/88.01	   POL(f_flat.0(x_1)) = x_1
169.36/88.01	   POL(g.0(x_1)) = 1 + x_1
169.36/88.01	   POL(g.1(x_1)) = 1 + x_1
169.36/88.01	   POL(g_flat.0(x_1)) = 1 + x_1
169.36/88.01	   POL(up.0(x_1)) = 1 + x_1
169.36/88.01	   POL(up.1(x_1)) = 1 + x_1
169.36/88.01	
169.36/88.01	
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(228)
169.36/88.01	Obligation:
169.36/88.01	Q DP problem:
169.36/88.01	The TRS P consists of the following rules:
169.36/88.01	
169.36/88.01	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))
169.36/88.01	
169.36/88.01	The TRS R consists of the following rules:
169.36/88.01	
169.36/88.01	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
169.36/88.01	   down.0(g.0(f.0(x))) -> up.0(f.0(g.0(x)))
169.36/88.01	   down.0(g.0(f.1(x))) -> up.0(f.0(g.1(x)))
169.36/88.01	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
169.36/88.01	   down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7)))
169.36/88.01	   down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7)))
169.36/88.01	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
169.36/88.01	   down.0(a.) -> up.0(f.0(a.))
169.36/88.01	
169.36/88.01	The set Q consists of the following terms:
169.36/88.01	
169.36/88.01	   down.0(a.)
169.36/88.01	   down.0(g.0(f.0(x0)))
169.36/88.01	   down.0(g.0(f.1(x0)))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(f.0(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(f.1(x0))))))
169.36/88.01	   down.0(f.0(a.))
169.36/88.01	   down.0(f.0(g.0(x0)))
169.36/88.01	   down.0(f.0(g.1(x0)))
169.36/88.01	   down.0(f.1(b.))
169.36/88.01	   down.0(f.0(fresh_constant.))
169.36/88.01	   down.0(g.0(a.))
169.36/88.01	   down.0(g.0(g.0(x0)))
169.36/88.01	   down.0(g.0(g.1(x0)))
169.36/88.01	   down.0(g.1(b.))
169.36/88.01	   down.0(g.0(fresh_constant.))
169.36/88.01	   down.0(f.0(f.0(a.)))
169.36/88.01	   down.0(f.0(f.0(g.0(x0))))
169.36/88.01	   down.0(f.0(f.0(g.1(x0))))
169.36/88.01	   down.0(f.0(f.1(b.)))
169.36/88.01	   down.0(f.0(f.0(fresh_constant.)))
169.36/88.01	   down.0(f.0(f.0(f.0(a.))))
169.36/88.01	   down.0(f.0(f.0(f.0(g.0(x0)))))
169.36/88.01	   down.0(f.0(f.0(f.0(g.1(x0)))))
169.36/88.01	   down.0(f.0(f.0(f.1(b.))))
169.36/88.01	   down.0(f.0(f.0(f.0(fresh_constant.))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(a.)))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(g.0(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(g.1(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.1(b.)))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(fresh_constant.)))))
169.36/88.01	   f_flat.0(up.0(x0))
169.36/88.01	   f_flat.0(up.1(x0))
169.36/88.01	   g_flat.0(up.0(x0))
169.36/88.01	   g_flat.0(up.1(x0))
169.36/88.01	
169.36/88.01	We have to consider all minimal (P,Q,R)-chains.
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(229) DependencyGraphProof (EQUIVALENT)
169.36/88.01	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(230)
169.36/88.01	Obligation:
169.36/88.01	Q DP problem:
169.36/88.01	The TRS P consists of the following rules:
169.36/88.01	
169.36/88.01	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))
169.36/88.01	
169.36/88.01	The TRS R consists of the following rules:
169.36/88.01	
169.36/88.01	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
169.36/88.01	   down.0(g.0(f.0(x))) -> up.0(f.0(g.0(x)))
169.36/88.01	   down.0(g.0(f.1(x))) -> up.0(f.0(g.1(x)))
169.36/88.01	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
169.36/88.01	   down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7)))
169.36/88.01	   down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7)))
169.36/88.01	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
169.36/88.01	   down.0(a.) -> up.0(f.0(a.))
169.36/88.01	
169.36/88.01	The set Q consists of the following terms:
169.36/88.01	
169.36/88.01	   down.0(a.)
169.36/88.01	   down.0(g.0(f.0(x0)))
169.36/88.01	   down.0(g.0(f.1(x0)))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(f.0(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(f.1(x0))))))
169.36/88.01	   down.0(f.0(a.))
169.36/88.01	   down.0(f.0(g.0(x0)))
169.36/88.01	   down.0(f.0(g.1(x0)))
169.36/88.01	   down.0(f.1(b.))
169.36/88.01	   down.0(f.0(fresh_constant.))
169.36/88.01	   down.0(g.0(a.))
169.36/88.01	   down.0(g.0(g.0(x0)))
169.36/88.01	   down.0(g.0(g.1(x0)))
169.36/88.01	   down.0(g.1(b.))
169.36/88.01	   down.0(g.0(fresh_constant.))
169.36/88.01	   down.0(f.0(f.0(a.)))
169.36/88.01	   down.0(f.0(f.0(g.0(x0))))
169.36/88.01	   down.0(f.0(f.0(g.1(x0))))
169.36/88.01	   down.0(f.0(f.1(b.)))
169.36/88.01	   down.0(f.0(f.0(fresh_constant.)))
169.36/88.01	   down.0(f.0(f.0(f.0(a.))))
169.36/88.01	   down.0(f.0(f.0(f.0(g.0(x0)))))
169.36/88.01	   down.0(f.0(f.0(f.0(g.1(x0)))))
169.36/88.01	   down.0(f.0(f.0(f.1(b.))))
169.36/88.01	   down.0(f.0(f.0(f.0(fresh_constant.))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(a.)))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(g.0(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(g.1(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.1(b.)))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(fresh_constant.)))))
169.36/88.01	   f_flat.0(up.0(x0))
169.36/88.01	   f_flat.0(up.1(x0))
169.36/88.01	   g_flat.0(up.0(x0))
169.36/88.01	   g_flat.0(up.1(x0))
169.36/88.01	
169.36/88.01	We have to consider all minimal (P,Q,R)-chains.
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(231) UsableRulesReductionPairsProof (EQUIVALENT)
169.36/88.01	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.
169.36/88.01	
169.36/88.01	No dependency pairs are removed.
169.36/88.01	
169.36/88.01	The following rules are removed from R:
169.36/88.01	
169.36/88.01	   down.0(g.0(f.1(x))) -> up.0(f.0(g.1(x)))
169.36/88.01	Used ordering: POLO with Polynomial interpretation [POLO]:
169.36/88.01	
169.36/88.01	   POL(TOP.0(x_1)) = x_1
169.36/88.01	   POL(a.) = 0
169.36/88.01	   POL(down.0(x_1)) = 1 + x_1
169.36/88.01	   POL(f.0(x_1)) = x_1
169.36/88.01	   POL(f.1(x_1)) = 1 + x_1
169.36/88.01	   POL(f_flat.0(x_1)) = x_1
169.36/88.01	   POL(g.0(x_1)) = x_1
169.36/88.01	   POL(g.1(x_1)) = 1 + x_1
169.36/88.01	   POL(g_flat.0(x_1)) = x_1
169.36/88.01	   POL(up.0(x_1)) = 1 + x_1
169.36/88.01	
169.36/88.01	
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(232)
169.36/88.01	Obligation:
169.36/88.01	Q DP problem:
169.36/88.01	The TRS P consists of the following rules:
169.36/88.01	
169.36/88.01	   TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))
169.36/88.01	   TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))
169.36/88.01	
169.36/88.01	The TRS R consists of the following rules:
169.36/88.01	
169.36/88.01	   down.0(g.0(f.0(x))) -> up.0(f.0(g.0(x)))
169.36/88.01	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
169.36/88.01	   down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7)))
169.36/88.01	   down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7)))
169.36/88.01	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
169.36/88.01	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
169.36/88.01	   down.0(a.) -> up.0(f.0(a.))
169.36/88.01	
169.36/88.01	The set Q consists of the following terms:
169.36/88.01	
169.36/88.01	   down.0(a.)
169.36/88.01	   down.0(g.0(f.0(x0)))
169.36/88.01	   down.0(g.0(f.1(x0)))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(f.0(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(f.1(x0))))))
169.36/88.01	   down.0(f.0(a.))
169.36/88.01	   down.0(f.0(g.0(x0)))
169.36/88.01	   down.0(f.0(g.1(x0)))
169.36/88.01	   down.0(f.1(b.))
169.36/88.01	   down.0(f.0(fresh_constant.))
169.36/88.01	   down.0(g.0(a.))
169.36/88.01	   down.0(g.0(g.0(x0)))
169.36/88.01	   down.0(g.0(g.1(x0)))
169.36/88.01	   down.0(g.1(b.))
169.36/88.01	   down.0(g.0(fresh_constant.))
169.36/88.01	   down.0(f.0(f.0(a.)))
169.36/88.01	   down.0(f.0(f.0(g.0(x0))))
169.36/88.01	   down.0(f.0(f.0(g.1(x0))))
169.36/88.01	   down.0(f.0(f.1(b.)))
169.36/88.01	   down.0(f.0(f.0(fresh_constant.)))
169.36/88.01	   down.0(f.0(f.0(f.0(a.))))
169.36/88.01	   down.0(f.0(f.0(f.0(g.0(x0)))))
169.36/88.01	   down.0(f.0(f.0(f.0(g.1(x0)))))
169.36/88.01	   down.0(f.0(f.0(f.1(b.))))
169.36/88.01	   down.0(f.0(f.0(f.0(fresh_constant.))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(a.)))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(g.0(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(g.1(x0))))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.1(b.)))))
169.36/88.01	   down.0(f.0(f.0(f.0(f.0(fresh_constant.)))))
169.36/88.01	   f_flat.0(up.0(x0))
169.36/88.01	   f_flat.0(up.1(x0))
169.36/88.01	   g_flat.0(up.0(x0))
169.36/88.01	   g_flat.0(up.1(x0))
169.36/88.01	
169.36/88.01	We have to consider all minimal (P,Q,R)-chains.
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(233) PisEmptyProof (SOUND)
169.36/88.01	The TRS P is empty. Hence, there is no (P,Q,R) chain.
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(234)
169.36/88.01	TRUE
169.36/88.01	
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(235)
169.36/88.01	Obligation:
169.36/88.01	Q DP problem:
169.36/88.01	The TRS P consists of the following rules:
169.36/88.01	
169.36/88.01	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.36/88.01	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.36/88.01	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.36/88.01	
169.36/88.01	The TRS R consists of the following rules:
169.36/88.01	
169.36/88.01	   down(g(f(x))) -> up(f(g(x)))
169.36/88.01	   down(g(a)) -> g_flat(down(a))
169.36/88.01	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.36/88.01	   f_flat(up(x_1)) -> up(f(x_1))
169.36/88.01	   g_flat(up(x_1)) -> up(g(x_1))
169.36/88.01	   down(a) -> up(f(a))
169.36/88.01	
169.36/88.01	The set Q consists of the following terms:
169.36/88.01	
169.36/88.01	   down(a)
169.36/88.01	   down(g(f(x0)))
169.36/88.01	   down(f(f(f(f(f(x0))))))
169.36/88.01	   down(f(a))
169.36/88.01	   down(f(g(x0)))
169.36/88.01	   down(f(b))
169.36/88.01	   down(f(fresh_constant))
169.36/88.01	   down(g(a))
169.36/88.01	   down(g(g(x0)))
169.36/88.01	   down(g(b))
169.36/88.01	   down(g(fresh_constant))
169.36/88.01	   down(f(f(a)))
169.36/88.01	   down(f(f(g(x0))))
169.36/88.01	   down(f(f(b)))
169.36/88.01	   down(f(f(fresh_constant)))
169.36/88.01	   down(f(f(f(a))))
169.36/88.01	   down(f(f(f(g(x0)))))
169.36/88.01	   down(f(f(f(b))))
169.36/88.01	   down(f(f(f(fresh_constant))))
169.36/88.01	   down(f(f(f(f(a)))))
169.36/88.01	   down(f(f(f(f(g(x0))))))
169.36/88.01	   down(f(f(f(f(b)))))
169.36/88.01	   down(f(f(f(f(fresh_constant)))))
169.36/88.01	   f_flat(up(x0))
169.36/88.01	   g_flat(up(x0))
169.36/88.01	
169.36/88.01	We have to consider all minimal (P,Q,R)-chains.
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(236) QReductionProof (EQUIVALENT)
169.36/88.01	We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN].
169.36/88.01	
169.36/88.01	   down(f(b))
169.36/88.01	   down(f(fresh_constant))
169.36/88.01	   down(g(b))
169.36/88.01	   down(g(fresh_constant))
169.36/88.01	   down(f(f(b)))
169.36/88.01	   down(f(f(fresh_constant)))
169.36/88.01	   down(f(f(f(b))))
169.36/88.01	   down(f(f(f(fresh_constant))))
169.36/88.01	   down(f(f(f(f(b)))))
169.36/88.01	   down(f(f(f(f(fresh_constant)))))
169.36/88.01	
169.36/88.01	
169.36/88.01	----------------------------------------
169.36/88.01	
169.36/88.01	(237)
169.36/88.01	Obligation:
169.36/88.01	Q DP problem:
169.36/88.01	The TRS P consists of the following rules:
169.36/88.01	
169.36/88.01	   TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))
169.36/88.01	   TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))
169.36/88.01	   TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))
169.36/88.01	
169.36/88.01	The TRS R consists of the following rules:
169.36/88.01	
169.36/88.01	   down(g(f(x))) -> up(f(g(x)))
169.36/88.01	   down(g(a)) -> g_flat(down(a))
169.36/88.01	   down(g(g(y7))) -> g_flat(down(g(y7)))
169.36/88.01	   f_flat(up(x_1)) -> up(f(x_1))
169.36/88.01	   g_flat(up(x_1)) -> up(g(x_1))
169.36/88.01	   down(a) -> up(f(a))
169.36/88.01	
169.36/88.01	The set Q consists of the following terms:
169.36/88.01	
169.36/88.01	   down(a)
169.36/88.01	   down(g(f(x0)))
169.36/88.01	   down(f(f(f(f(f(x0))))))
169.36/88.01	   down(f(a))
169.36/88.01	   down(f(g(x0)))
169.36/88.01	   down(g(a))
169.36/88.01	   down(g(g(x0)))
169.36/88.01	   down(f(f(a)))
169.36/88.01	   down(f(f(g(x0))))
169.36/88.01	   down(f(f(f(a))))
169.36/88.01	   down(f(f(f(g(x0)))))
169.36/88.01	   down(f(f(f(f(a)))))
169.36/88.01	   down(f(f(f(f(g(x0))))))
169.36/88.01	   f_flat(up(x0))
169.36/88.01	   g_flat(up(x0))
169.36/88.01	
169.36/88.01	We have to consider all (P,Q,R)-chains.
170.71/89.30	EOF